EQUIVARIANT INTERSECTION THEORY AND SURGERY THEORY FOR MANIFOLDS WITH MIDDLE DIMENSIONAL SINGULAR SETS

ANTHONY BAK AND MASAHARU MORIMOTO

In memory of Yu Solovyov

Abstract. Let G denote a finite group and n = 2k ≧ 6 an even integer. Let X denote a simply connected, compact, oriented, smooth G-manifold of dimension n. Let L denote a union of connected, compact, neat submanifolds in X of dimension ≦ k. We invoke the hypothesis that L is a G-subcomplex of a G-equivariant smooth triangulation of X and contains the singular set of the action of G on X. If the dimension of the G-singular set is also k then the ordinary equivariant self-intersection form is not well defined, although the equivariant intersection form is well defined. The first goal of the paper is to eliminate the deficiency above by constructing a new, well defined, equivariant, self-intersection form, called the generalized (or doubly parametrized) equivariant self-intersection form. Its value at a given element agrees with that of the ordinary equivariant self-intersection form when the latter value is well defined. Let F denote a finite family of immersions with trivial normal bundle of k-dimensional, connected, closed, orientable, smooth manifolds into X. Assume that the integral (and mod 2) intersection forms applied to members of F and to orientable (and nonorientable) k-dimensional members of L are trivial. Then the vanishing of the equivariant intersection form on F × F and the generalized equivariant self-intersection form on F is a necessary and sufficient condition that F is regularly homotopic to a family of disjoint embeddings, each of which is disjoint from L. This property, when F is a finite family of immersions of the k-dimensional sphere S^{k} into X, is just what is needed for constructing an equivariant surgery theory for G-manifolds X as above whose G-singular set has dimension less than or equal to k. What is new for surgery theory is that the equivariant surgery obstruction is defined for an almost arbitrary singular set of dimension ≦ k and in particular, the k-dimensional components of the singular set can be nonorientable.

1. Introduction

Equivariant surgery solves the following problem. Let G denote a finite group and n = 2k ≧ 6. Let f : X → Y denote a k-connected, degree-one, G-framed map of oneconnected, compact, n-dimensional, smooth G-manifolds such that the fixed point set of X satisfies the following assumption.

(A1) Any connected component of a fixed point set X^{g}, where g ∈ G ∖ {1}, has dimension ≦ k and no connected component of dimension (k − 1) of a fixed point

2000 Mathematics Subject Classification. Primary: 57S17, 57S25. Secondary: 19G12, 19J25, 55M35, 57M60, 57R67.

Key words and phrases. Equivariant surgery, surgery obstruction, form parameter, quadratic module, group action, fixed point, homotopy equivalence.

set X^{H}, H ⫅ G, is totally contained in a connected component of dimension k of some fixed point set X^{K}, K ⫅ G.

Assume further (A2) ∂f (:= f|_{∂X}) : ∂X → ∂Y is a homology equivalence, and (A3) f^{H} (:= f|_{X^{H}}) : X^{H} → Y^{H} is a homology equivalence for each nontrivial hyperele-

mentary subgroup H ⫅ G. Since f is G-framed, it comes equipped with a G-vector bundle isomorphism b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η), where T (X) denotes the tangent bundle over X with its natural Gaction and η and ξ are G-vector bundles over Y. Let ε_{Y} (ℝ^{n}) = (Y × ℝ^{n} → Y) denote the product G-vector bundle on Y where G acts trivially on ℝ^{n}. After replacing η by η ⊕ ε_{Y} (ℝ^{n}), it can be assumed and will be assumed throughout the paper that (A4) η contains the product bundle ε_{Y} (ℝ^{n}) as a G-vector subbundle. The problem of equivariant surgery is finding necessary and sufficient conditions when

(f, b) is G-framed cobordant rel ∂X ∪X_{sing} to a G-framed map (f′, b′) such that f′ : X′ → Y is a homotopy equivalence, where X_{sing} denotes the singular set ⋃_{g∈G∖{1}} X^{g} of X. This problem is solved along the usual lines by constructing a group W (G, Y, X_{sing})_{free} called the surgery obstruction group and an element σ(f, b) in W (G, Y, X_{sing})_{free} called the surgery obstruction of (f, b), having the property that σ(f, b) = 0 if and only if (f, b) is G-framed cobordant rel ∂X ∪ X_{sing} to a G-framed map (f′, b′) such that f′ : X′ → Y is a homotopy equivalence. However the construction of the group W (G, Y, X_{sing})_{free} and the surgery obstruction σ(f, b) is much more intricate than those of previous theories, because the singular set is now much more general.

We summarize next the literature on equivariant surgery and then state our main surgery results.

The best results to date in the literature are in [6] and [30]. They are obtained when Assumption (A1) above is strengthened (considerably) by adding the conditions that each k-dimensional connected component of X^{H}, H ⫅ G, is orientable, that for each g ∈ G the translation X^{H} → X^{gGg^{−1}}, x ↦ gx, is orientation preserving, and that 3 further technical assumptions [6, (2.1.2)–(2.1.4)] are satisfied. The surgery theory in [25] and [26] is obtained when the assumptions above are further strengthened by adding the condition dim X^{g} ≦ k − 1 for all g ∈ G ∖ {1}. Finally the surgery theories of T. Petrie [39], [40], [12], [41] and W. Lück–I. Madsen [19], [20] are obtained when the assumptions above are strengthened again by adding either the condition dim X^{g} ≦ k − 2 for all g ∈ G ∖ {1} or the condition that G has odd order. Applications of the surgery results above to transformation groups are found in [4], [7], [16], [17], [22], [24], [27], [28], [32], [33], [40],

[41] and have motivated all of the work above on equivariant surgery theory. Applications of results of the current paper will be forthcoming.

Our main surgery results are as follows.

Theorem 1.1. Let X and Y be compact, connected, oriented, smooth G-manifolds of dimension n = 2k ≧ 6. Suppose X satisfies (A1) and Y is simply connected. Let fff = (f, b) be a degree-one G-framed map satisfying (A2)–(A4), where f : (X, ∂X) → (Y, ∂Y) and b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η). Then one can perform G-surgery rel ∂X ∪ X_{sing} on fff to obtain a G-framed map fff′ = (f′, b′) with f′ : (X′, ∂X′) → (Y, ∂Y) and b′ : T (X′)⊕f′^{∗}η → f′^{∗}(ξ ⊕ η) such that f′ : X′ → Y is a homotopy equivalence if and only if the element σ(fff) in W (G, Y, X_{sing})_{free} is trivial.

This follows from Theorem 7.8.

The subscript free on W (G, Y, X_{sing})_{free} signifies that the underlying modules used in constructing the Witt group W (G, Y, X_{sing})_{free} are free (or stably free) over the integral group ring ℤ[G]. Under a weaker assumption than (A3), namely (A3′) f^{P} : X^{P} → Y^{P} is a ℤ_{(p)}-homology equivalence for each prime p and any nontrivial

p-subgroup P of G, we get also an equivariant surgery theory, except the modules used in constructing the surgery obstruction group here are only projective over ℤ[G]. Accordingly, the surgery obstruction group here is denoted by W (G, Y, X_{sing})_{proj}.

Theorem 1.2. Let X and Y be as in the previous theorem. Let fff = (f, b) denote a degree-one G-framed map satisfying (A2), (A3′) and (A4), where f : (X, ∂X) → (Y, ∂Y) and b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η). If the element σ(fff) in W (G, Y, X_{sing})_{proj} is trivial, then one can perform G-surgery rel ∂X ∪ X_{sing} on fff to obtain a G-framed map fff′ = (f′, b′) where f′ : (X′, ∂X′) → (Y, ∂Y) and b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η) such that f′ : X′ → Y is a homotopy equivalence.

This follows from Theorem 6.3.

It is worth noting that there is a canonical homomorphism

W (G, Y, X_{sing})_{free} → W (G, Y, X_{sing})_{proj}

so that a part of Theorem 1.1 follows directly from Theorem 1.2. Moreover the homomorphism is injective, according to Lemma 3.34.

We describe how the rest of the paper is organized, while at the same time providing insight how the surgery obstruction groups W (G, Y, X_{sing})_{ε}, ε = free or proj, and the surgery obstruction σ(fff) are constructed.

The key to constructing an equivariant, smooth, surgery theory on 2k-dimensional (k ≧ 3) G-manifolds X is having good equivariant, geometric, intersection tools. This means the following: Given a finite set F of smooth immersions with trivial normal bundle of 1-connected, k-dimensional, closed, oriented manifolds into X, the tools should provide necessary and sufficient conditions that F is regularly homotopic to a set of equivariantly disjoint embeddings, each of which is disjoint from the G-singular set X_{sing} of the action of G on X. The usual tools for doing this are the equivariant, geometric, intersection form #_{G} and the equivariant, geometric, self-intersection form μ_{G}. However since X_{sing} can have k-dimensional components, μ_{G} is not always defined. To get around this problem, we construct in Section 4 a generalized, equivariant, self-intersection form ♮_{G}, which is always defined and agrees with μ_{G} whenever it is defined. The construction of ♮_{G} uses 2 parameters instead of just one form parameter as in the case of μ_{G}. The forms #_{G} and ♮_{G} are related by certain equalities which generalize the relationship between #_{G} and μ_{G} and the pair (#_{G}, ♮_{G}) defines a so-called doubly parametrized form on the surgery kernel K_{k}(X) := Ker[f_{∗} : H_{k}(X) → H_{k}(Y)] of a G-framed map fff = (f, b) such that f : X → Y is k-connected. Section 2 develops for reference purposes, several algebraic concepts including that of doubly parametrized forms on modules. Section 5 constructs the surgery module M_{fff} of fff, the surgery obstruction σ(fff) of fff, and the surgery obstruction group W (G, Y, X_{sing})_{ε}, ε = proj or free. By definition, σ(fff) is the class of M_{fff} in W (G, Y, X_{sing})_{ε}. To define M_{fff}, we need to take into account additional structure on K_{k}(X), arising from X_{sing}. This is done as follows. Let Θ̃ (resp. Θ_{2}) denote the G-set of all connected, k-dimensional, oriented (resp. nonoriented) components of X_{sing}. There are canonically defined a G × {±1}-map θ : Θ̃ → K_{k}(X) and a G-map θ_{2} : Θ_{2} → K_{k}(X)/2K_{k}(X) called positioning maps, making K_{k}(X) into a so-called positioning module. Section 2 also develops the algebraic concepts of positioning module and doubly parametrized positioning module. The 5-tuple M_{fff} = (K_{k}(X), #_{G}, ♮_{G}, θ, θ_{2}) is an example of a doubly parametrized positioning module and is by definition the surgery module of fff. One might at this point that the surgery obstruction group is the Witt group of all nonsingular doubly parametrized positioning modules, but this is not the case. It turns out that a subtle invariant ∇ is playing a role here. This invariant vanishes on all surgery modules M_{fff} and the correct definition of the surgery obstruction group is the Witt group of all nonsingular, doubly parametrized positioning modules with trivial ∇-invariant. (This vanishing of ∇ affects the concept of Lagrangian and gives a distinctly different Witt group.) The algebraic concepts underlying ∇ are developed in Section 3, as well as the algebraic constructions of the kinds of Witt groups needed in surgery theory.

Our main surgery result Theorem 1.2 is proved in Section 6. The cobordism invariance of the surgery obstruction and Theorem 1.1 are proved in Section 7.

Acknowledgements. The first author gratefully acknowledges the support of INTAS 00-566 and the second author would like to acknowledge the support of the Grant-in-Aid for Scientific Research (Kakenhi) No. 15540076.

Table of Contents

1. Introduction

2. Doubly parametrized forms on positioning modules

3. Invariants of forms and G-surgery obstruction groups

4. Geometric intersection theory

5. G-surgery obstruction modules 6. G-surgery theorems

7. G-framed Θ Θ Θ-cobordism invariance

2. Doubly parametrized forms on positioning modules

This section introduces the concepts of doubly parametrized form and positioning module and develops their elementary K-theory. Both concepts are motivated by and needed in the surgery theory developed in Sections 5–7.

The concept of a doubly parametrized form is a nonobvious synthesis of the concepts of Λ-quadratic form and Λ-Hermitian form in [2]. Both older concepts are special cases of the new one and the K-theory developed in this section for the new concept generalizes that in Section 2 and Sections 5–6 of [2] for the older ones. The geometry in Sections 5–7 requires that the underlying modules of our doubly parametrized forms contain positioning information. So we shall develop, right from the beginning, our K-theory of doubly parametrized forms, over positioning modules rather than just ordinary modules. We shall see that ordinary modules are the special case when the positioning information is empty. This has the consequence that our results for doubly parametrized forms on positioning modules are also valid for doubly parametrized forms on ordinary modules.

For functorial reasons, it is better to define our forms over associative algebras rather than associative rings, because in practice a ground ring in the center of our ring will play a special role and it will be necessary to restrict ring homomorphisms to those which preserve the ground rings. By definition, an associative algebra A will mean an associative ring A with identity together with a fixed commutative ring R ⫅ Center(A), called the ground ring of A. An R-algebra is an associative algebra whose ground ring is R. An algebra homomorphism A → A′ is any ring homomorphism f : A → A′ which preserves the identity and has the property that f(R) ⫅ R′ where R and R′ are the ground rings of A and A′, respectively. Throughout this article, we shall use the word algebra to mean associative algebra in the sense above.

An involution on an R-algebra A is a map A → A, a ↦ \overline{a}, such that \overline{r} ∈ R for all r ∈ R and such that \overline{\overline{a}} = a and \overline{ab} = \overline{b}\overline{a} for all a, b ∈ A. An antistructure in the sense of Wall [44] on an R-algebra A is a pair (−, λ) consisting of a map A → A, a ↦ \overline{a}, and a unit λ ∈ A, which we shall call the symmetry of the antistructure, such that \overline{r} ∈ R for all r ∈ R and such that \overline{\overline{a}} = λ^{−1}aλ and \overline{ab} = \overline{b}\overline{a} for all a, b ∈ A. A morphism (A, (−, λ)) → (A′, (−, λ′)) of algebras with antistructure is an algebra homomorphism f : A → A′ such that f(\overline{a}) = \overline{f(a)} for all a ∈ A and f(λ) = λ′.

Let A be an R-algebra with antistructure (−, λ). The additive subgroups

\begin{matrix}min_{λ}(A) = \{a − λ\overline{a} | a ∈ A\}, \\ max_{λ}(A) = \{a ∈ A | a = −λ\overline{a}\} \end{matrix}

are introduced in Section 1 of [2].

Definition 2.1. A (−, λ)-form parameter in the sense of [2, §13, p.255] on the R-algebra A is an additive subgroup Λ ⫅ A such that (2.1.1) min_{λ}(A) ⫅ Λ ⫅ max_{λ}(A), (2.1.2) ax\overline{a} ∈ Λ for all x ∈ Λ and a ∈ A, and (2.1.3) R_{0}Λ ⫅ Λ, where R_{0} = \{r ∈ R | r = \overline{r}\}.

One checks easily that min_{λ}(A) and max_{λ}(A) satisfy the closure conditions (2.1.2–3) so that both min_{λ}(A) and max_{λ}(A) are form parameters.

Note that if (−, λ) is an antistructure on the R-algebra A then (−, −λ) is also an antistructure on A.

Definition 2.2. We define a (−, λ)-symmetric parameter on the R algebra A to be a (−, −λ)-form parameter on A.

The next definition is motivated by geometric considerations in Sections 4–5.

Definition 2.3. A prepared algebra is a quadruple (A, (−, λ), G, A_{s}) where (2.3.1) A is an R-algebra with antistructure (−, λ), (2.3.2) G is a multiplicative subset of A such that A is generated over R by G and for

each g ∈ G, there are elements r ∈ R and g′ ∈ G with the property that \overline{g} = rg′, and

(2.3.3) A_{s} is an R-submodule of A which is closed under the operations a ↦ λ\overline{a} and

a ↦ ga\overline{g} for all a ∈ A_{s} and g ∈ G. A morphism (A, (−, λ), G, A_{s}) → (A′, (−, λ′), G′, A′_{s}) of prepared algebras is a homomorphism f : (A, (−, λ)) → (A′, (−, λ′)) of algebras with antistructure such that f(G) ⫅ G′ and f(A_{s}) ⫅ A′_{s}.

In the geometric situation, each element a ∈ A_{s} has the property a = λ\overline{a}, i.e. is λ-symmetric. This is our justification for introducing the notation A_{s}.

Let (A, (−, λ), G, A_{s}) be a prepared algebra. Define

max_{λ,A_{s}}(A) = \{a ∈ A| a ≡ −λ\overline{a} mod A_{s}\}.

Since the operations a ↦ λ\overline{a} and a ↦ ga\overline{g}(g ∈ G) on A preserve A_{s}, they induce operations on the quotient module A/A_{s}. This fact is used in making the following definition.

Definition 2.4. Let (A, (−, λ), G, A_{s}) be a prepared algebra. A generalized form parameter or ((−, λ), G, A_{s})-form parameter on A is an additive subgroup Λ of A such that (2.4.1) A_{s} + min_{λ}(A) ⫅ Λ ⫅ max_{λ,A_{s}}(A), (2.4.2) gx\overline{g} ∈ Λ for all x ∈ Λ and g ∈ G, and (2.4.3) R_{0}Λ ⫅ Λ, where R_{0} = \{r ∈ R| r = \overline{r}\}.

One checks easily that A_{s} + min_{λ}(A) and max_{λ,A_{s}}(A) satisfy the closure conditions ((2.4.2–3) so that both A_{s} + min_{λ}(A) and max_{λ,A_{s}}(A) are generalized form parameters.

Recall that a form algebra [2, §1, B] is a triple (A, (−, λ), Λ) where (A, (−, λ)) is an Ralgebra with antistructure and Λ is a (−, λ)-form parameter in the sense of Definition 2.1 on A. The next concept generalizes that of a form algebra.

Definition 2.5. A double parameter algebra or simply parameter algebra is a 6-tuple (A, (−, λ), Γ, G, A_{s}, Λ) where Γ a (−, λ)-symmetric parameter, (A, (−, λ), G, A_{s}) is a prepared algebra, and Λ a ((−, λ), G, A_{s})-form parameter. A morphism

(A, (−, λ), Γ, G, A_{s}, Λ) ⟶ (A′, (−, λ′), Γ′, G′, A′_{s}, Λ′)

of parameter algebras is a morphism f : (A, (−, λ), G, A_{s}) → (A′, (−, λ′), G′, A′_{s}) of prepared algebras such that f(Γ) ⫅ Γ′ and f(Λ) ⫅ Λ′.

Let (A, (−, λ)) be an algebra with antistructure. Let M be a left A-module. Recall that a sesquilinear form on M is a biadditive map B : M × M → A such that B(am, bn) = bB(m, n)\overline{a} for all a, b ∈ A and m, n ∈ M. Let B be a sesquilinear form on M. Define \overline{B} : M × M → A; (m, n) ↦ \overline{B(n, m)}. Although \overline{B} is not necessarily a sesquilinear form on M, one checks easily

λ\overline{B} : M × M → A; (m, n) ↦ λ(\overline{B}(m, n)),

is a sesquilinear form on M. B is called (−, λ)-Hermitian or simply λ-Hermitian if B = λ\overline{B}. Let Γ be a (−, λ)-symmetric parameter on A. A (−, λ)-Hermitian form is called Γ-Hermitian in the sense of [2, §1, C] or Γ-symmetric if B(m, m) ∈ Γ for all m ∈ M. A Γ-Hermitian form B on M is called nonsingular if M is finitely generated and projective over A and the map M → M^{∗}; m ↦ B(m, −) is bijective, where

M^{∗} = Hom_{A}(M, A).

The next definition generalizes [2, p.251, 13.1].

Definition 2.6. Let (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra. Let M be a left Amodule. An (A, (−, λ), Γ, G, A_{s}, Λ)-form or simply (Γ, Λ)-form on M is a pair (B, q) where B is a Γ-Hermitian form on M and q : M → A/Λ is a map called a 𝔮-form, satisfying the following: (2.6.1) q(am) = aq(m)\overline{a} for all m ∈ M and a ∈ R ∪ G. (2.6.2) q(m + n) − q(m) − q(n) = B(m, n) in A/Λ for all m, n ∈ M. (2.6.3) If α ∈ A/Λ, let α^{∼} denote a lifting of α to A. Then for all m ∈ M,

{ \atop q(m)} + λ\overline{{ \atop q(m)}} ≡ B(m, m) mod A_{s}.

The triple (M, B, q) is called an (A, (−, λ), Γ, G, A_{s}, Λ)-module or a (Γ, Λ)-module over (A, (−, λ), G, A_{s}). A doubly parametrized form is a (Γ, Λ)-form for some pair (Λ, Γ). A doubly parametrized module is a (Γ, Λ)-module for some pair (Γ, Λ).

A (Γ, Λ)-module (M, B, q) is called nonsingular if the Hermitian module (M, B) is nonsingular. This includes the condition that M is finitely generated and projective over A. The orthogonal sum of two (Γ, Λ)-modules is defined by

(M, B, q) ⊕ (M′, B′, q′) = (M ⊕ M′, B ⊕ B′, q ⊕ q′)

where M ⊕ M′ is the direct sum of M and M′, (q ⊕ q′)(m, m′) = q(m) + q′(m′), and B ⊕ B′((m, m′), (n, n′)) = B(m, n) + B′(m′, n′) for all m, n ∈ M and m′, n′, ∈ M′. A morphism (M, B, q) → (M′, M′, q′) is an A-linear map f : M → M′ which preserves the Hermitian forms and 𝔮-forms, i.e. B′(f(m), f(n)) = B(m, n) and q′(f(m)) = q(m) for all m, n ∈ M.

The symmetric monoidal category of all nonsingular, doubly parametrized modules over (A, (−, λ), Γ, A_{s}, Λ), with morphisms restricted to all isomorphisms will be denoted by

Q Q Q(A, (−, λ), Γ, G, A_{s}, Λ).

The next result generalizes Theorem 1.1 of [2].

Lemma 2.7. Let (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra such that Λ = max_{λ,A_{s}}(A). Then an A-linear map f : M → M′ of A-modules defines a morphism (M, (B, q)) → (M′, (B′, q′)) of (Γ, Λ)-modules if and only if f preserves the Hermitian forms.

Proof. The only if part is trivial. We prove now the if part. Let m ∈ M. Let { \atop q(m)} and { \atop q′(f(m))} be liftings of q(m) and q′(f(m)), respectively, to A. It suffices to show that

{ \atop q(m)} − { \atop q′(f(m))} ∈ Λ. By (2.6.3),

{ \atop q(m)} + λ\overline{{ \atop q(m)}} ≡ B(m, m) mod A_{s}

and

{ \atop q′(f(m))} + λ\overline{{ \atop q′(f(m))}} ≡ B′(f(m), f(m)) mod A_{s}.

By assumption, B(m, m) ≡ B′(f(m), f(m)). Thus,

{ \atop q(m)} − \overline{{ \atop q′(f(m))}} ≡ −λ(\overline{{ \atop q(m)}} − \overline{{ \atop q′(f(m)))}} mod A_{s}.

Hence, { \atop q(m)} − { \atop q′(f(m))} ∈ Λ. □

Lemma 2.8. Let (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra. Let (M_{1}, B_{1}, q_{1}) and (M_{2}, B_{2}, q_{2}) be (Γ, Λ)-modules. Let B be a Γ-Hermitian form on M_{1} ⊕ M_{2} such that B|_{M_{1}} = B_{1} and B|_{M_{2}} = B_{2}. Then

q : M_{1} ⊕ M_{2} → A/Λ; (m_{1}, m_{2}) ↦ q_{1}(m_{1}) + q_{2}(m_{2}) + [B(m_{1}, m_{2})]

is the unique 𝔮-form such that q|_{M_{1}} = q_{1}, q|_{M_{2}} = q_{2} and (M, B, q) is a (Γ, Λ)-module.

Proof. It is easy to check that (M, B, q) is a (Γ, Λ)-module. The uniqueness follows from property (2.6.2) of (Γ, Λ)-modules. □

Lemma 2.9. Let A A A = (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter R-algebra. If either Λ = max_{λ,A_{s}}(A) or each element of A is a unique R-linear combination ∑_{g∈G} r_{g}g then the following property is satisfied:

(2.9.1) If x ∈ A_{s} ∩ Γ and ∑_{i=1}^{n} r_{i}g_{i} = 0, r_{i} ∈ R, g_{i} ∈ G, then ∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j} ∈ Λ.

Furthermore, if A A A satisfies (2.9.1) then every quotient of A A A and every localization

S^{−1}A A A := (S^{−1}A, (−, λ), S^{−1}Γ, G, S^{−1}A_{s}, S^{−1}Λ)

of A A A where S is a multiplicative set in R_{0} = \{r ∈ R | r = \overline{r}\} also satisfies (2.9.1).

Proof. If each element of A is a unique R-linear combination of elements of G then ∑_{i=1}^{n} r_{i}g_{i} = 0 implies each r_{i} = 0. Thus, ∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j} = 0 ∈ Λ. Suppose now that Λ = max_{λ,A_{s}}(A). By \overline{\overline{g_{j}}} = λ^{−1}g_{j}λ,

\begin{aligned}0 & = (∑_{i=1}^{n} r_{i}g_{i}) x\overline{(∑_{i=1}^{n} r_{i}g_{i})} \\ & = ∑_{i=1}^{n} r_{i}\overline{r}_{i}g_{i}x\overline{g}_{i} + ∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j} + λ\overline{∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j}}. \end{aligned}

Since each r_{i}\overline{r}_{i}g_{i}x\overline{g}_{i} ∈ A_{s} by (2.3.3), it follows that

∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j} + λ\overline{∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j}} ≡ 0 mod A_{s}.

Thus, ∑_{i<j} r_{i}\overline{r}_{j}g_{i}x\overline{g}_{j} ∈ Λ. The second assertion in the lemma is clear. □

The next result is a generalization of [2, Lemma 13.6].

Lemma 2.10. Let A A A = (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra. Suppose that A A A satisfies (2.9.1) and Γ + A_{s} = min_{−λ}(A) + A_{s}. If (M, B) is a Γ-Hermitian module and M is finitely generated and projective over A then there is a map q : M → A/Λ such that (M, B, q) is a (Γ, Λ)-module.

Proof. Let M′ be an A-module such that M ⊕ M′ is finitely generated and free over A and let B′ be the Γ-Hermitian form on M′ with constant value 0. Let (N, C) = (M ⊕ M′, B ⊕ B′). If there is a map q : N → A/Λ such that (N, C, q) is a (Γ, Λ)-module then (M, B, q|_{M}) is a (Γ, Λ)-module. Thus, we can reduce to the case M is finitely generated and free over A, with A-basis {v_{i}, …, v_{n}}. Let B_{0} and B_{k} (1 ≦ k ≦ r) denote the unique A-sesquilinear forms on M such that

B_{0}(v_{i}, v_{j}) = \begin{cases}B(v_{i}, v_{j}) & (i < j) \\ 0 & (otherwise) \end{cases}

B_{k}(v_{i}, v_{j}) = \begin{cases}B(v_{i}, v_{j}) & (k = i = j) \\ 0 & (otherwise). \end{cases}

One checks easily that B = B_{0} + λ\overline{B}_{0} + ∑_{i=1}^{r} B_{i} where λ\overline{B}_{0}(m, m′) = λ(\overline{B_{0}(m′.m)}).

We define now 𝔮-forms for B_{0} + λ\overline{B}_{0}, B_{1}, …, B_{r}. Define q_{0} : M → A/Λ by q(m) = [B_{0}(m, m)]. For 1 ≦ i ≦ r, it follows from the assumption on Γ and the fact that B is Γ-Hermitian that B_{i}(v_{i}, v_{i}) = c_{i} + λ\overline{c}_{i} + d_{i} for some c_{i} ∈ A and d_{i} ∈ A_{s}. Since

c_{i} + λ\overline{c}_{i} + d_{i} ∈ Γ and c_{i} + λ\overline{c}_{i} ∈ min_{−λ}(A) ⫅ Γ,

we have that d_{i} ∈ Γ. Let a ∈ A and suppose a = ∑ r_{k}g_{k} (r_{k} ∈ R, g_{k} ∈ G). By (2.9.1), the class in A/Λ of the element ∑_{k<ℓ} r_{ℓ}\overline{r}_{k}g_{ℓ}d_{i}\overline{g}_{k} does not depend on the decomposition of the element a. For 1 ≦ i ≦ r, define

q_{i} : M → A/Λ; ∑_{j=1}^{r} a_{j}v_{j} ⟼ [a_{i}c_{i}\overline{a}_{i} + ∑_{k<ℓ} r_{ℓ}\overline{r}_{k}g_{ℓ}d_{i}\overline{g}_{k}]

where a_{i} = ∑_{k} r_{k}g_{k}. One shows straightforward that (M, q_{0}, B_{0} +λ\overline{B}_{0}) is a (Γ, Λ)-module and using the manipulations in the proof of Lemma 2.9 that (M, B_{i}, q_{i}) (1 ≦ i ≦ r) is a (Γ, Λ)-module. Thus,

(M, B, ∑_{i=0}^{r} q_{i}) = (M, B_{0} + λ\overline{B}_{0} + ∑_{i=1}^{r} B_{i}, ∑_{i=0}^{r} q_{i})

is a (Γ, Λ)-module. □

Let (A, (−, λ)) be an R-algebra with antistructure and let Λ (resp. Γ) be a (−, λ)-form parameter (resp. (−, λ)-symmetric parameter) on A. Let

Q Q Q(A, (−, λ), Λ)

denote the category of nonsingular, Λ-quadratic modules (cf. Sections 1, B and 13 of [2]) and let

H H H(A, (−, λ), Γ)

denote the category of nonsingular Γ-Hermitian modules. Both categories above are symmetric monoidal categories under orthogonal sum.

Lemma 2.11. Let A A A = (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra.

(2.11.1) If A_{s} = 0, there is a canonical equivalence

\begin{matrix}Q Q Q(A, (−, λ), Λ) & ⟶ Q Q Q(A A A) \\ (M, B, q) & ⟼ (M, B, q) \\ [f : (M, B, q) → (M_{1}, B_{1}, q_{1})] & ⟼ [f : (M, B, q) → (M_{1}, B_{1}, q_{1})] \end{matrix}

of symmetric monoidal categories.

(2.11.2) Let A A A′ = (A, (−, λ), G, Γ′, A′_{s}, Λ′) be a parameter algebra such that Γ ⫅ Γ′, A_{s} ⫅ A′_{s}, and Λ ⫅ Λ′. Let π : A/Λ → A/Λ′ denote the canonical map. Then there is a canonical functor

\begin{matrix}Q Q Q(A A A) & ⟶ Q Q Q(A A A′) \\ (M, B, q) & ⟼ (M, B, πq) \\ [f : (M, B, q) → (M_{1}, B_{1}, q_{1})] & ⟼ [f : (M, B, πq) → (M_{1}, B_{1}, πq_{1})] \end{matrix}

of symmetric monoidal categories. (2.11.3) If A_{s} = A, there is a canonical equivalence

\begin{matrix}Q Q Q(A A A) & ⟶ H H H(A, (−, λ), Γ) \\ (M, B, q) & ⟼ (M, B) \\ [f : (M, B, q) ⟶ (M_{1}, B_{1}, q_{1})] & ⟼ [f : (M, B) ⟶ (M_{1}, B_{1})] \end{matrix}

of symmetric monoidal categories.

The proof is straightforward.

Lemma 2.12. Let A A A = (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra. Let

F : Q Q Q(A A A) ⟶ H H H(A, (−, λ), Γ); (M, q, B) ⟼ (M, B)

denote the forgetful functor. If A A A satisfies (2.9.1) (resp. Λ = max_{λ,A_{s}}(A)) then F is surjective on isomorphism classes of objects (resp. an isomorphism of symmetric monoidal categories).

Proof. The assertions are an immediate consequence of Lemmas 2.7, 2.9, and 2.10. □

Corollary 2.13. Let A A A = (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra. If there is an element x ∈ Center(A) such that x + \overline{x} = 1 then Γ = max_{−λ}(A), Λ = max_{λ,A_{s}}(A), and the forgetful functor

Q Q Q(A A A) ⟶ H H H(A, (−, λ), Γ); (M, q, B) ⟼ (M, B)

is an isomorphism of symmetric monoidal categories.

Proof. By the proof of Lemma 13.8 of [2], Γ = max_{−λ}(A) and Λ = max_{λ,A_{s}}(A). The last assertion of the corollary follows now from Lemmas 2.9 and 2.12. □

Our next goal is to introduce the notion of positioning module and to study doubly parametrized forms on positioning modules. A positioning module with empty positioning data is essentially just an ordinary module, so that our study of doubly parametrized forms on positioning modules will include as a special case the doubly parametrized modules in Definition 2.6.

Suppose G is a monoid, i.e. a nonempty set with an associative, binary operation having an identity element 1_{G}. Recall that a G-set is a set X together with a map G × X → X; (g, x) ↦ gx, such that (gf)x = g(fx) and 1_{G}x = x for all g, f ∈ G and x ∈ X.

Definition 2.14. A positioning algebra is a triple

(A, G, Θ)

where A is an algebra, G a monoidal subset of A, and Θ a G-set. A morphism (A, G, Θ) → (A′, G′, Θ′) of positioning algebras is a pair (f_{A}, τ) where f_{A} : A → A′ is a homomorphism of algebras such that f_{A}(G) ⫅ G′ and τ : Θ → Θ′ an equivariant map, i.e. τ(gx) = f_{A}(g)τ(x) for all g ∈ G and x ∈ Θ.

Definition 2.15. Let (A, G, Θ) be a positioning algebra. A positioning module over (A, G, Θ) is a pair

(M, θ : Θ → M)

where M is a left A-module and θ : Θ → M a G-equivariant map, i.e. θ(gx) = gθ(x) for all g ∈ G and x ∈ X. This θ will be called a positioning map of the module. Define

P P P(A, G, Θ)

to be the category whose objects are all positioning modules (P, θ : Θ → P) over (A, G, Θ) where P is a finitely generated, projective A-module. A morphism (P, θ : Θ → P) → (P′, θ′ : Θ → P′) of positioning modules is an A-linear isomorphism f : P → P′ such that fθ = θ′. (We use only isomorphisms so that we can define later the hyperbolic functor.) The category P P P(A, G, Θ) has a sum operation

(P, θ) ⊕ (P′, θ′) = (P ⊕ P′, θ ⊕ θ′)

where P ⊕ P′ denotes the direct sum of P and P′ and θ ⊕ θ′ is defined by

(θ ⊕ θ′)(x) = (θ(x), θ′(x)).

Let

P P P(A)

denote as usual the category whose objects are all finitely generated, projective A-modules and whose morphisms are all A-linear isomorphisms between such modules. P P P(A) has a sum operation defined by direct sum.

Lemma 2.16. Let (A, G, Θ) be a positioning algebra such that Θ is the empty G-set. Then the canonical functor

\begin{matrix}P P P(A) & ⟶ P P P(A, G, Θ) \\ P & ⟼ (P, ϕ) \\ [f : P → P′] & ⟼ [f : (P, ϕ) → (P′, ϕ)] \end{matrix}

where ϕ : Θ → P denotes the empty map is an equivalence of symmetric monoidal categories.

This is evident.

If M M M is a symmetric monoidal category, let

K_{0}(M M M)

denote its Grothendieck group.

Lemma 2.17. Let (A, G, Θ) be a positioning algebra. Then the functor

\begin{matrix}P P P(A) & ⟶ P P P(A, G, Θ) \\ P & ⟼ (P, 0) \\ [f : P → P′] & ⟼ [f : (P, 0) → (P′, 0)] \end{matrix}

where 0 : Θ → P denotes the constant map with value 0 preserves sums and the forgetful functor

\begin{matrix}P P P(A, G, Θ) & ⟶ P P P(A) \\ (P, θ) & ⟼ P \\ [f : (P, θ) → (P′, θ′)] & ⟼ [f : P → P′] \end{matrix}

is a sum preserving retract. Moreover, the induced homomorphism

K_{0}(P P P(A)) ⟶ K_{0}(P P P(A, G, Θ)); [P] ⟼ [P, 0]

of Grothendieck groups is an isomorphism.

Proof. The assertions for the functor P P P(A) → P P P(A, G, Θ); P ↦ (P, 0), and the forgetful functor P P P(A, G, Θ) → P P P(A); (P, θ) ↦ P, are clear. Thus, the induced homomorphism K_{0}(P P P(A)) → K_{0}(P P P(A, G, Θ)); [P] ↦ [P, 0], is injective, since it has a retract defined by the forgetful functor. Let (P, θ) ∈ P P P(A, G, Θ). The A-linear isomorphism P ⊕P → P ⊕P; (p, p′) ↦ (p, p′ + p), defines an isomorphism

(2.17.1)  (P, θ) ⊕ (P, 0) ⟶^{≅} (P, θ) ⊕ (P, θ)

of positioning modules. Thus, [P, 0] = [P, θ]. Thus, the homomorphism K_{0}(P P P(A)) → K_{0}(P P P(A, G, Θ)) is surjective. □

Definition 2.18. A parameter algebra with positioning data is a 7-tuple

(A, (−, λ), Γ, G, A_{s}, Λ, Θ)

where (A, (−, λ), Γ, G, A_{s}, Λ) is a parameter algebra and (A, G, Θ) a positioning algebra. A morphism of parameter algebras with positioning date is defined in the obvious way, i.e. it is an algebra homomorphism which induces morphisms of parameter and positioning algebras.

Definition 2.19. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. A doubly parametrized positioning module over (A A A, Θ) is a quadruple (M, B, q, θ) where (M, θ) is a positioning module and (B, q) a doubly parametrized form on M over A A A. Morphisms (resp. orthogonal sums) of doubly parametrized positioning modules over (A A A, Θ) are defined in the obvious way, i.e. so that they induce morphism (resp. sums) on the underlying doubly parametrized modules and positioning modules.

The symmetric monoidal category of all nonsingular, doubly parametrized positioning modules over (A A A, Θ), with morphisms restricted to all isomorphisms will be denoted by

Q Q Q(A A A, Θ).

Lemma 2.20. If (M, B, q, θ) be a doubly parametrized positioning module over (A A A, Θ) then

(M, B, q, θ) ≅ (M, B, q, −θ).

Proof. The map M → M; m ↦ −m, is an isomorphism (M, B, q, θ) → (M, B, q, −θ). □

Lemma 2.21. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. If Θ is the empty set then the canonical functor

Q Q Q(A A A) → Q Q Q(A A A, Θ); (M, B, q) ↦ (M, B, q, ϕ)

where ϕ : Θ → M is the empty map is an equivalence of symmetric monoidal categories.

This is clear.

Lemma 2.22. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Then the functor

Q Q Q(A A A) → Q Q Q(A A A, Θ); (M, B, q) ↦ (M, B, q, 0)

where 0 : Θ → M denotes the constant map with value 0 preserves sums and the forgetful functor

Q Q Q(A A A, Θ) → Q Q Q(A A A); (M, B, q, θ) ↦ (M, B, q)

is a sum preserving retract. In particular, the induced homomorphism

K_{0}(Q Q Q(A A A)) → K_{0}(Q Q Q(A A A, Θ)); [M, B, q] ↦ [M, B, q, 0]

of Grothendieck groups has a retract (but is not in general an isomorphism, as in Lemma 2.17.

The proof is straightforward.

In order to obtain the analogs of Lemmas 2.11, 2.12, and Corollary 2.13 for positioning modules, we introduce notions of quadratic and Hermitian forms on positioning modules.

Definition 2.23. A form algebra with positioning data is a quintuple

(A, (−, λ), Λ, G, Θ)

where (A, (−, λ), Λ) is a form algebra and (A, G, Θ) a positioning algebra. A morphism of form algebras with positioning data is a homomorphism of algebras which induces morphisms of form algebras and positioning algebras.

Definition 2.24. A symmetric parameter algebra with positioning data is a quintuple

(A, (−, λ), Γ, G, Θ)

such that (A, (−, −λ), Γ, G, Θ) is a form algebra with positioning data.

Definition 2.25. Let (A, (−, λ), Λ, G, Θ) be a form algebra with positioning data. A quadratic positioning module over (A, (−, λ), Λ, G, Θ) is a quintuple (M, B, q, θ) where (M, θ) is a positioning module and (B, q) a Λ-quadratic form on M. Morphisms (resp. orthogonal sums) of quadratic positioning modules over (A, (−, λ), Λ, G, Θ) are defined in the usual way, i.e. so that they induce morphisms (resp. sums) on the underlying quadratic modules and positioning modules.

The symmetric monoidal category of all nonsingular, quadratic forms on positioning modules over (A, (−, λ), Λ, G, Θ), with morphisms restricted to all isomorphisms will be denoted by

Q Q Q(A, (−, λ), Λ, G, Θ).

Definition 2.26. Let (A, (−, λ), Γ, G, Θ) be a symmetric parameter algebra with positioning data. A Hermitian positioning module over (A, (−, λ), Γ, G, Θ) is a triple (M, B, θ) where (M, θ) is a positioning module and B a Γ-Hermitian form on M. Morphsims (resp. orthogonal sums) of Hermitian positioning modules over (A, (−, λ), Γ, G, Θ) are defined in the usual way, i.e. so that they induce morphisms (resp. sums) on the underlying Hermitian modules and positioning modules.

The symmetric monoidal category of all nonsingular, Hermitian forms on positioning modules over (A, (−, λ), Γ, G, Θ), with morphisms restricted to all isomorphisms will be denoted by

H H H(A, (−, λ), Γ, G, Θ).

The next result generalizes Lemma 2.11.

Lemma 2.27. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. (2.27.1) If A_{s} = 0, there is a canonical equivalence

Q Q Q(A, (−, λ), Λ, G, Θ) → Q Q Q(A A A, Θ); (M, B, q, θ) ↦ (M, B, q, θ)

of symmetric monoidal categories. (2.27.2) Let (A A A′, Θ) = (A, (−, λ), Γ′, G, A′_{s}, Λ′, Θ) be a parameter algebra with positioning

data such that Γ ⫅ Γ′, A_{s} ⫅ A′_{s}, and Λ ⫅ Λ′. Let π : A/Λ → A/Λ′ denote the canonical map. Then there is a canonical functor

Q Q Q(A A A, Θ) → Q Q Q(A A A′, Θ); (M, B, q, θ) ↦ (M, B, π ∘ q, θ)

of symmetric monoidal categories.

(2.27.3) If A_{s} = A, there is a canonical equivalence

Q Q Q(A A A, Θ) → H H H(A, (−, λ), Γ, G, Θ); (M, B, q, θ) ↦ (M, B, θ)

of symmetric monoidal categories.

The proof is straightforward.

The next result generalizes Lemma 2.12.

Lemma 2.28. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let

F : Q Q Q(A A A, Θ) → H H H(A, (−, λ), Γ, G, Θ); (M, B, q, θ) ↦ (M, B, θ)

denote the forgetful functor. If the underlying parameter algebra A A A satisfies (2.9.1) (resp. Λ = max_{λ,A_{s}}(A)) then F is surjective on isomorphism classes of objects (resp. an isomorphism of symmetric monoidal categories).

Proof. The proof is the same as that of Lemma 2.12. The positioning maps are only extra baggage. □

The next result generalizes Corollary 2.13.

Corollary 2.29. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. If there is an element x ∈ Center(A) such that x+\overline{x} = 1 then Γ = max_{−λ}(A), Λ = max_{λ,A_{s}}(A), and the forgetful functor

Q Q Q(A A A, Θ) → H H H(A, (−, λ), Γ, G, Θ); (M, (B, q), θ) ↦ (M, B, θ)

is an isomorphism of symmetric monoidal categories.

Proof. The proof is the same as that of Corollary 2.13, except the reference to Lemma 2.12 is replaced by one to Lemma 2.28. □

Let C C C be a subcategory of P P P(A) satisfying the next assumption.

Assumption 2.30. C C C contains A and is closed under isomorphisms, direct sums, and dualization P ↦ P^{∗}.

Definition 2.31. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let (P, θ) ∈ P P P(A, G, Θ). Let B′_{P} denote the sesquilinear form on P ⊕ P^{∗} defined by B′_{P} ((p, f), (p′, f′)) = f(p′). Let

B_{P} = B′_{P} + λ\overline{B}′_{P}

where λ\overline{B}′_{P} (m, m′) = λ\overline{(B′_{P} (m′, m))}. Let

q_{P} : P ⊕ P^{∗} → A/Λ; m ↦ [B′_{P} (m, m)].

One checks easily that the construction (P, θ) ↦ ℍ(P, θ), where

ℍ(P, θ) = (P ⊕ P^{∗}, B_{P} , q_{P} , θ),

defines a sum preserving functor

\begin{matrix}ℍ : P P P(A, G, Θ) & ⟶ Q Q Q(A A A, Θ) \\ (P, θ) & ⟼ ℍ(P, θ) \\ [f : (P, θ) → (P′, θ′)] & ⟼ [f ⊕ f^{∗}^{−1} : ℍ(P, θ) → ℍ(P′, θ′)] \end{matrix}

called the hyperbolic functor. The object ℍ(P, θ) is called the hyperbolic module on (P, θ) and ℍ(A, θ) is a called a hyperbolic plane. ℍ(P, θ) is called a C C C-hyperbolic module whenever P ∈ C C C.

Definition 2.32. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let

P P P(A A A, Θ)

denote the symmetric monoidal category whose objects are all quadruples (P, b_{0}, q_{0}, θ) where (P, θ) ∈ P P P(A, G, Θ) and (b_{0}, q_{0}) is a doubly parametrized form on P^{∗} over A A A. A morphism (P, b_{0}, q_{0}, θ) → (P′, b′_{0}, q′_{0}, θ′) is an A-linear isomorphism f : P → P′ such that f defines a morphism (P, θ) → (P′, θ′) of positioning modules and f^{∗}^{−1} defines a morphism (P^{∗}, b_{0}, q_{0}) → (P′^{∗}, b′_{0}, q′_{0}) of doubly parametrized modules over A A A. Sums are defined as in Definitions 2.6 and 2.15.

Let (P, b_{0}, q_{0}, θ) ∈ P P P(A A A, Θ). Let B_{P,b_{0}} denote the Γ-Hermitian form on P ⊕ P^{∗} defined by

B_{P,b_{0}}((p, f), (p′, f′)) = B_{P} ((p, f), (p′, f′)) + b_{0}(f, f′)

where B_{P} is defined as in the paragraph above. Define

q_{P,q_{0}} : P ⊕ P^{∗} → A/Λ; (p, f) ↦ [f(p)] + q_{0}(f).

By Lemma 2.8, (B_{P,b_{0}}, q_{P,q_{0}}) is a doubly parametrized form on P ⊕P^{∗} over A A A. One checks easily that the construction (P, b_{0}, q_{0}, θ) ↦ 𝕄(P, b_{0}, q_{0}, θ), where

𝕄(P, b_{0}, q_{0}, θ) = (P ⊕ P^{∗}, B_{P,b_{0}}, q_{P,q_{0}}, θ),

defines a sum preserving functor

\begin{matrix}𝕄 : P P P(A A A, Θ) & ⟶ Q Q Q(A A A, Θ) \\ (P, b_{0}, q_{0}, θ) & ⟼ 𝕄(P, b_{0}, q_{0}, θ) \\ [f : (P, b_{0}, q_{0}, θ) → (P′, b′_{0}, q′_{0}, θ′)] & ⟼ [f ⊕ f^{∗}^{−1} : 𝕄(P, b_{0}, q_{0}, θ) → 𝕄(P′, b′_{0}, q′_{0}, θ′)] \end{matrix}

called the metabolic functor. The object 𝕄(P, b_{0}, q_{0}, θ) is called the metabolic module on (P, b_{0}, q_{0}, θ) and 𝕄(A, b_{0}, q_{0}, θ) is called a metabolic plane. 𝕄(P, b_{0}, q_{0}, θ) is called a C C C-metabolic module whenever P ∈ C C C.

Lemma 2.33. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Then the diagram

\xymatrix{P P P(A, G, Θ) \ar[rr]^{I} \ar[dr]_{ℍ} &  & P P P(A A A, Θ) \ar[dl]^{𝕄} \\  & Q Q Q(A A A, Θ)}

of symmetric monoidal categories commutes, where I((P, θ)) = (P, 0, 0, θ).

This lemma is obvious.

Definition 2.34. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let (M, B, q, θ) ∈ Q Q Q(A A A, Θ). An A-submodule L ⫅ M is called totally isotropic if B and q are trivial on L, i.e. B(m, m′) = 0 and q(m) = 0 for all m, m′ ∈ L. The positioning function θ is called totally isotropic if B(θ(x), θ(x′)) and q(θ(x)) = 0 for all x, x′ ∈ Θ. A totally isotropic, A-direct summand L ⫅ M which contains Image(θ) is called a sublagrangian. If, in addition, L ∈ C C C then L is called a C C C-sublagrangian. A sublagrangian (resp. C C C-sublagrangian) L such that L = L^{⊥}, where

L^{⊥} = {m ∈ M | B(m, ℓ) = 0 ∀ℓ ∈ L},

is called a Lagrangian (resp. C C C-Lagrangian). (M, B, q, θ) is called a null module (resp. C C C-null module), if it contains a Lagrangian (resp. C C C-Lagrangian).

The next result generalizes Lemma 2.8 of [2].

Lemma 2.35. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let P ⫅ M be a Lagrangian in (M, B, q, θ) and let i : P ↪ M denote the natural embedding. Then the A-linear map M ↦ P^{∗}; m ↦ B(m, −), has a splitting i′ : P^{∗} → M and for any splitting i′, if b_{0} = B|_{Image(i′)} and q_{0} = q|_{Image(i′)} then the A-linear map i ⊕ i′ : P ⊕ P^{∗} → M defines an isomorphism 𝕄(P, b_{0}, q_{0}, θ) → (M, B, q, θ) of doubly parametrized, positioning modules.

Proof. Since B is nonsingular and P is a direct summand of M, it follows that the A-linear map M → P^{∗}; m ⟼ B(m, −), is surjective. Since M is projective, it follows that P is projective and hence, P^{∗} is projective. Thus, the map M → P above has a splitting, say i′. Let K = Ker[M → P^{∗}]. By definition,

K = {m ∈ M | B(m, p) = 0 ∀p ∈ P } = P^{⊥}.

Since P is a Lagrangian, it follows that P = P^{⊥} = K. Thus, the A-linear map i ⊕ i′ : P ⊕P^{∗} → M is an isomorphism of A-modules. Defining (b_{0}, q_{0}) as in the lemma, one checks that i ⊕ i′ defines an isomorphism 𝕄(P, b_{0}, q_{0}, θ) → (M, B, q, θ) of doubly parametrized, positioning modules. □

The next result generalizes Lemmas 2.6 and 2.7 of [2].

Corollary 2.36. Let (M, B, q, θ) ∈ Q Q Q(A A A, Θ) and let P ⫅ M be a sublagrangian. Let i : P ↪ M denote the natural embedding and let i′ : P^{∗} → M be a splitting of the Alinear map M → P^{∗}; m ↦ B(m, −). Let b_{0} = B|_{Image(i′)} and q_{0} = q|_{Image(i′)}. Then the A-linear map i ⊕ i′ : P ⊕ P^{∗} → M defines an embedding 𝕄(P, b_{0}, q_{0}, θ) → (M, B, q, θ) and if N = Image(i ⊕ i′) then

(M, B, q, θ) = (N, B|_{N}, q|_{N}, θ) ⊕ (N^{⊥}, B|_{N^{⊥}}, q|_{N^{⊥}}, 0).

Furthermore, the Hermitian form B and the map q are well defined on P^{⊥}/P and if j : P^{⊥} ↪ M denotes the natural embedding then j induces an isomorphism

(P^{⊥}/P, B|_{P^{⊥}}, q|_{P^{⊥}}, 0) → (N^{⊥}, B|_{N^{⊥}}, q|_{N^{⊥}}, 0).

Proof. As in the proof of Lemma 2.35, one shows that i ⊕ i′ defines an embedding

𝕄(P, b_{0}, q_{0}, θ) → (M, B, q, θ).

By Corollary 2.3 of [2] and Lemma 2.8,

(M, B, q, θ) = (N, B|_{N}, q|_{N}, θ) ⊕ (N^{⊥}, B|_{N^{⊥}}, q_{N^{⊥}}, 0).

It follows that P^{⊥} = P ⊕ N^{⊥}. One checks easily that the isomorphism j : P^{⊥} → P ⊕ N^{⊥} followed by the canonical projection P ⊕ N^{⊥} → N^{⊥} defines an isomorphism (P^{⊥}/P, B|_{P^{⊥}}, q|_{P^{⊥}}, 0) → (N^{⊥}, B|_{N^{⊥}}, q|_{N^{⊥}}, 0). □

The next result generalizes Lemma 2.9 of [2].

Lemma 2.37. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let (M, B, q, θ) ∈ Q Q Q(A A A, Θ). Then the diagonal submodule {(m, m)| m ∈ M} of M ⊕ M is a Lagrangian in (M, B, q, θ) ⊕ (M, −B, −q, θ).

The proof is straightforward.

Corollary 2.38. The family of metabolic planes in Q Q Q(A A A, Θ) is cofinal in Q Q Q(A A A, Θ), i.e. given M M M ∈ Q Q Q(A A A, Θ), there is an M M M′ ∈ Q Q Q(A A A, Θ) and metabolic planes M M M_{1}, …, M M M_{r} ∈ Q Q Q(A A A, Θ) such that

M M M ⊕ M M M′ ≅ M M M_{1} ⊕ ⋯ ⊕ M M M_{r}.

Proof. From Lemmas 2.37 and 2.35, it follows that metabolic modules 𝕄(P′, b′_{0}, q′_{0}, θ) are cofinal in Q Q Q(A A A, Θ). Choosing P ∈ P P P(A) such that P′ ⊕ P is free and forming the orthogonal sum 𝕄(P′, b′_{0}, q′_{0}, θ) ⊕ 𝕄(P, 0, 0, 0), one obtains that metabolic modules 𝕄(F, b_{0}, q_{0}, θ) on finitely generated, free modules F are cofinal in Q Q Q(A A A, Θ). We shall show that 𝕄(F, b_{0}, q_{0}, θ) is isomorphic to an orthogonal sum of metabolic planes. Let (B, q) denote the doubly parametrized form (B_{F,b_{0}}, q_{F,q_{0}}) on 𝕄(F, b_{0}, q_{0}, θ).

Let {e_{1}, …, e_{r}} be a basis for F and {f_{1}, …, f_{r}} ∈ F^{∗} its dual basis. Let c denote the (unique) sesquilinear form on F^{∗} such that

c(f_{i}, f_{j}) = \begin{cases}−b_{0}(f_{i}, f_{j}) & (i < j) \\ 0 & (otherwise). \end{cases}

Let α : F^{∗} → F denote the (unique) A-linear map such that

B(α(f_{i}), f_{j}) = c(f_{i}, f_{j}) (1 ≦ i, j ≦ r).

Let id : F → F^{∗} denote the identity map. Define f′_{i} = (id + α)f_{i}, M_{i} = Ae_{i} + Af′_{i}, B_{i} = B|_{M_{i}} and q_{i} = q|_{M_{i}} (1 ≦ i ≦ r). Further define θ_{i} : Θ → Ae_{i} such that θ(x) = ∑_{i=1}^{r} θ_{i}(x) for all x ∈ Θ. Then 𝕄(F, b_{0}, q_{0}, θ) = (F ⊕ F^{∗}, B, q, θ) = ⨁_{i=1}^{r}(M_{i}, B_{i}, q_{i}, θ_{i}). Moreover, one sees easily (see the proof of Lemma 2.35) that each (M_{i}, B_{i}, q_{i}, θ_{i}) is isomorphic to a metabolic plane. □

Definition 2.39. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let (M, B, q, θ) be a doubly parametrized, positioning module over (A A A, Θ). An ordered pair (e, f) of elements in M is called a metabolic pair if

B(e, e) = 0, B(f, e) = 1, q(e, e) = q(f, f) = 0,

and

B(θ(x), e) = 0 (∀x ∈ Θ).

If a metabolic pair (e, f) generates M as an A-module then it is called a metabolic basis for (M, B, q, θ). In this case, (M, B, q, θ) is a metabolic plane in the sense of the paragraph subsequent to Definition 2.32.

Lemma 2.40. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. (2.40.1) A doubly parametrized, positioning module over (A A A, Θ) has a metabolic basis if and

only if it is isomorphic to a metabolic plane. (2.40.2) If (e, f) is a metabolic basis for (M, B, q, θ) and a ∈ A_{s} then (e, f + ae) is a

metabolic basis for (M, B, q, θ) and B(f + ae, f + ae) = B(f, f) + a + λ\overline{a}.

Proof. We prove the only if part of (2.40.1). Let (e, f) be a metabolic basis for the doubly parametrized, positioning module (M, B, q, θ). Let x = 1 ∈ A and let y ∈ A^{∗} be its dual. Let α : A⊕A^{∗} → M; cx+dy ↦ ce+df. Let b_{0} : A^{∗} ×A^{∗} → A; (dy, d′y) ↦ B(df, d′f) and let q_{0} : A^{∗} → A/Λ; dy ↦ q(df). Let τ : Θ → A be the unique map such that the diagram

\xymatrix{Θ \ar[r]^{τ} \ar[dr]_{θ} & A ⊕ A^{∗} \ar[d]^{α}_{≅} \\  & M}

commutes. Then (b_{0}, q_{0}) is a doubly parametrized form on A^{∗} and α defines an isomorphism 𝕄(A, b_{0}, q_{0}, τ) → (M, B, q, θ).

Next we prove the if part of (2.40.1). Let (B, q) denote the doubly parametrized form on the metabolic plane 𝕄(A, b_{0}, q_{0}, θ). Let e = 1 ∈ A and y ∈ A its dual. By definition, B(e, e) = 0, q(e) = 0, and B(θ(t), e) = 0 for all t ∈ Θ. Let a ∈ A_{q} be a representative of q(y) and let f = y − ae. Then

B(f, e) = B(y, e) − B(e, e)\overline{a} = B(y, e) = 1 and
q(f) = q(y) + q(−ae) + [B(y, −ae)] (by (2.6.2))
= q(y) + [−aB(y, e)] = q(y) − [a] = 0.

Thus, (e, f) is a metabolic basis for 𝕄(A, b_{0}, q_{0}, θ).

The assertion (2.40.2) can be proved by a straightforward computation. □

Lemma 2.41. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let (P, b_{0}, q_{0}, θ) ∈ P P P(A A A, Θ). Let β : P^{∗} → P denote the (unique) A-linear map such that f′(β(f)) = b_{0}(f′, f) for all f′ ∈ P^{∗}. Then the A-linear isomorphism

(\begin{matrix}I_{P} & 0 & 0 & 0 \\ 0 & I_{P^{∗}} & 0 & I_{P^{∗}} \\ 0 & 0 & I_{P} & 0 \\ 0 & 0 & 0 & I_{P^{∗}} \end{matrix} ) (\begin{matrix}I_{P} & 0 & 0 & 0 \\ 0 & I_{P^{∗}} & 0 & 0 \\ −I_{P} & 0 & I_{P} & 0 \\ 0 & 0 & 0 & I_{P^{∗}} \end{matrix} ) (\begin{matrix}I_{P} & 0 & 0 & 0 \\ 0 & I_{P^{∗}} & 0 & 0 \\ 0 & −β & I_{P} & 0 \\ 0 & 0 & 0 & I_{P^{∗}} \end{matrix} ) :

\begin{matrix}P ⊕ P^{∗} ⊕ P ⊕ P^{∗} & ⟶ P ⊕ P^{∗} ⊕ P ⊕ P^{∗}; \\ (\begin{matrix}p \\ f \\ p′ \\ f′ \end{matrix}) & ⟼ (\begin{matrix}p \\ f + f′ \\ p′ − p − β(f) \\ f′ \end{matrix}) , \end{matrix}

defines an isomorphism

𝕄(P, b_{0}, q_{0}, θ) ⊕ ℍ(P, 0) ⟶^{≅} 𝕄(P, b_{0}, q_{0}, θ) ⊕ 𝕄(P, −b_{0}, −q_{0}, −θ)

of doubly parametrized, positioning modules over (A A A, Θ).

Proof. One checks first that the A-linear map in the lemma preserves the positioning maps and Hermitian forms. The one can use Lemma 2.8 to simplify checking that it preserves the 𝔮-forms. □

Let A be an algebra with antistructure (−, λ).

Definition 2.42. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Define

P P P(A A A, Θ)_{C C C}
Q Q Q(A A A, Θ)_{C C C}

to be the full, symmetric monoidal subcategories of P P P(A A A, Θ) and Q Q Q(A A A, Θ), respectively, such that the underlying A-module of each object lies in C C C.

Let

D D D ⫅ C C C

be a subcategory which contains A and has the same closure properties as C C C (see Assumption 2.30).

Definition 2.43. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Define

KQ_{0}(A A A, Θ)_{C C C} = K_{0}(Q Q Q(A A A, Θ)_{C C C})

W Q_{0}(A A A, Θ)_{D D D,C C C} = KQ_{0}(A A A, Θ)_{C C C}/⟨D D D-null modules⟩
W Q_{0}(A A A, Θ)_{C C C} = W Q_{0}(A A A, Θ)_{C C C,C C C}
KQ_{0}(A A A, Θ) = KQ_{0}(A A A, Θ)_{P P P(A A A)} and
W Q_{0}(A A A, Θ) = W Q_{0}(A A A, Θ)_{P P P(A A A)}.

Proposition 2.44. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let ℍ : D D D → Q Q Q(A A A, Θ)_{C C C}; P ↦ ℍ(P, 0) = 𝕄(P, 0, 0). Then the canonical homomorphisms below are isomorphisms:

Coker[ℍ : K_{0}(D D D) → KQ_{0}(A A A, Θ)_{C C C}] ⟶^{≅} Coker[𝕄 : K_{0}(P P P(A A A, Θ)_{D D D}) → KQ_{0}(A A A, Θ)_{C C C}]

⟶^{≅} W Q_{0}(A A A, Θ)_{D D D,C C C}.

Proof. The first isomorphism follows from Lemma 2.41 and the second from Lemma 2.35.

□

3. Invariants of forms and G-surgery obstruction groups

In geometry, not all nonsingular, doubly parametrized, positioning modules can arise. Precisely those modules which are trivial under a certain invariant will occur. This means that from a geometric point of view, we must study K-groups of nonsingular, doubly parametrized, positioning modules vanishing under a certain kind of invariant. This section describes first in a general context the notion of invariant that will be used and then sets up the K-theoretic framework needed to deal with forms of invariant zero. This is done abstractly in order to focus attention on essentials. Then we define the specific invariant that is required in the geometric context and classify the metabolic planes of invariant zero. The classification is crucial for proving our main surgery results. At the end of the section, we use the geometric invariant to define the G-equivariant surgery obstruction groups.

Definition 3.1. Let Q Q Q be a symmetric monoidal category. (We have in mind the symmetric monoidal category Q Q Q(A, (−, λ), Γ, G, A_{s}, Λ, Θ) of all nonsingular, doubly parametrized, positioning modules.) Let ((abelian groups)) denote the symmetric monoidal category of all abelian groups, under direct sum. An invariant ∇ on Q Q Q with values in a symmetric monoidal subcategory A A A of ((abelian groups)) consists of a functor

α : Q Q Q ⟶ A A A

(covariant or contravariant) of symmetric monoidal categories and for each object M ∈ Q Q Q, an element

∇(M) ∈ α(M)

such that the following holds: (3.1.1) If f : M → M′ is a morphisms in Q Q Q then α(f)(∇(M)) = ∇(M′) if α is covariant

and ∇(M) = α(f)(∇(M′)) if α is contravariant. (3.1.2) The natural identification α_{M⊕M′} : α(M ⊕ M′) → α(M) ⊕ α(M′) included in the

definition of α has the property that α_{M⊕M′}(∇(M ⊕ M′)) = (∇(M), ∇(M′)).

If ∇ is an invariant on Q Q Q, let

∇Q Q Q

denote the full subcategory of Q Q Q consisting of all objects M ∈ Q Q Q such that ∇(M) = 0. Call an invariant ∇ on Q Q Q a trivial invariant, if ∇(M) = 0 ∈ α(M) for all objects M ∈ Q Q Q, in which case ∇Q Q Q = Q Q Q.

Below, we shall be confronted with a family ∇ of invariants ∇_{i} where i ranges through an index set I. In this situation, one constructs in the obvious way an invariant

∇′ = ⨁_{i∈I}∇_{i} by ∇′(M) = (∇_{i}(M))_{i∈I} ∈ ⨁_{i∈I} α_{i}(M)

and defines

∇Q Q Q = ∇′Q Q Q.

Lemma 3.2. Let Q Q Q be a symmetric monoidal category and ∇ invariant on Q Q Q. Then ∇Q Q Q is closed under isomorphism classes, sums and summands. In particular, ∇Q Q Q is a symmetric monoidal subcategory of Q Q Q.

Proof. The assertions follow immediately from the definition of an invariant. □

Lemma 3.3. Let Q Q Q denote the symmetric monoidal category Q Q Q(A, (−, λ), Γ, G, A_{s}, Λ, Θ). Let M M M denote the full subcategory of Q Q Q generated under orthogonal sum by all metabolic planes. Let ∇ be an invariant on Q Q Q. Suppose that for each object (M, B, q, θ) ∈ Q Q Q, ∇(M, −B, −q, θ) = −∇(M, B, q, θ) and ∇(ℍ(A, 0)) = 0. Then ∇M M M is cofinal in ∇Q Q Q. (In practice, this means that ∇Q Q Q has a good class of cofinal objects.)

Proof. Given an object (M, B, q, θ) ∈ ∇Q Q Q, we must show there is an object (M′, B′, q′, θ′) ∈ ∇Q Q Q such that

(M, B, q, θ) ⊕ (M′, B′, q′, θ′) ∈ ∇M M M.

Suppose (M, B, q, θ) ∈ ∇Q Q Q. By assumption, ∇(M, −B, −q, θ) = −∇(M, B, q, θ) = 0. Thus, by Lemma 3.2,

∇((M, B, q, θ) ⊕ (M, −B, −q, θ)) = 0.

By Lemma 2.37, (M, B, q, θ)⊕(M, −B, −q, θ) has a Lagrangian and hence, by Lemma 2.35, is isomorphic to a metabolic module. The proof of Corollary 2.38 shows there is a hyperbolic module ℍ(P, 0) and metabolic planes M M M_{1}, …, M M M_{r} such that

(M, B, q, θ) ⊕ (M, −B, −q, θ) ⊕ ℍ(P, 0) ≅ M M M_{1} ⊕ ⋯ ⊕ M M M_{r}.

By assumption and Lemma 3.2, ∇(ℍ(P, 0)) = 0 and

∇((M, B, q, θ) ⊕ (M, −B, −q, θ) ⊕ ℍ(P, 0)) = 0.

Thus, again by Lemma 3.2, ∇(M M M_{i}) = 0 for all 1 ≦ i ≦ r. □

Lemma 3.4. Let Q Q Q denote the symmetric monoidal category Q Q Q(A, (−, λ), Γ, G, A_{s}, Λ, Θ) of nonsingular, doubly parametrized, positioning modules over the parameter algebra (A, (−, λ), Γ, G, A_{s}, Λ, Θ) with positioning data. Let ∇ be an invariant on Q Q Q. If ∇ vanishes on the metabolic module 𝕄(P, b_{0}, q_{0}, θ) and on the hyperbolic module ℍ(P, 0) then ∇ vanishes on 𝕄(P, −b_{0}, −q_{0}, −θ).

Proof. By Lemma 2.41, there is an isomorphism

𝕄(P, b_{0}, q_{0}, θ) ⊕ ℍ(P, 0) ⟶^{≅} 𝕄(P, b_{0}, q_{0}, θ) ⊕ 𝕄(P, −b_{0}, −q_{0}, −θ).

Since ∇ vanishes on 𝕄(P, b_{0}, q_{0}, θ) and ℍ(P, 0), it follows from Lemma 3.2 that ∇ vanishes on 𝕄(P, −b_{0}, −q_{0}, −θ). □

Let A be an algebra with antistructure (−, λ) and let D D D ⫅ C C C be as in Definition 2.43. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data.

Let ∇ be an invariant on Q Q Q(A A A, Θ). Define

∇Q Q Q(A A A, Θ)_{C C C}

to be the full, symmetric monoidal subcategory of ∇Q Q Q(A A A, Θ) such that the underlying A-module of each object lies in C C C, i.e. ∇Q Q Q(A A A, Θ)_{C C C} = ∇(Q Q Q(A A A, Θ)_{C C C}).

Definition 3.5. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let ∇ be an invariant on Q Q Q(A A A, Θ) such that ∇(ℍ(A, 0)) = 0. Define

∇KQ_{0}(A A A, Θ)_{C C C} = K_{0}(∇Q Q Q(A A A, Θ)_{C C C})
∇W Q_{0}(A A A, Θ)_{D D D,C C C} = ∇KQ_{0}(A A A, Θ)_{C C C}/⟨D D D-null modules in ∇Q Q Q(A A A, Θ)⟩.

Define

∇W Q_{0}(A A A, Θ)_{C C C} = ∇W Q_{0}(A A A, Θ)_{C C C,C C C}

and

∇KQ_{0}(A A A, Θ) = ∇KQ_{0}(A A A, Θ)_{P P P (A)}
∇W Q_{0}(A A A, Θ) = ∇W Q_{0}(A A A, Θ)_{P P P (A)}.

Proposition 3.6. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ) be a parameter algebra with positioning data. Let ∇ be an invariant on Q Q Q(A A A, Θ) such that ∇(ℍ(A, 0)) = 0. Then ∇ vanishes on all D D D-null modules in Q Q Q(A A A, Θ) and the canonical homomorphism

Coker[ℍ : K_{0}(D D D) → ∇KQ_{0}(A A A, Θ)_{C C C}] ⟶^{≅} ∇W Q_{0}(A A A, Θ)_{D D D,C C C}

is an isomorphism.

Proof. The assertions follow directly from Lemmas 2.41, 3.2 and 3.4. □

In order to define surgery obstruction groups, we shall need a refinement of the situation above. For this, we adopt the following notation and assumption.

Assumption 3.7. We shall assume that (A A A, Θ̃) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ̃) is a parameter algebra with positioning data such that Θ̃ has a free action of the multiplicative group {±1} of two elements, which commutes with the action of G. (Hence Θ̃ is regarded as a G × {±1}-set.) Let

(A A A_{2}, Θ_{2}) = (A_{2}, (−λ), Γ_{2}, G_{2}, A_{2,s}, Λ_{2}, Θ_{2})

be a parameter algebra with positioning data where A_{2} = A/2A, (−, λ) denotes the antistructure on A_{2} induced by that on A, Γ_{2} = Image[Γ → Λ_{2}], G_{2} = Image[G → A_{2}], A_{2,s} = Image[A_{s} → A_{2}], Λ_{2} = Image[Λ → A_{2}], and Θ_{2} is a G_{2}-set. We shall assume that the pair (Θ̃, Θ_{2}) is equipped with an equivariant map

p : Θ̃ → Θ_{2},

namely p((g, ϵ)x) = [g]p(x) for g ∈ G, ϵ ∈ {±1} and x ∈ Θ̃, where [g] ∈ A_{2} is the image of g. (The action of {±1} on Θ̃ and Θ_{2} is introduced in order to handle orientation considerations in the next section.) If M is an A-module, let M_{2} = M/2M. If (M, B, q) is a doubly parametrized module over (A, (−, λ), Γ, G, A_{s}, Λ), let (M_{2}, B_{2}, q_{2}) be the induced doubly parametrized module over (A_{2}, (−, λ), Γ_{2}, A_{2,s}, G_{2}, Λ_{2}).

Let (A A A, Θ̃) = (A, (−, λ), Γ, G, A_{s}, Λ, Θ̃) be a parameter algebra with positioning data. Let

Q̃ Q Q(A A A, Θ̃)

denote the full subcategory of Q Q Q(A A A, Θ̃) of all objects (M, B, q, θ) such that θ is {±}- equivariant, i.e. θ(ϵx) = ϵθ(x) for all ϵ ∈ {±1} and x ∈ Θ. Then Q̃ Q Q(A A A, Θ̃) is closed under orthogonal sums and orthogonal summands and in particular, is a symmetric monoidal subcategory of Q Q Q(A A A, Θ̃). Finally, if ∇ is an invariant on Q Q Q(A A A, Θ̃) then ∇ is defined on Q̃ Q Q(A A A, Θ̃) and so the symmetric monoidal subcategory

∇Q̃ Q Q(A A A, Θ̃)

of Q̃ Q Q(A A A, Θ̃) is defined by Definition 3.1 and closed under orthogonal summands by Lemma 3.2. If Θ is a G-set and Θ̃ = Θ × {±1} then

Q̃ Q Q(Ã A A, Θ̃) = Q Q Q(A A A, Θ).

Let A A A = (A, (−, λ), Γ, G, A_{s}, Λ) be a parameter algebra and

Θ Θ Θ = (Θ̃, p, Θ_{2}).

Let

Q Q Q(A A A,Θ Θ Θ)

denote the symmetric monoidal category whose objects are all quintuples (M, B, q, θ, θ_{2}) such that (M, B, q, θ) ∈ Q̃ Q Q(A A A, Θ̃), (M_{2}, B_{2}, q_{2}, θ_{2}) ∈ Q Q Q(A A A_{2}, Θ_{2}), and the diagram

\xymatrix{Θ̃ \ar[r]^{θ} \ar[d]^{p} & M \ar[d] \\ Θ_{2} \ar[r]_{θ_{2}} & M_{2}}

commutes. A morphism α : (M, B, q, θ, θ_{2}) → (M′, B′, q′, θ′, θ′_{2}) is an A-linear map α : M → M′ which defines a morphism (M, B, q, θ) → (M′, B′, q′, θ′) in Q̃ Q Q(A A A, Θ̃) and makes the diagram

\xymatrix{Θ_{2} \ar[r]^{θ_{2}} \ar[dr]_{θ′_{2}} & M_{2} \ar[d]^{α_{2}} \\  & M′_{2}}

commute where α_{2} is the A_{2}-linear map M_{2} → M′_{2} induced by α. The orthogonal sum of objects is defined in the obvious way, namely

(M, B, q, θ, θ_{2}) ⊕ (M′, B′, q′, θ′, θ′_{2}) = (M ⊕ M′, B ⊕ B′, q ⊕ q′, θ ⊕ θ′, θ_{2} ⊕ θ′_{2}).

Furthermore, if ∇ is an invariant on Q Q Q(A A A,Θ Θ Θ) then the symmetric monoidal subcategory

∇Q Q Q(A A A,Θ Θ Θ)

of Q Q Q(A A A,Θ Θ Θ) is defined in Definition 3.1 and is closed under orthogonal summands by Lemma 3.2.

If the equivariant map p : Θ̃ → Θ_{2} is surjective, the forgetful functor

Q Q Q(A A A,Θ Θ Θ) → Q̃ Q Q(A A A, Θ̃)

is an isomorphism of symmetric monoidal categories.

Our next goal is to prove the analogs of Lemmas 3.3, 3.4 and Proposition 3.6, after replacing Q Q Q(A A A, Θ) by Q Q Q(A A A,Θ Θ Θ). This requires extending the language we have for the category Q Q Q(A A A, Θ) to the category Q Q Q(A A A,Θ Θ Θ).

Let D D D ⫅ C C C be the symmetric monoidal subcategories of P P P (A) appearing in Definition 2.43.

Definition 3.8. Let (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ). An A-submodule L ⫅ M is called totally isotropic if q and B are trivial on L (and so B_{2} and q_{2} are trivial on Image[L → M_{2}]). The positioning function θ (resp. θ_{2}) is called totally isotropic if B(θ(t), θ(t′)) = 0 and q(θ(t)) = 0 for all t, t′ ∈ Θ̃ (resp. B_{2}(θ_{2}(u), θ_{2}(u′)) = 0 and q(θ_{2}(u)) = 0 for all u, u′ ∈ Θ_{2}). A totally isotropic, A-direct summand L ⫅ M such that Image(θ) ⫅ L and Image(θ_{2}) ⫅ Image[L → M_{2}] is called a sublagrangian. If, in addition, L ∈ C C C then L is called a C C Csublagrangian. A sublagrangian (resp. C C C-sublagrangian) L such that L = L^{⊥} := {m ∈ M | B(m, ℓ) = 0 ∀ ℓ ∈ L} is called a Lagrangian (resp. C C C-Lagrangian). (M, B, q, θ, θ_{2}) is called a null module (resp. C C C-null module), if it contains a Lagrangian (resp. C C C-Lagrangian). (M, B, q, θ, θ_{2}) is called hyperbolic, if (M, B, q, θ) = ℍ(P, θ) and Image(θ_{2}) ⫅ P_{2}; in this case, we write (M, B, q, θ, θ_{2}) = ℍ(P, θ, θ_{2}). (M, B, q, θ, θ_{2}) is called metabolic, if (M, B, q, θ) = 𝕄(P, b_{0}, q_{0}, θ) and Image(θ_{2}) ⫅ P_{2}; in this case, we write (M, B, q, θ, θ_{2}) = 𝕄(P, b_{0}, q_{0}, θ, θ_{2}). Objects of the kind ℍ(A, θ, θ_{2}) (resp. 𝕄(A, b_{0}, q_{0}, θ, θ_{2})) are called hyperbolic (resp. metabolic) planes.

Definition 3.9. Let (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ). An ordered pair (e, f) of elements of M is called a metabolic pair if it is a metabolic pair in the sense of Definition 2.39 for (M, B, q, θ) and if B_{2}(θ_{2}(y), e) = 0 for all y ∈ Θ_{2}. If a metabolic pair (e, f) generates M as an A-module then it is called a metabolic basis for (M, B, q, θ, θ_{2}).

Proposition 3.10. The analog of anyone of Lemmas 2.35, 2.37, 2.40, 2.41 and Corollaries 2.36, 2.38 is valid when Q Q Q(A A A, Θ) is replaced by Q Q Q(A A A,Θ Θ Θ).

The proofs are analogous to those of the original results. There are no pitfalls. Details are left to the reader.

Lemma 3.11. Let M M M denote the full subcategory of Q Q Q(A A A,Θ Θ Θ) generated under orthogonal sum by all metabolic planes. Let ∇ be an invariant on Q Q Q(A A A,Θ Θ Θ). Suppose that for each object (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ), ∇(M, −B, −q, θ, θ_{2}) = −∇(M, B, q, θ, θ_{2}) and ∇(ℍ(A, 0, 0)) = 0. Then ∇M M M is cofinal in ∇Q Q Q(A A A,Θ Θ Θ).

Proof. Thanks to Proposition 3.10, the proof is the same as that of Lemma 3.3. □

Lemma 3.12. Let ∇ be an invariant on Q Q Q(A A A,Θ Θ Θ). If ∇ vanishes on the metabolic module 𝕄(P, b_{0}, q_{0}, θ, θ_{2}) and on the hyperbolic module ℍ(P, 0, 0) then ∇ vanishes also on 𝕄(P, −b_{0}, −q_{0}, −θ, −θ_{2}) and

𝕄(P, b_{0}, q_{0}, θ, θ_{2}) ⊕ ℍ(P, 0, 0) ≅ 𝕄(P, b_{0}, q_{0}, θ, θ_{2}) ⊕ 𝕄(P, −b_{0}, −q_{0}, −θ, −θ_{2})

under the isomorphism in Lemma 2.41.

Proof. Thanks to Proposition 3.10, the proof is the same as that of Lemma 3.4. □

Let ∇ be an invariant on Q Q Q(A A A,Θ Θ Θ). Let

∇Q Q Q(A A A,Θ Θ Θ)_{C C C}

denote the full, symmetric monoidal subcategory of ∇Q Q Q(A A A,Θ Θ Θ) such that the underlying A-module of each object lies in C C C. Let

Q Q Q(A A A,Θ Θ Θ)_{C C C} = TQ Q Q(A A A,Θ Θ Θ)_{C C C}

where T denotes a trivial invariant.

Definition 3.13. Let ∇ be an invariant on Q Q Q(A A A,Θ Θ Θ) such that ∇(ℍ(A, 0, 0)) = 0. Define

∇KQ_{0}(A A A,Θ Θ Θ)_{C C C} = K_{0}(∇Q Q Q(A A A,Θ Θ Θ)_{C C C})

∇W Q_{0}(A A A,Θ Θ Θ)_{D D D,C C C} = ∇KQ_{0}(A A A,Θ Θ Θ)_{C C C}/⟨D D D-null modules in ∇Q Q Q(A A A,Θ Θ Θ)⟩.

Define

∇W Q_{0}(A A A,Θ Θ Θ)_{C C C} = ∇W Q_{0}(A A A,Θ Θ Θ)_{C C C,C C C}

and

∇KQ_{0}(A A A,Θ Θ Θ) = ∇KQ_{0}(A A A,Θ Θ Θ)_{P P P (A)}
∇W Q_{0}(A A A,Θ Θ Θ) = ∇W Q_{0}(A A A,Θ Θ Θ)_{P P P (A)}.

The next proposition is a generalization of Proposition 3.6.

Proposition 3.14. Let ∇ be an invariant on Q Q Q(A A A,Θ Θ Θ) such that ∇(ℍ(A, 0, 0)) = 0 where P ∈ D D D, θ = 0, and θ_{2} = 0. Then ∇ vanishes on all D D D-null modules in Q Q Q(A A A,Θ Θ Θ) and the canonical homomorphism

Coker[ℍ : K_{0}(D D D) → ∇KQ_{0}(A A A,Θ Θ Θ)_{C C C}] ⟶ ∇W Q_{0}(A A A,Θ Θ Θ)_{D D D,C C C}

is an isomorphism.

Proof. The assertions follow directly from Lemmas 3.2 and 3.12. □

Corollary 3.15. Let ∇ be an invariant on Q Q Q(A A A,Θ Θ Θ) such that for any (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ), ∇(M, B, q, θ, θ_{2}) = −∇(M, −B, −q, θ, θ_{2}) and ∇(ℍ(A, 0, 0)) = 0. Let B B B denote a subcategory of P P P (A), which contains A, is closed under isomorphisms, direct sums, and dualization, and C C C ⫅ B B B. Then the canonical homomorphisms

∇KQ_{0}(A A A,Θ Θ Θ)_{D D D,C C C} ⟶ ∇KQ_{0}(A A A,Θ Θ Θ)_{D D D,B B B} and
∇W Q_{0}(A A A,Θ Θ Θ)_{D D D,C C C} ⟶ ∇W Q_{0}(A A A,Θ Θ Θ)_{D D D,B B B}

are injective.

Proof. The injectivity of the first homomorphism follows from Lemma 3.11 and of the second from Lemma 3.11 and Proposition 3.14. □

The rest of this section will be concerned with certain invariants which arise geometrically. A prototype for these invariants is given in the following lemma.

Lemma 3.16. Let (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, Λ, , Θ) be a parameter algebra with positioning data. Let Γ′ ⫅ Γ be a (−, λ)-symmetric parameter on A. Give Γ/Γ′ the Amodule structure defined by a[γ] = [aγ\overline{a}] for all a ∈ A and γ ∈ Γ. For each object (M, B, q, θ) ∈ Q Q Q(A A A, Θ), let

α_{Γ′}(M, B, q, θ) = Hom_{A}(M, Γ/Γ′) and
∇_{Γ′}(M, B, q, θ) : M ⟶ Γ/Γ′; m ⟼ [B(m, m)].

Then the pair (∇_{Γ′}, α_{Γ′}) defines an invariant on Q Q Q(A A A, Θ).

The proof is straightforward.

Geometric invariants are obtained by modifying the basic idea above and taking into account the positioning functions. To simplify details, we shall restrict to the case A is a group ring. We make for the rest of this section, the following assumption concerning our setup.

Assumption 3.17. Let G denote a group, w : G → {±1} a group homomorphism (which will be referred to as an orientation homomorphism), R a commutative ring with 1 (unity) and with involution r ↦ \overline{r}, and A the group ring R[G] with compatible involution a ↦ \overline{a} such that \overline{g} = w(g)g^{−1} for all g ∈ G. Let

ε : A → R; ∑_{g∈G} r_{g}g ↦ r_{1}

denote the projection to the coefficient of 1 ∈ G. Let

G(2) = {g ∈ G| g^{2} = 1, g ≠ 1}.

Let S denote a subset of G(2) ∪ {1} such that S is closed under conjugation by elements of G. Let

A_{s} = R[S] (:= {∑_{g∈S} r_{g}g | r_{g} ∈ R (r_{g} = 0 except for finitely many g’s)}) .

Let λ ∈ R such that λ\overline{λ} = 1, Λ a (−, λ)-form parameter on A, Γ a (−, λ)-symmetric parameter on A, Θ a finite G-set, and (A A A, Θ) = (A, (−, λ), Γ, G, A_{s}, A_{s} + Λ, Θ). Let

Γ′ ⫅ Γ

denote a (−, λ)-symmetric parameter on A and

Γ′_{g} = ε(Γ′g^{−1}) for g ∈ G.

Clearly, if g ∈ G(2) ∪ {1} then Γ′_{g} is a (−, λw(g))-symmetric parameter on R and if g ∉ G(2) ∪ {1} then Γ′_{g} = R. Let S(G) denote the G-set of all subgroups of G with G-action given by conjugation, i.e.

G × S(G) → S(G); (g, H) ↦ gHg^{−1}.

Let

ρ : Θ → S(G)

denote a G-equivariant map such that for each s ∈ S, the set

Θ|_{s} = {t ∈ Θ | ρ(t) ∋ s}

is finite.

Definition 3.18. If (M, B, q, θ) ∈ Q Q Q(A A A, Θ) and s ∈ S, let

Σ_{ρ,s}^{θ} = ∑_{x∈Θ|_{s}} θ(x)

α_{Γ′,s}(M, B, q, θ) = Hom_{ℤ}(M, R/Γ′_{s})

∇_{Γ′,s}(M, B, q, θ) : M → R/Γ′_{s}; m ⟼ w(s)[ε (B(sm, Σ_{ρ,s}^{θ} − m))]

∇_{Γ′} = ⨁_{s∈S} ∇_{Γ′,s} and α_{Γ′} = ⨁_{s∈S} α_{Γ′,s}.

Lemma 3.19. Each pair (∇_{Γ′,s}, α_{Γ′,s}) (s ∈ S) defines an invariant in the sense of Definition 3.1 on Q Q Q(A A A, Θ). Thus, the pair (∇′_{Γ}, α_{Γ′}) defines an invariant on Q Q Q(A A A, Θ). Moreover, if g ∈ G then Γ′_{s} = Γ′_{gsg^{−1}}, α_{Γ′,s} = α_{Γ′,gsg^{−1}}, and

∇_{Γ′,s}(M, B, q, θ)(m) = w(g) ∇_{Γ′,gsg^{−1}}(M, B, q, θ)(gm)

for any (M, B, q, θ) ∈ Q Q Q(A A A, Θ) and m ∈ M.

Proof. Fix (M, B, q, θ) ∈ Q Q Q(A A A, Θ) and set f_{s} = ∇_{Γ′,s}(M, B, q, θ). First, we show that f_{s} is an additive map, which immediately implies f_{s} ∈ α_{Γ′,s}(M, B, q, θ). It then follows routinely that ∇_{Γ′,s} is an invariant on Q Q Q(A A A, Θ) and hence, ∇_{Γ′} is an invariant on Q Q Q(A A A, Θ).

Let b = ε ∘ B. Let m, m′ ∈ M. Then

f_{s}(m + m′) − f_{s}(m) − f_{s}(m′)

= w(s)[b(s(m + m′), Σ_{ρ,s}^{θ} − (m + m′)) − b(sm, Σ_{ρ,s}^{θ} − m) − b(sm′, Σ_{ρ,s}^{θ} − m′)] = w(s)[b(s(m + m′), −(m + m′)) − b(sm, −m) − b(sm′, −m′)] (biadditivity of b) = w(s)[b(sm′, −m) + b(sm, −m′)] (biadditivity of b) = w(s)[b(sm′, −m) + λ\overline{b(−m′, sm)}] (λ-Hermitian property of b) = w(s)[b(sm′, −m) + λw(s)\overline{b(−sm′, m)}] = w(s)[b(sm′, −m) + λw(s)\overline{b(sm′, −m)}] = 0.

It is clear Γ′_{g} = Γ′_{gsg^{−1}}. Thus, α_{Γ′,s} = α_{Γ′,gsg^{−1}}. To complete the proof of the lemma, it suffices to show that for each m ∈ M, f_{gsg^{−1}}(gm) = w(g)f_{s}(m). But,

f_{gsg^{−1}}(gm) = w(gsg^{−1})[b((gsg^{−1})gm, Σ_{ρ,gsg^{−1}}^{θ} − gm)]
= w(s)[b(gsm, gΣ_{ρ,s}^{θ} − gm)] (because Σ_{ρ,gsg^{−1}}^{θ} = gΣ_{ρ,s}^{θ})
= w(g)w(s)[b(sm, Σ_{ρ,s}^{θ} − m)]
= w(g)f_{s}(m).
□

Lemma 3.20. The invariant ∇_{Γ′} on Q Q Q(A A A, Θ) is trivial on modules (M, B, q, θ) such that B is Γ′-Hermitian (i.e. B(m, m) ∈ Γ′ for all m ∈ M) and θ = 0. In particular, ∇_{Γ′} vanishes on all hyperbolic modules ℍ(P, θ) such that θ = 0.

If suffices to prove the assertions for the invariants ∇_{Γ′,s} (s ∈ S). This follows from a straightforward computation which is left to the reader.

Lemma 3.21. Let M(Θ, A)^{G} denote the set of all G-equivariant maps Θ → A. Let M(Θ, Θ)_{G} denote the set of all maps f : Θ → R such that for each x ∈ Θ, there exists a finite subset L ⫅ G satisfying f(gx) = 0 for all g ∈ G ∖ L. Then the maps

tr : M(Θ, A)^{G} → M(Θ, R)_{G}; tr(θ) = ε ∘ θ

sr : M(Θ, R)_{G} → M(Θ, A)^{G}; sr(c)(x) = ∑_{g∈G} c(g^{−1}x)g (x ∈ Θ)

are mutually inverse isomorphisms.

Let M(Θ̃, A)^{G×{±1}} denote the set of all G × {±1}-equivariant maps Θ̃ → A. Let M(Θ, R)_{G}^{{±1}} denote the set of all {±1}-equivariant maps f : Θ̃ → R such that for each x ∈ Θ̃, there exists a finite subset L ⫅ G satisfying f(gx) = 0 for all g ∈ G ∖ L. Then the maps

tr : M(Θ, A)^{G×{±1}} → M(Θ, R)_{G}^{{±1}}; tr(θ) = ε ∘ θ

sr : M(Θ, R)_{G}^{{±1}} → M(Θ, A)^{G×{±1}}; sr(c)(x) = ∑_{g∈G} c(g^{−1}x)g (x ∈ Θ)

are mutually inverse isomorphisms.

This is proven by a straightforward computation.

Lemma 3.22. Let 𝕄(A, b_{0}, q_{0}, θ) be a metabolic plane in Q Q Q(A A A, Θ). Let B = B_{A,b_{0}} denote the Hermitian form on 𝕄(A, b_{0}, q_{0}, θ). Then 𝕄(A, b_{0}, q_{0}, θ) has a metabolic basis (e, f) such that if B(f, f) = ∑_{g∈G} r_{g}g (r_{g} ∈ R) then r_{g} = 0 for all g ∉ S.

Proof. Let (e, f) be a metabolic basis. By Assumption 3.17 and (2.6.3), we know that ∑_{g∉S} r_{g}g = a + λ\overline{a} for some a ∈ A. Thus, by (2.40.2), (e, f − ae) is a metabolic basis with the desired property. □

Lemma 3.23. Suppose that the involution on R is trivial and that for each r ∈ R, r^{2} = r, e.g. R = ℤ/2ℤ. Let 𝕄(A, b_{0}, q_{0}, θ) be a metabolic plane in Q Q Q(A A A, Θ) with metabolic basis (e, f). Let B = B_{A,b_{0}} denote the Hermitian form on 𝕄(A, b_{0}, q_{0}, θ). Let

B(f, f) = ∑_{g∈G} r_{g}g

(r_{g} ∈ R). Then ∇_{Γ′}(𝕄(A, b_{0}, q_{0}, θ)) = 0 if and only if for each s ∈ S,

∑_{x∈Θ|_{s}} tr(θ)(x) ≡ r_{s} mod Γ′_{s}.

If additionally Γ′ = min_{−λ}(A) then there is a metabolic basis (e, f) such that r_{g} = 0 for all g ∉ S and

r_{s} = ∑_{x∈Θ|_{s}} tr(θ)(x)

for all s ∈ S, where min_{−λ}(A) = \{a + λ\overline{a} | a ∈ A\}.

Proof. Let ∇ = ∇_{Γ′}(𝕄(A, b_{0}, q_{0}, θ)) and ∇_{s} = ∇_{Γ′,s}(𝕄(A, b_{0}, q_{0}, θ)) for each s ∈ S. By definition, ∇ = 0 ⟺ for each s ∈ S, ∇_{s} = 0. Since ∇_{s} : Ae ⊕ Af ⟶ R/Γ′_{s} is an additive function by Lemma 3.19 and ∇_{s} vanishes on Ae, it follows that ∇_{s} = 0 ⟺ for each γ ∈ R and g ∈ G, ∇_{s}(γgf) = 0. But, by Lemma 3.19, ∇_{s}(γgf) = w(g)∇_{g^{−1}sg}(γf). Thus, ∇ = 0 ⟺ for each γ ∈ R and s ∈ S, ∇_{s}(γf) = 0. But

∇_{s}(γf) = w(s)[ε (B(sγf, Σ_{ρ,s}^{θ} − γf))]

= w(s)[ε (B(sγf, Σ_{ρ,s}^{θ}) − B(sγf, γf))] (using bilinearity of B)

= w(s) [ε (B(sγf, ∑_{x∈Θ|_{s}}θ(x)) − B(sγf, γf))]

= w(s) [ε (B(sγf, ∑_{x∈Θ|_{s}} ∑_{g∈G} tr(θ)(g^{−1}x)ge) − γB(f, f)\overline{γ}\overline{s})]

= w(s) [ε ( ∑_{x∈Θ|_{s}} tr(θ)(s^{−1}x)s\overline{sγ}) − γr_{s}\overline{γ}s\overline{s}]

= w(s) [ε ( ∑_{x∈Θ|_{s}} tr(θ)(s^{−1}x)\overline{γ}w(s)) − γ\overline{γ}r_{s}w(s)]

= [γ ( ∑_{x∈Θ|_{s}} tr(θ)(s^{−1}x) − r_{s})] .

The first assertion in the lemma follows.

Let (e, f) be a metabolic basis for 𝕄(A, b_{0}, q_{0}, θ). By Lemma 3.21, we can assume r_{g} = 0 for all g ∉ S. Since Γ′ = min_{−λ}(A), we can find an element a = ∑_{s∈S} r′_{s}s ∈ A such that for each s ∈ S,

∑_{x∈Θ|_{s}} tr(θ)(x) − r_{s} = a + λ\overline{a}.

But, then by (2.40.2) the metabolic basis (e, f + ae) has the desired properties. □

We extend our list of assumptions under Assumptions 3.7 and 3.17 concerning the setup to include the following

Assumption 3.24. Let R_{2} = R/2R and A_{2} = A/2A. Let ε_{2} : A_{2} → R_{2}; ∑_{g∈G} r_{g}g ↦ r_{1} denote the projection to the coefficient of 1 ∈ G. Let Θ Θ Θ = (Θ̃, p, Θ_{2}). Let ρ : Θ̃ → S(G) and ρ_{2} : Θ_{2} → S(G) denote G-equivariant functions such that ρ((−1)x) = ρ(x) for all x ∈ Θ̃ and the diagram

\xymatrix{Θ̃ \ar[d]_{p} \ar[r]^{ρ} & S(G) \\ Θ_{2} \ar@{-->}[ur]_{ρ_{2}}}

commutes. We shall assume that Θ_{2}|_{s} = {x ∈ Θ_{2} | ρ(x) ∋ s} is a finite set for every s ∈ S. Let Γ′_{2} denote a (−, λ)-symmetric parameter on A_{2} such that Γ′_{2} ⫅ Image[Γ → A_{2}] and Γ′_{2,g} = ε_{2}(Γ′_{2}g^{−1}), for g ∈ G. If (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ) and g ∈ G, let

Σ_{ρ_{2},g}^{θ_{2}} = ∑_{x∈Θ_{2}|_{g}} θ_{2}(x).

If (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ) and s ∈ S, let α_{Γ′_{2,s}}(M, B, q, θ, θ_{2}) = Hom_{ℤ} (M_{2}, R_{2}/Γ′_{2,s}) and

∇_{Γ′_{2,s}}(M, B, q, θ, θ_{2}) : M_{2} ⟶ R_{2}/Γ′_{2,s}; m ⟼ [ε_{2} (B_{2}(sm, Σ_{ρ_{2},s}^{θ_{2}} − m)]) .

(We remark that w(s) = 1 in R_{2}.) Set ∇_{Γ′_{2}} = ⨁_{s∈S}∇_{Γ′_{2,s}} and α_{Γ′_{2}} = ⨁_{s∈S}α_{Γ′_{2,s}}.

Lemma 3.25. Each pair (∇_{Γ_{2,s}}, α_{Γ_{2,s}}) (s ∈ S) defines an invariant in the sense of Definition 3.1 on Q Q Q(A A A_{2}, Θ_{2}). Thus, the pair (∇_{Γ′_{2}}, α_{Γ′_{2}}) defines an invariant on Q Q Q(A A A,Θ Θ Θ). Moreover, if g ∈ G then Γ′_{2,s} = Γ′_{2,gsg^{−1}}, α_{Γ′_{2,s}} = α_{Γ′_{2,gsg^{−1}}}, and for each m ∈ M_{2} (where (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ)), ∇_{Γ′_{2,s}}(M, B, q, θ, θ_{2})(m) = w(g)∇_{Γ′_{2},gsg^{−1}} (M, B, q, θ, θ_{2})(gm).

Proof. By Lemma 3.19, the constructions ∇_{Γ′_{2,s}} and α_{Γ′_{2,s}} define an invariant on

Q Q Q(A_{2}, (−, λ), Γ_{2}, G, A_{2,s}, Λ_{2}, Θ_{2}).

These constructions composed with the forgetful functor

Q Q Q(A A A,Θ Θ Θ) ⟶ Q Q Q(A_{2}, (−, λ), Γ_{2}, G, A_{2,s}, Λ_{2}, Θ_{2})

yield the constructions in the lemma. The assertions in the lemma follow now from Lemma 3.19. □

Lemma 3.26. The invariant ∇_{Γ′_{2}} on Q Q Q(A A A,Θ Θ Θ) is trivial on modules (M, B, q, θ, θ_{2}) such that B_{2} is Γ′_{2}-Hermitian (i.e. B_{2}(m, m) ∈ Γ′_{2} for all m ∈ M_{2}) and θ_{2} = 0. In particular, ∇_{Γ′_{2}} vanishes on all hyperbolic modules ℍ(P, θ, θ_{2}) such that θ_{2} = 0. Furthermore, for any module (M, B, q, θ, θ_{2}), ∇_{Γ′_{2}}(M, B, q, θ, θ_{2}) = −∇_{Γ′_{2}}(M, −B, −q, θ, θ_{2}).

Proof. The first assertion follows from Lemma 3.20. The second and third ones are obvious. □

Lemma 3.27. Suppose the involution on R_{2} is trivial and for each r ∈ R_{2}, r^{2} = r, e.g.R_{2} = ℤ/2ℤ. Let 𝕄(A, b_{0}, q_{0}, θ, θ_{2}) be a metabolic plane in Q Q Q(A A A,Θ Θ Θ) with metabolic basis (e, f). Let B = B_{A,b_{0}} denote the Hermitian form on 𝕄(A, b_{0}, q_{0}, θ). Let

B(f, f) = ∑_{g∈G} r_{g}g (r_{g} ∈ R).

Then the following hold.

(3.27.1) ∇_{Γ′_{2}}(𝕄(A, b_{0}, q_{0}, θ, θ_{2})) = 0 if and only if for each s ∈ S,

r_{s} = ∑_{x∈Θ_{2}|_{s}} tr(θ_{2})(x) in R_{2}/Γ′_{2,s}.

(3.27.2) If additionally Γ′_{2} = min_{−λ}(A_{2}) := \{a + λ\overline{a}| a ∈ A_{2}\} then there is a metabolic basis (e, f) such that for each g ∉ S, r_{g} = 0, and for every s ∈ S,

r_{s} = ∑_{x∈Θ_{2}|_{s}} tr(θ_{2})(x) in R_{2}.

(3.27.3) If furthermore R is a complete set of representatives in R for the set R_{2} and the subset ∑_{s∈S}2Rs of A is included in min_{−λ}(A) then there is a metabolic basis (e, f) such that for each g ∉ S, r_{g} = 0, and for every s ∈ S,

r_{s} = { \atop ∑_{x∈Θ_{2}|_{s}} tr(θ_{2})(x)} in R

where r̃ ∈ R denotes the lifting of r ∈ R_{2}.

The proof is the same as that of Lemma 3.23, except for a few small changes in detail. We leave the checking to the reader.

Definition 3.28. Let Θ Θ Θ = (Θ̃, p, Θ_{2}), ρ : Θ̃ → S(G), and ρ_{2} : Θ_{2} → S(G) be as in Assumption 3.24. Let ccc = (c, c_{2}) be a pair consisting of {±1}-equivariant map c : Θ̃ → R and a map c_{2} : Θ_{2} → R_{2} such that

\xymatrix{Θ̃ \ar[r]^{c} \ar[d]_{p} & R \ar[d] \\ Θ_{2} \ar[r]_{c_{2}} & R_{2}}

commutes. Let c̃: Θ_{2} ⟶ R denote any map such that the diagram

\xymatrix{Θ_{2} \ar[r]^{c̃} \ar[dr]_{c_{2}} & R \ar[d] \\  & R_{2}}

commutes. Recall the definition of metabolic plane in the paragraph subsequent to Definition 2.32 and the definition of metabolic basis in Definition 2.39. Let 𝕄(R[G],ccc, c̃) denote the metabolic plane 𝕄(R[G], b_{0,c̃}, q_{0}, sr(c), sr(c_{2})) defined (up to isomorphism) by a metabolic basis (e, f) such that

q_{0} = 0,

b_{0,c̃}(f, f) = ∑_{s∈S} ∑_{x∈Θ_{2}|_{s}} c̃(x)s ∈ R[G],

sr(c)(x) = ∑_{g∈G} c(g^{−1}x)ge ∈ R[G]e (x ∈ Θ̃), and

sr(c_{2})(x) = ∑_{g∈G} c_{2}(g^{−1}x)ge ∈ R_{2}[G]e (x ∈ Θ_{2}).

If the involution on R_{2} is trivial and if r^{2} = r for any r ∈ R_{2}, then 𝕄(R[G],ccc, c̃) ∈ ∇_{Γ′_{2}}Q Q Q(A A A,Θ Θ Θ) by Lemma 3.27.

Lemma 3.29. Suppose the involution on R_{2} is trivial, r^{2} = r for any r ∈ R_{2}, ∑_{s∈S}2Rs ⫅

min_{−λ}(R[G]), and Γ′_{2} = min_{−λ}(R_{2}[G]). Let (ccc, c̃) and (ccc′, c̃′) be as in Definition 3.28. If ccc = ccc′ then 𝕄(R[G],ccc, c̃) ≅ 𝕄(R[G],ccc′, c̃′).

Proof. Let (e, f) and (e′, f′) be the metabolic bases defining 𝕄(R[G],ccc, c̃) and 𝕄(R[G], ccc′, c̃′), respectively, as in Definition 3.28. Let R denote a complete set of representatives in R for the set R_{2}. By (3.27.3) and the proof of Lemma 3.23, there exists a metabolic basis (e, f + ae) (resp. (e′, f′ + ae′)), where a (resp. a′) ∈ R[G], of 𝕄(R[G],ccc, c̃) (resp. 𝕄(R[G],ccc′, c̃′)) such that

r_{s} = { \atop ∑_{x∈Θ_{2}|_{s}} c_{2}(x)} (resp. r′_{s} = { \atop ∑_{x∈Θ_{2}|_{s}} c′_{2}(x)}) ∈ R,

where B(f + ae, f + ae) = ∑_{s∈S}r_{s}s and B(f′ + a′e′, f′ + a′e′) = ∑_{s∈S}r′_{s}s. Since c_{2} = c′_{2}, it

follows that r_{s} = r′_{s}. Thus 𝕄(R[G],ccc, c̃) ≅ 𝕄(R[G],ccc′, c̃′). □

Definition 3.30. Let R and Γ′_{2} be as in Lemma 3.29. Set 𝕄(R[G],ccc) = 𝕄(R[G],ccc, c̃) for some c̃. By Lemma 3.29, the isomorphism class of 𝕄(R[G],ccc) is uniquely defined.

Lemma 3.31. Let R and Γ′_{2} be as in Lemma 3.29. For any invertible element r ∈ R, 𝕄(R[G],ccc) ≅ 𝕄(R[G], rccc), where rccc = (rc, rc_{2}).

Proof. The maps sr(c_{2}) and sr(rc_{2}) are defined in Lemma 3.21. Let (e, f) be a metabolic basis defining 𝕄(R[G],ccc, c̃) as in Definition 3.28. Since r^{2} ≡ r mod 2R and r is invertible, r ≡ 1 mod 2R. Set e′ = r^{−1}e and f′ = rf. Then sr(c)(x)e = sr(rc)(x)e′, sr(c_{2})(x)e = sr(rc_{2})(x)e′ and B(f, f) ≡ B(f′, f′) mod min_{−λ}(R[G]). This proves Lemma 3.31. □

The next lemma classifies metabolic planes vanishing under ∇_{Γ′_{2}} and plays a key role in proving our G-surgery results.

Lemma 3.32. Let R and Γ′_{2} be as in Lemma 3.29. If 𝕄(R[G], b_{0}, q_{0}, θ, θ_{2}) ∈ ∇_{Γ′_{2}}Q Q Q(A A A,Θ Θ Θ) is a metabolic plane then

𝕄(R[G], b_{0}, q_{0}, θ, θ_{2}) ≅ 𝕄(R[G],ccc)

for some ccc = (c, c_{2}).

Proof. This follows immediately from Lemma 3.27. □

To close this section, we use the notation developed above to define G-surgery obstruction groups. They will play an important role in the rest of the paper.

Definition 3.33. We make even more precise now, the setup described under Assumptions 3.7, 3.17 and 3.24. Let

λ = (−1)^{k}

G(2)_{λ} = {g ∈ G(2) | w(g) = −λ}

Q = a subset of G(2)_{λ} ∪ {1} ∖ S, which is closed under conjugation

by elements of G Λ(Q) = the (−, (−1)^{k})-form parameter on A generated by Q Γ(S) = the (−, (−1)^{k})-symmetric parameter on A generated by S ∖ G(2)_{λ} Γ′_{2} = the minimum symmetric parameter min_{(−1)^{k}}(A_{2}) on A_{2} F F F(A) = the full, symmetric monoidal subcategory of P P P(A) consisting of all

finitely generated, free A-modules.

Define the G-equivariant surgery obstruction groups W_{2k}(R, G, Q, S,Θ Θ Θ)_{C C C} = ∇_{Γ′_{2}}W Q_{0}(A A A,Θ Θ Θ)_{F F F(A),C C C}

W_{2k}(R, G, Q, S,Θ Θ Θ)_{proj} = W_{2k}(R, G, Q, S,Θ Θ Θ)_{P P P(A)}
W_{2k}(R, G, Q, S,Θ Θ Θ)_{free} = W_{2k}(R, G, Q, S,Θ Θ Θ)_{F F F(A)}.

Lemma 3.34. Let B B B be a subcategory of P P P(A), which contains A, is closed under isomorphisms, direct sums, and dualization, and C C C ⫅ B B B. Then the canonical homomorphism

W_{2k}(R, G, Q, S,Θ Θ Θ)_{C C C} ⟶ W_{2k}(R, G, Q, S,Θ Θ Θ)_{B B B}

is injective.

Proof. The assertion follows immediately from Corollary 3.15 and Lemma 3.26. □

Remark 3.35. Let S S S(A) denote the full, symmetric monoidal subcategory of P P P(A) consisting of all finitely generated, stably free A-modules. Then the canonical maps

∇_{Γ′_{2}}W Q_{0}(A A A,Θ Θ Θ)_{F F F(A),C C C} ⟶ ∇_{Γ′_{2}}W Q_{0}(A A A,Θ Θ Θ)_{S S S(A),C C C}, and
W_{2k}(R, G, Q, S,Θ Θ Θ)_{F F F(A)} ⟶ W_{2k}(R, G, Q, S,Θ Θ Θ)_{S S S(A)}

are isomorphisms. Henceforth we shall identify groups related by a canonical isomorphism above.

4. Geometric intersection theory

For the rest of the paper, let G denote a finite group, k an integer ≧ 3, and n = 2k. Unless specifically mentioned otherwise, the term submanifold means smooth neat sub-

manifold (cf. [15]). Similarly the term embedding (resp. immersion) means smooth neat embedding (resp. immersion). Let Z be a compact, connected, simply connected, n-dimensional, oriented, smooth G-manifold such that

dim Z^{g} ≦ k for all g ∈ G ∖ {1} .

On the G-manifold Z, G-surgery of dimension k is the process of equivariantly replacing embedded G-handles (G × S^{k} × D^{k})_{α}, α ∈ A, by handles (G × D^{k+1} × S^{k−1})_{α} where the latter are obtained from the former by filling in {g} × S^{k} × D^{k}, g ∈ G, to get {g} × D^{k+1} × D^{k} and emptying out {g} × D^{k+1} × Interior(D^{k}) to get {g} × D^{k+1} × S^{k−1}. However, one can make this replacement simultaneously for all embedded G-handles only when all handles are mutually disjoint. To get the resulting space as a manifold, the G-handles (G × S^{k} × D^{k})_{α} must be disjoint from the singular set of Z:

\begin{aligned}Sing(G, Z) & = ⋃_{g∈G∖\{1\}}Z^{g} \\ & = ⋃_{H∈S(G)∖\{\{1\}\}} ( ⋃_{γ∈π_{0}(Z^{H})} Z_{γ}) , \end{aligned}

where Z_{γ} is the underlying space of the connected component γ. Let us suppose this is the case. Let h_{α} : S^{k} = {1} × S^{k} × {0} → Z, α ∈ A, and h_{γ} : Z_{γ} → Z, γ ∈ π_{0}(Z^{H}) with H ≠ {1}, denote the canonical inclusions.

By performing G-surgery on the G-handles above, we kill the elements h_{α}_{∗}[S^{k}] of H_{k}(Z; ℤ). However the problem we shall be facing starts not with the h_{α}’s above, but with a set of elements a_{α} in Image[π_{k}(Z) → H_{k}(Z; ℤ)]. We are asked to realize the a_{α}’s by embeddings h_{α} : S^{k} → Z which extend to G-handles (G×S^{k} ×D^{k})_{α} that are mutually disjoint from one another and from the singular set. On the other hand, we know only that each a_{α} can be realized by an immersion h_{α} (cf. [15]). Getting the conditions above satisfied will rest on establishing a G-equivariant geometric self-intersection form which is defined on all immersions. Establishing such a form and showing that it does the job are the main goals of this section.

In applications outside the current paper, we shall need that the G-handles above are disjoint not only from the singular set, but also from possibly larger G-subsets L. We describe these G-subsets next.

Assumption 4.1. Let L be a G-subcomplex of a G-equivariant smooth triangulation of Z satisfying the following. (4.1.1) L ⫆ Sing(G, Z).

(4.1.2) L = L^{(k−1)}∪⋃_{β∈B} B_{β} where L^{(k−1)} denotes the (k−1)-skeleton of L and (forgetting

the G-action on L) each B_{β} is a subcomplex of L. (4.1.3) For each β ∈ B, B_{β} is a compact connected k-dimensional submanifold of Z. (4.1.4) For each β ∈ B, L^{(k−1)} ∩ B_{β} ⫅ L^{(k−2)}. (4.1.5) For every β, β′ ∈ B, B_{β} ∩ B_{β′} is a submanifold of Z. (4.1.6) For every β, β′ ∈ B, if B_{β} ≠ B_{β′} as subsets of Z then B_{β} ∩ B_{β′} ⫅ L^{(k−2)}. (4.1.7) If B_{β}, β ∈ B, is orientable then B_{β} is oriented.

The assumption above obviously implies the next assumption on Z.

Assumption 4.2. If Z_{γ} ≠ Z_{γ′} for connected components γ ∈ π_{0}(Z^{H}) and γ′ ∈ π_{0}(Z^{H′}), where H, H′ are nontrivial subgroups of G, then dim(Z_{γ} ∩ Z_{γ′}) ≦ k − 2.

It is remarkable that if Z satisfies Assumption 4.2 then Assumption 4.1 is fulfilled by L = Sing(G, Z) where (forgetting orientations) the B_{β}’s range over all underlying spaces Z_{γ} such that γ ∈ π_{0}(Z^{H}) and dim Z^{H} = k. Let h_{β} : B_{β} → Z, β ∈ B, denote the canonical inclusion maps. Since L^{(k−1)} is not crucial for the general position argument for h_{α}, the problem we actually face is that of separating the h_{α}’s, α ∈ A, from one another and from the h_{β}’s, β ∈ B. In case each B_{β}, β ∈ B, is orientable, it is to be expected one can do this if and only if the geometric intersection numbers #(h_{α}, h_{α′}) and #(h_{α}, h_{β}) and the equivariant self-intersection numbers #_{g}(h_{α}) are zero for all α, α′ ∈ A, β ∈ B, and g ∈ G. However, this requires that the equivariant self-intersection numbers #_{g}(h_{α}) are defined. The main goal of this section is to define for certain g and arbitrary α, a replacement ♮_{g}(h_{α}) for #_{g}(h_{α}), which agrees with #_{g}(h_{α}) whenever the latter is defined, and to show (Theorems 4.19 and 4.21) that the vanishing of the geometric intersection numbers and the replacements ♮_{g} above provides a necessary and sufficient criteria for solving the problem above. Of course, this problem is not only interesting for immersions of k-dimensional spheres, but for any finite family of immersions of closed, connected, oriented, k-dimensional manifolds and so, we shall treat the problem above in this context.

The rest of the section is organized as follows. Looking ahead, we shall want in the next section to pack all of the information above into one module called the surgery obstruction module. In order to deal with this module, it is convenient to have an algebraic description of the geometric intersection form. We begin below by recalling the algebraic intersection form on H_{k}(Z; R) and establish some basic facts concerning it. Then we recall the geometric intersection form and compare it with the algebraic one. Next, selfintersection forms #_{g} are recalled and the self-intersection forms ♮_{g} are defined. Results relating the forms # and ♮_{g} are established. Then results concerning singular sets are proved. Finally, the main theorems are proved.

Let R be a commutative ring with unit element. For 0 ≦ ℓ ≦ n, let

P_{Z,∂Z} : H^{n−ℓ}(Z, ∂Z; R) → H_{ℓ}(Z; R)

and

P_{Z} : H^{n−ℓ}(Z; R) → H_{ℓ}(Z, ∂Z; R)

denote the Poincaré-Lefschetz duality isomorphisms (cf. Theorems 18 and 20 of [43, VI §2]). Let

\begin{matrix}∪ : H^{n−ℓ}(Z, ∂Z; R) × H^{ℓ}(Z; R) → H^{n}(Z, ∂Z,; R) and \\ ∪ : H^{n−ℓ}(Z, ∂Z; R) × H^{ℓ}(Z, ∂Z; R) → H^{n}(Z, ∂Z; R) \end{matrix}

denote the cup products [43, V §6]. If A ∈ H^{n}(Z, ∂Z; R) and a ∈ H_{n}(Z, ∂Z; R), let A(a) denote the evaluation of A on a. Let [Z, ∂Z] denote the orientation class of Z in H_{n}(Z, ∂Z; ℤ) and [Z, ∂Z]_{R} the image of [Z, ∂Z] in H_{n}(Z, ∂Z; R), under the canonical homomorphism. Define the mixed R-intersection pairing

\begin{matrix}Int_{Z,∂Z;R} : H_{k}(Z; R)× & H_{k}(Z, ∂Z; R) → R; \\ (a, b) ↦ (−1)^{k}(P_{Z,∂Z;R}^{−1}(a) ∪ P_{Z;R}^{−1}(b))([Z, ∂Z]_{R}) \end{matrix}

and the R-intersection pairing

\begin{matrix}Int_{Z;R} : H_{k}(Z; R)× & H_{k}(Z; R) → R; \\ (a, b) ↦ (−1)^{k}(P_{Z,∂Z;R}^{−1}(a) ∪ P_{Z,∂Z;R}^{−1}(b))([Z, ∂Z]_{R}). \end{matrix}

Of course, if ∂Z = ∅ then by definition Int_{Z,∂Z;R} = Int_{Z,R}.

The action of G on Z induces a G-module structure on the homology groups H_{ℓ}(Z; R) and H_{ℓ}(Z, ∂Z; R). Namely if g ∈ G and x ∈ H_{ℓ}(Z; R) or H_{ℓ}(Z, ∂Z; R) then gx = ℓ_{g}_{∗}(x), where ℓ_{g} denotes the left translation map Z → Z; z ↦ gz. Let R[G] denote the group ring of G with coefficients in R. Define the mixed G-equivariant R-intersection pairing

Int_{G,Z,∂Z;R} : H_{k}(Z; R) × H_{k}(Z, ∂Z; R) → R[G]

by

Int_{G,Z,∂Z;R}(a, b) = ∑_{g∈G} Int_{Z,∂Z;R}(a, g^{−1}b)g,

and the G-equivariant R-intersection pairing

Int_{G,Z;R} : H_{k}(Z; R) × H_{k}(Z; R) → R[G]

by

Int_{G,Z;R}(a, b) = ∑_{g∈G} Int_{Z;R}(a, g^{−1}b)g.

Lemma 4.3. The diagram

\xymatrix{H_{k}(Z; R) × H_{k}(Z; R) \ar[r]^{^{Int^{G,Z;R}}} \ar[d] & R[G] \\ H_{k}(Z, R) × H_{k}(Z, ∂Z; R) \ar[ur]_{_{Int_{G,Z,∂Z;R}}}}

commutes.

Proof. The case of an arbitrary finite group G follows from the case G = {1}. The case G = {1} follows from the fact that the cup product diagram corresponding to the diagram in the lemma commutes. □

Define the orientation homomorphism w_{Z} : G ⟶ {±1} by

w_{Z}(g) = \begin{cases}1, & if g preserves the orientation of Z \\ −1, & if g reverses the orientation of Z. \end{cases}

Thus,

g[Z, ∂Z] = w_{Z}(g)[Z, ∂Z].

When working in the context of Z, we shall give the group ring R[G] the involution a ↦ \overline{a} such that for all r ∈ R and g ∈ G,

\overline{rg} = rw_{Z}(g)g^{−1}

and let λ = (−1)^{k}. The G-actions on the cohomology groups H^{ℓ}(Z; R) and H^{ℓ}(Z, ∂Z; R) are afforded by gx = w_{Z}(g)ℓ_{g^{−1}}^{∗}(x) for all g ∈ G and x ∈ H^{ℓ}(Z; R) or H^{ℓ}(Z, ∂Z; R).

If f : (A, B) → (A′, B′) is a continuous map of pairs of topological spaces (B ⫅ A, B′ ⫅ A′ and f(B) ⫅ B′) then for each integer ℓ, define

K_{ℓ}(A, B; R) = Ker[f_{∗} : H_{ℓ}(A, B; R) ⟶ H_{ℓ}(A′, B′; R)]

K_{ℓ}(A; R) = Ker[f_{∗} : H_{ℓ}(A; R) ⟶ H_{ℓ}(A′; R)]
K^{ℓ}(A, B; R) = Coker[f^{∗} : H^{ℓ}(A′, B′; R) ⟶ H^{ℓ}(A, B; R)]
K^{ℓ}(A; R) = Coker[f^{∗} : H^{ℓ}(A′; R) ⟶ H^{ℓ}(A; R)].

Let f : (X, ∂X) → (Y, ∂Y) be a continuous map of compact, connected, n-dimensional manifolds. Define

f̂_{∗} : H_{ℓ}(Y; R) ⟶ H_{ℓ}(X; R)

to be the composite of the solid arrows in the diagram

\xymatrix{H_{ℓ}(Y; R) \ar@{.>}[r]^{f̂_{∗}} \ar[d]_{P_{Y,∂Y}^{−1}} & H_{ℓ}(X; R) \\ H^{n−ℓ}(Y, ∂Y; R) \ar[r]_{f^{∗}} & H^{n−ℓ}(X, ∂X; R) \ar[u]_{P_{X,∂X}}}

and define

f̂_{∗} : H_{ℓ}(Y, ∂Y; R) ⟶ H_{ℓ}(X, ∂X; R)

to be the composite of the solid arrows in the diagram

\xymatrix{H_{ℓ}(Y, ∂Y; R) \ar@{.>}[r]^{f̂_{∗}} \ar[d]_{P_{Y}^{−1}} & H_{ℓ}(X, ∂X; R) \\ H^{n−ℓ}(Y; R) \ar[r]_{f^{∗}} & H^{n−ℓ}(X; R). \ar[u]_{P_{X}}}

Recall that f is called a degree-one map, if X and Y are oriented and f_{∗}[X, ∂X] = [Y, ∂Y] in H_{n}(Y, ∂Y; ℤ). In the remainder of this section, let f : (X, ∂X) → (Y, ∂Y) be a degree-one G-map of compact, connected, n-dimensional, oriented, smooth G-manifolds X and Y. Since f^{∗} and the Poincaré-Lefschetz duality maps in the diagrams above are R[G]-homomorphisms, so is f̂_{∗}.

Lemma 4.4. The following holds: (4.4.1) The homomorphism f̂_{∗} is a G-equivariant splitting for f_{∗}. (4.4.2) The sequence of R[G]-modules

\xymatrix{⋯ \ar[r]^{∂} & K_{ℓ}(∂X; R) \ar[r] & K_{ℓ}(X; R) \ar[r] & K_{ℓ}(X, ∂X; R) \ar[r]^{∂} & K_{ℓ−1}(∂X; R) \ar[r] & ⋯}

is exact. (4.4.3) If n = 2k then

Int_{G,X;R}(K_{k}(X; R), f̂_{∗}H_{k}(Y; R)) = 0, Int_{G,X;R}(f̂_{∗}H_{k}(Y; R), K_{k}(X; R)) = 0,

Int_{G,X,∂X;R}(K_{k}(X; R), f̂_{∗}H_{k}(Y, ∂Y; R)) = 0, Int_{G,X,∂X;R}(f̂_{∗}H_{k}(Y; R), K_{k}(X, ∂X; R)) = 0.

Proof. (4.4.1): Let a ∈ H_{ℓ}(Y; R). Then there exists an element b ∈ H^{n−ℓ}(Y, ∂Y; R) such that P_{Y,∂Y} (b) = a. Thus

f_{∗}f̂_{∗}(a) = f_{∗}(P_{X,∂X}(f^{∗}b)) = f_{∗}([X, ∂X]_{R} ∩ f^{∗}b) = [Y, ∂Y]_{R} ∩ b = P_{Y,∂Y} (b) = a.

Similarly f_{∗}f̂_{∗}(a′) = a′ for all a′ ∈ H_{ℓ}(X, ∂X; R).

(4.4.2): The assertion follows from the definition of K_{ℓ}(−; R) and (4.4.1).

(4.4.3): This is proved by straightforward computation, cf. [8, I.2.9]. □

Lemma 4.5. Suppose K_{k}(X; R) and K_{k}(X, ∂X; R) are free R-modules. Then the pairing Int_{X,∂X;R} : K_{k}(X; R) × K_{k}(X, ∂X; R) → R is nonsingular.

Let A denote a connected, closed, oriented, smooth manifold of dimension k, and B a connected, compact, smooth manifold of dimension k. We assume that B is oriented if B is orientable. In the following, immersions and embeddings will always mean smooth, neat immersions and embeddings (cf. [15]). Let h_{α} : A → Z and h_{β} : B → Z be immersions such that h_{β}|_{∂B} is an embedding. (Since A is closed, it follows from the definition of neat immersion that Image(h_{α}) ⫅ Interior(Z).)

Let ℤ_{β} = ℤ or ℤ_{2} (= ℤ/2ℤ) depending on whether B is oriented or nonorientable. Let

#(h_{α}, h_{β}) ∈ ℤ_{β}

denote the geometric intersection number of h_{α} and h_{β}. Since A is oriented, the normal bundle ν(h_{α}) of h_{α} is oriented such that a local frame of T (A) with positive orientation followed by a local frame of ν(h_{α}) with positive orientation becomes a local frame of h_{α}^{∗}T (Z) with positive orientation. If both h_{α} and h_{β} are embeddings and B is oriented then our number #(h_{α}, h_{β}) above is the same as the number Image(h_{α}) · Image(h_{β}) in Definition 6.1 of Milnor [21]. In this connection, we stress that our #(h_{α}, h_{β}) is the intersection number not “of Image(h_{α}) and Image(h_{β})” but “of Image(h_{β}) and Image(h_{α})” in the language of [21].

The geometric intersection form #( , ) has the property that #(h_{α}, h_{β}) = 0 ⟺ there is a regular homotopy h_{α} ∼ h′_{α} such that Image(h′_{α})∩Image(h_{β}) = ∅. (A regular homotopy, by definition, is one such that at each level the maps are immersions.) The result above is an easy corollary of Theorem 6.6 of [21].

Define the G-equivariant geometric intersection number #_{G}(h_{α}, h_{β}) by

#_{G}(h_{α}, h_{β}) = ∑_{g∈G} #(h_{α}, g^{−1}h_{β})g ∈ ℤ_{β}[G].

Lemma 4.6 (cf. [1, p.15, Theorem 1]). Let [A] denote the orientation class of A and [B, ∂B] the orientation class of B. Then

#_{G}(h_{α}, h_{β}) = Int_{G,Z,∂Z;ℤ_{β}}(h_{α}_{∗}[A], h_{β}_{∗}[B, ∂B])

(if B is closed then this reads

#_{G}(h_{α}, h_{β}) = Int_{G,Z;ℤ_{β}}((h_{α})_{∗}[A], (h_{β})_{∗}[B]).)

Proof. For G = {1}, the first statement is deduced from the definitions of the cup product, the Poincaré-Lefschetz duality isomorphism as derived from the dual cell decomposition, and the ordinary geometric intersection number. The result for arbitrary G follows directly from the above and the definitions of #_{G}( , ) and Int_{G,Z,∂Z;ℤ_{β}}( , ). □

We want to define next the generalized (or doubly parametrized) G-equivariant geometric self-intersection form. This is a new construction, since we are going to allow fixed point manifolds of dimension k. Our definition will reduce to the one of [25, §3], when the fixed point manifolds of nontrivial subgroups have dimension ≦ k − 1 and to [44, Part I, §5], when the fixed point manifolds of nontrivial subgroups are empty.

For an integer ℓ, let G(Z, ℓ) = {g ∈ G | Z^{g} has an ℓ-dimensional, connected component} G(2) = {g ∈ G | g ≠ 1, g^{2} = 1} G(> 2) = {g ∈ G | g^{2} ≠ 1}.

Lemma 4.7. If g ∈ G(Z, ℓ) then w_{Z}(g) = (−1)^{n−ℓ}.

Proof. Let L be an ℓ-dimensional, connected component of Z^{g}. Let p ∈ L be an interior point of L. Let T_{p}(L) (resp. T_{p}(Z)) denote the tangent space at p in L (resp. Z). Since G is finite and Z is compact, there is a G-invariant Riemannian metric on Z. Fix such a metric. Denote by ⟨g⟩ the subgroup generated by g. The exponential map Exp from a certain neighborhood of 0 in T_{p}(Z) to Z (with respect to the Riemannian metric) is automatically G-equivariant and a local diffeomorphism as long as its image lies in the interior of Z. Let B_{p}(Z) ⫅ T_{p}(Z) denote a tiny ⟨g⟩-invariant ball centered at 0 ∈ T_{p}(Z) such that Exp|_{B_{p}} : B_{p}(Z) → Z is a diffeomorphism of B_{p}(Z) onto a neighborhood U of p. Since Exp is G-equivariant, B_{p}(Z)^{g} is diffeomorphic to U^{g} and U^{g} ⫅ L. Clearly, ℓ = dim L = dim T_{p}(L) ≦ dim T_{p}(Z)^{g} = dim B_{p}(Z)^{g} = dim U^{g} ≦ dim L = ℓ. Thus, T_{p}(L) = T_{p}(Z)^{g}. Let ν_{p}(L) denote the normal space of T_{p}(L), i.e. the orthogonal complement to T_{p}(L) in T_{p}(Z) with respect to the Riemannian metric. Clearly ν_{p}(L) is ⟨g⟩-invariant. Since T_{p}(L) = T_{p}(Z)^{g}, the group ⟨g⟩ acts freely on ν_{p}(L) \ 0. Thus, the ⟨g⟩- module ν_{p}(L) decomposes as a product of nontrivial, 1-dimensional real representations of ⟨g⟩ and realifications of nontrivial, 1-dimensional complex representations of ⟨g⟩. In the first case, the determinant over ℝ of the ℝ-linear automorphism defined by g is (−1) and in the second case (+1). The determinant d of the ℝ-linear automorphism of ν_{p}(L) defined by g satisfies the relation d = (−1)^{n−ℓ}. But d = w_{Z}(g). □

Corollary 4.8. (Recall λ = (−1)^{k}.) If g ∈ G(Z, ℓ) ∩ G(2) then

g = \begin{cases}λw_{Z}(g)g^{−1} (= λ\overline{g}), & if ℓ ≡ k mod 2 \\ −λw_{Z}(g)g^{−1} (= −λ\overline{g}), & if ℓ ≢ k mod 2. \end{cases}

Proof. By Lemma 4.7, λ = w_{Z}(g) ⟺ ℓ ≡ k mod 2, and λ = −w_{Z}(g) ⟺ ℓ ≢ k mod 2. □

Corollary 4.9. If ℓ ≢ ℓ′ mod 2 then G(Z, ℓ) ∩ G(Z, ℓ′) = ∅.

Proof. Suppose g ∈ G(Z, ℓ) ∩ G(Z, ℓ′). By Lemma 4.7 (−1)^{n−ℓ} = w_{Z}(g) = (−1)^{n−ℓ′}, and hence n − ℓ ≡ n − ℓ′ mod 2. This contradicts the assumption ℓ ≢ ℓ′ mod 2. □

Recall the notion of self-intersection form introduced in [44, pp.45–46]. It is naturally generalized as follows. Let A be a connected, closed, k-dimensional, orientable, smooth manifold and h_{α} : A → Z an immersion. If H is a subgroup of G, let

Free(H, Z) = {x ∈ Z | H_{x} = {1}},

where H_{x} is the isotropy subgroup at x of the H-action on Z. Note that Free(H, Z) is 1-connected. Suppose Image(h_{α}) ⊂ Free(H, Z). Since H acts freely on Free(H, Z), Free(H, Z)/H is a manifold with the fundamental group ≅ H, and the canonical map π : Free(H, Z) → Free(H, Z)/H is a universal covering map. Using the composite mapping π ∘ h_{α} : A → Free(H, Z)/H, one defines the H-equivariant geometric self-intersection number μ_{H}(h_{α}) ∈ ℤ[H]/min_{λ}(ℤ[H]), exactly as in [44, pp.45–46], where min_{λ}(ℤ[H]) is the minimum λ-form parameter on ℤ[H]. Thus, setting

ℤ_{g} = \begin{cases}ℤ/(1 − λw_{Z}(g))ℤ, & if g ∈ G and g^{2} = 1 \\ ℤ, & if g ∈ G and g^{2} ≠ 1, \end{cases}

one can regard μ_{H}(h_{α}) as an |H|-tuple (μ_{H}(h_{α})_{g})_{g∈H} such that μ_{H}(h_{α})_{g} ∈ ℤ_{g} and μ_{H}(h_{α})_{g} = λw_{Z}(g)μ_{H}(h_{α})_{g^{−1}}. If H′ is another subgroup of G such that Free(H′, Z) is 1-connected and Image(h_{α}) ⊂ Free(H′, Z), and if g ∈ H ∩ H′, then one can check that μ_{H}(h_{α})_{g} = μ_{H′}(h_{α})_{g}. For any g ∈ H, we define the g-th geometric self-intersection number

#_{g}(h_{α}) = μ_{H}(h_{α})_{g} ∈ ℤ_{g}.

Thus, #_{g}(h_{α}) is defined only when Free(⟨g⟩, Z) is 1-connected and Image(h_{α}) ⊂ Free(⟨g⟩, Z). We shall partially overcome this restriction in Definition 4.11 below. The geometric selfintersection numbers #_{g}( ) (g ∈ G) have the property that #_{1}(h_{α}) = 0 ⟺ h_{α} is regularly homotopic to an embedding h′_{α}, and the property that #_{1}(h_{α}) = 0 and #_{g}(h_{α}) = 0 ⟺ h_{α} is regularly homotopic to an embedding h′_{α} such that Image(h′_{α})∩Image(gh′_{α}) = ∅ (cf. [1, p.17, Corollary 1]).

Let ν(h_{α}) denote the normal bundle of the immersion h_{α} : A → Z and χ(ν(h_{α})) the Euler number of ν(h_{α}). If the normal bundle ν(h_{α}) is trivial then χ(ν(h_{α})) = 0. There is a relation (cf. Theorem 4.11 (iii) of [44, p.5]) between the equivariant geometric intersection number and the equivariant geometric self-intersection number given by the equation

#_{H}(h_{α}, h_{α}) = { \atop μ_{H}(h_{α})} + λ\overline{{ \atop μ_{H}(h_{α})}} + χ(ν(h_{α})),

where { \atop μ_{H}(h_{α})} is a lifting of μ_{H}(h_{α}) to ℤ[H]. From this, one deduces easily the next lemma.

Lemma 4.10. Let h_{α} : A → Z be an immersion such that #_{g}(h_{α}) is defined. If g ∈ \{1\} ∪ G(2), let { \atop #_{g}(h_{α})} denote a lifting of #_{g}(h_{α}) to ℤ. Then

#(h_{α}, g^{−1}h_{α}) = \begin{cases}#_{g}(h_{α}), & if order(g) ≧ 3 \\ { \atop #_{g}(h_{α})} + λw_{Z}(g){ \atop #_{g}(h_{α})}, & if g ∈ G(2) \\ { \atop #_{g}(h_{α})} + λ{ \atop #_{g}(h_{α})} + χ(ν(h_{α})), & if g = 1. \end{cases}

In particular, χ(ν(h_{α})) = 0 whenever #(h_{α}, h_{α}) = 0 and #_{1}(h_{α}) = 0.

In the following, let

\begin{matrix}Q(G, Z) = G(Z, k − 1) ∩ G(2) \\ S(G, Z) = G(Z, k) ∩ G(2). \end{matrix}

Let Λ(Q(G, Z)) denote the λ-form parameter on ℤ[G] generated by Q(G, Z), namely

Λ(Q(G, Z)) = min_{λ}(ℤ[G]) + ℤ[Q(G, Z)].

Let

Z(k − 1, G(2)) = ⋃_{g∈G(2)} ( ⋃_{γ∈π_{0}(Z^{g},k−1)} Z_{γ})

where π_{0}(Z^{g}, k − 1) is the subset of π_{0}(Z^{g}) consisting of all (k − 1)-dimensional connected components, and Z_{γ} stands for the underlying space of γ. Set

Ẑ = Z \ Z(k − 1, G(2)).

Lemma 4.10 motivates making the following definition.

Definition 4.11. Let A be a connected, closed, k-dimensional, oriented, smooth manifold and h_{α} : A → Z (resp. h_{α} : A → Ẑ) an immersion. If g ∈ G ∖ (Q(G, Z) ∪ S(G, Z)) (resp. G ∖ S(G, Z)), define

♮_{g}(h_{α}) = \begin{cases}#(h_{α}, g^{−1}h_{α}), & if g ∈ G(> 2) \\ #_{g}(h′_{α}), & if g ∈ G(2) with dim Z^{g} ≦ k − 2 \\ ( & resp. g ∈ G(2) with dim Z^{g} ≦ k − 1 & ) \\ #_{1}(h_{α}), & if g = 1 \end{cases}

where h′_{α} : A → Free(⟨g⟩, Z) (resp. h′_{α} : A → Free(⟨g⟩, Ẑ)) is regularly homotopic in Z (resp. Ẑ) to h_{α}. If h′′_{α} : A → Free(⟨g⟩, Z), g ∈ G(2) with dim Z^{g} ≦ k − 2, (resp. h′′_{α} : A → Free(⟨g⟩, Ẑ), g ∈ G(2) with dim Z^{g} ≦ k − 1,) is regularly homotopic in Z (resp. Ẑ) to h_{α} then h′′_{α} is regularly homotopic in Free(⟨g⟩, Z) to h′_{α}, so that ♮_{g}(h_{α}) is well defined.

Theorem 4.12. Let h_{α} : A → Z (resp. h_{α} : A → Ẑ) be as in Definition 4.11 and g ∈ G ∖ (Q(G, Z) ∪ S(G, Z)) (resp. G ∖ S(G, Z)). Then ♮_{g}(h_{α}) is defined and ♮_{g}(h_{α}) = #_{g}(h_{α}) whenever #_{g}(h_{α}) is defined, i.e. Image(h_{α}) ⫅ Free(⟨g⟩, Z). Furthermore, if { \atop ♮_{g}(h_{α})} denotes a lifting of ♮_{g}(h_{α}) to ℤ in the case g ∈ {1} ∪ G(2) then

#(h_{α}, g^{−1}h_{α}) = \begin{cases}♮_{g}(h_{α}), & if g ∈ G(> 2) \\ { \atop ♮_{g}(h_{α})} + λw_{Z}(g){ \atop ♮_{g}(h_{α})}, & if g ∈ G(2) \ (Q(G, Z,) ∪ S(G, Z)) \\ ( & resp. g ∈ G(2) ∖ S(G, Z) & ) \\ { \atop ♮_{g}(h_{α})} + λ{ \atop ♮_{g}(h_{α})} + χ(ν(h_{α})), & if g = 1. \end{cases}

Proof. By definition, ♮_{g}(h_{α}) is always defined. The first equation for #(h_{α}, g^{−1}h_{α}) holds by definition and the others follow from the corresponding equations in Lemma 4.10. The assertion that ♮_{g}(h_{α}) = #_{g}(h_{α}) whenever the latter is defined follows now from the equations just proved and those in Lemma 4.10. □

Whereas the G-equivariant, self-intersection form #_{G}( ) is defined only on immersions h_{α} : A → Free(G, Z), the generalized G-equivariant self-intersection form ♮_{G}( ) (resp. ˆ♮_{G}( )) to be constructed below will be defined on all immersions h_{α} : A → Z (resp. h_{α} : A → Ẑ). To make this construction, we note first that each element g ∈ Q(G, Z) (resp. S(G, Z)) satisfies by Corollary 4.8 the equation g = −λ\overline{g} (resp. g = λ\overline{g}).

Definition 4.13. Let h_{α} : A → Z (resp. h_{α} : A → Ẑ) be as in Definition 4.11. Let G(> 2) = C ∪ C^{−1} be a disjoint decomposition. For each g ∈ {1} ∪ (G(2) ∖ (Q(G, Z) ∪ S(G, Z))) ∪ C (resp. \{1\} ∪ (G(2) \ ∪S(G, Z)) ∪ C), let { \atop ♮_{g}(h_{α})} be a lifting of ♮_{g}(h_{α}) to ℤ. Define the generalized G-equivariant self-intersection number

♮_{G}(h_{α}) = ∑_{g} { \atop ♮_{g}(h_{α})}g ∈ ℤ[G]/(Λ(Q(G, Z)) + ℤ[S(G, Z)]))

where g runs over {1} ∪ (G(2) ∖ (Q(G, Z) ∪ S(G, Z))) ∪ C (resp.

ˆ♮_{G}(h_{α}) = ∑_{g} { \atop ♮_{g}(h_{α})}g ∈ ℤ[G]/(min_{λ}(ℤ[G]) + ℤ[S(G, Z)])

where g runs over {1} ∪ (G(2) ∖ S(G, Z)) ∪ C).

The number ♮_{G}(h_{α}) (resp. ˆ♮_{G}(h_{α})) does not depend on the liftings { \atop ♮(h_{α})}, and not on the choice of C, because for g ∈ G(> 2),

♮_{g^{−1}}(h_{α}) = λw_{Z}(g)♮_{g}(h_{α})

and therefore, ♮_{g}(h_{α})g ≡ ♮_{g^{−1}}(h_{α})g^{−1} mod (min_{λ}(ℤ[G])). The number ♮_{G}(h_{α}) (resp. ˆ♮_{G}(h_{α})) is invariant under regular homotopy of h_{α} in Z (resp. Ẑ).

Theorem 4.14. Let h_{α} : A → Z (resp. h_{α} : A → Ẑ) be as in Definition 4.11 and let { \atop ♮_{G}(h_{α})} denote a lifting of ♮_{G}(h_{α}) (resp. ˆ♮_{G}(h_{α})) to ℤ[G]/ℤ[S(G, Z)]. Then

#_{G}(h_{α}, h_{α}) = { \atop ♮_{G}(h_{α})} + λ\overline{{ \atop ♮_{G}(h_{α})}} + χ(ν(h_{α})) in ℤ[G]/ℤ[S(G, Z)].

Proof. It follows from the definitions of #_{G} and ♮_{G} (resp. ˆ♮_{G}) that the assertion of the theorem is equivalent to the assertion of Theorem 4.12. □

Definition 4.15. Let h_{i} : A_{i} → Z (resp. h_{i} : A_{i} → Ẑ), i = 0, 1, be immersions as in Definition 4.11. Then the connected sum h_{0}#h_{1} of h_{0} and h_{1} is defined up to regular homotopy in Z (resp. Ẑ) as follows. Let D_{i} ⊂ A_{i}, i = 0, 1, be k-dimensional disks such that h_{i}|_{D_{i}} are injective. Let I denote the closed unit interval [0, 1], D^{k} a kdimensional disk, and identify the k-dimensional disks {i} × D^{k} ⊂ I × D^{k}, i = 0, 1, with the k-dimensional disks D_{i}, respectively, by orientation preserving diffeomorphisms. Let φ : I × D^{k} → Z (resp. φ : I × D^{k} → Ẑ) be an immersion such that φ|_{D_{i}} = h_{i}. Let A_{0}#A_{1} denote the usual connected sum of A_{0} and A_{1} formed by cutting out the interior of the disks D_{i} from A_{i}, and let T denote the tube in A_{0}#A_{1} connecting A_{0} and A_{1}. Thus one has a diffeomorphism ψ : I × S^{k−1} → T such that the maps ψ|_{{i}×S^{k−1}}, i = 0, 1, agree with the identifications made above. Let h = h_{0}#h_{1} : A_{0}#A_{1} → Z denote the immersion such that h|_{A_{i}\D_{i}} = h_{i}|_{A_{i}\D_{i}} and

h|_{T} (ψ(t, x)) = φ(t, x) ((t, x) ∈ I × S^{k−1}).

If h′_{i} : A_{i} → Z (resp. h′_{i} : A_{i} → Ẑ), i = 0, 1, are immersions which are regularly homotopic in Z (resp. Ẑ) to h_{i}, i = 0, 1, respectively, and if h′ : A_{0}#A_{1} → Z (resp. h′ : A_{0}#A_{1} → Ẑ ) is an immersion enjoying the same properties relative to h′_{0} and h′_{2} as h relative to h_{0} and h_{1}, then h′ is regularly homotopic in Z (resp. Ẑ) to h, because A_{0} and A_{1} are connected and Z (resp. Ẑ) is 1-connected. We define ♮_{G}(h_{0}#h_{1}) = ♮_{G}(h) (resp. ˆ♮_{G}(h_{0}#h_{1}) = ˆ♮_{G}(h)) and conclude that ♮_{G}(h_{0}#h_{1}) (resp. ˆ♮_{G}(h_{0}#h_{1})) is well defined and invariant under regular homotopy of h_{0} and h_{1} in Z (resp. Ẑ).

Theorem 4.16. Let h_{i} : A_{i} → Z (resp. h_{i} : A_{i} → Ẑ), i = 0, 1, be immersions as in Definition 4.11. Then

\begin{matrix}#_{G}(h_{0}, h_{1}) & = ♮_{G}(h_{0}#h_{1}) − ♮_{G}(h_{0}) − ♮_{G}(h_{1}) \\ in ℤ[G]/(Λ(Q(G, Z)) + ℤ[S(G, Z)]) \end{matrix}

(resp. #_{G}(h_{0}, h_{1}) = ˆ♮_{G}(h_{0}#h_{1}) − ˆ♮_{G}(h_{0}) − ˆ♮_{G}(h_{1})
in ℤ[G]/(min_{λ}(ℤ[G]) + ℤ[S(G, Z)])).

Proof. The proof is similar to Theorem 5.2 (iv) of [44]. □

In applications, it will be sometimes necessary to perform G-surgery relative to a Gsubset which is strictly larger than the singular set Sing(G, Z). This is of interest also in the case G = 1. In order to handle such situations, we introduced already in Assumption 4.1 a generalization L of the singular set. For the rest of this section, L denotes such a set. If U is a subset of Z, define

ρ_{G}(U) = ⋂_{x∈U} G_{x}.

If β ∈ B, set ρ_{G}(β) = ρ_{G}(B_{β}). For a subgroup H of G, define

U^{=H} = {x ∈ U | G_{x} = H}.

Proposition 4.17. For any β ∈ B,

B_{β}^{=ρ_{G}(β)}

is connected and open dense in B_{β}.

Proof. This follows from the property that any element x ∈ B_{β} such that G_{x} ≠ ρ_{G}(β) lies in L^{(k−2)}. □

Proposition 4.18. If dim Z_{γ} = k − 1 or k, where γ ∈ π_{0}(Z^{H}) and H ∈ S(G), then |ρ_{G}(Z_{γ}) ∩ G(2)| ≦ 1.

Proof. Let p ∈ Interior(Z_{γ}). Let T_{p}(Z_{γ}) (resp. T_{p}(Z)) denote the tangent space at p in γ (resp. Z). Then T_{p}(Z_{γ}) has an orthogonal complement ν(Z_{γ}) in T_{p}(Z) with respect to some ρ_{G}(Z_{γ})-invariant inner product. Note dim ν(Z_{γ}) = n − dim(Z_{γ}) ≧ 1. We claim ρ_{G}(Z_{γ}) acts freely on ν_{p}(Z_{γ}) \ {0}. Suppose this has been shown. Then the group ρ_{G}(Z_{γ}) injects into Aut_{ℝ[ρ_{G}(Z_{γ})]}(ν_{p}(Z_{γ})). Let f ∈ Aut_{ℝ[ρ_{G}(Z_{γ})]}(ν_{p}(Z_{γ})) denote an element such that f^{2} = id. Each eigenvalue of f must be 1 or −1. Moreover if f acts freely on ν_{p}(Z_{γ}) \ {0} then all eigenvalues of f are −1, hence f = −id. In other words, Aut_{ℝ[ρ_{G}(Z_{γ})]}(ν_{p}(Z_{γ})) has exactly one element which is of order 2 and acts freely on ν_{p}(Z_{γ}) \ {0}, namely −id. Thus, |ρ_{G}(Z_{γ})∩G(2)| ≦ 1. If dim Z_{γ} = k then it follows from Sing(G, Z) ≦ k that ρ_{G}(Z_{γ}) acts freely on elements ν_{p}(Z_{γ}) \ {0}. If dim γ = k − 1 then there are no β ∈ B such that B_{β} ⫌ Z_{γ}. Thus, ρ_{G}(Z_{γ}) acts freely on ν_{p}(Z_{γ}) \ {0}. □

Let h_{β} : B_{β} → Z, β ∈ B, denote the canonical inclusion map afforded by L in Assumption 4.1.

Theorem 4.19. Let {A_{α} | α ∈ A} be a finite set of connected, closed, k-dimensional, oriented, smooth manifolds and for each α ∈ A, let h_{α} : A_{α} → Ẑ be an immersion. Then (4.19.1) and (4.19.2) below are equivalent. (4.19.1) (i) #_{G}(h_{α}, h_{α′}) = 0 for all α ≠ α′ ∈ A.

(ii) #(h_{α}, h_{β}) = 0 for all (α, β) ∈ A × B.

(iii) #_{G}(h_{α}, h_{α}) = 0 and ˆ♮_{G}(h_{α}) = 0 for all α ∈ A. (4.19.2) There is a family of regular homotopies h_{α} ∼ h′_{α} in Ẑ where α ranges through A

such that the following holds:

(a) Image(h′_{α}) ∩ gImage(h′_{α′}) = ∅ for all α ≠ α′ ∈ A and all g ∈ G.

(b) Image(h′_{α}) ∩ B_{β} = ∅ for all (α, β) ∈ A × B.

(c) h′_{α} is an embedding such that χ(ν(h′_{α})) = 0 and Image(h′_{α}) ∩ gImage(h′_{α}) = ∅ for all α ∈ A and all g ∈ G ∖ {1}.

Proof. Since each h_{α}, α ∈ A, is regularly homotopic in Ẑ to an immersion h′′_{α} : A_{α} → Ẑ \ L^{(k−1)}, it suffices to prove the equivalence (4.19.1) ⟺ (4.19.2) when h_{α} is replaced by h′′_{α}.

It is routine to establish (i) ⟺ (a), (ii) ⟺ (b), (iii) ⇐ (c), and (i) ∪ (ii) ∪ (iii) ⇐ (a) ∪ (b) ∪ (c). To prove (i) ∪ (ii) ∪ (iii) ⇒ (a) ∪ (b) ∪ (c), it suffices to show (i) ∪ (ii) ∪ (iii) ⇒ (c). By (i) ∪ (ii), there is a family of regular homotopies h′′_{α} ∼ h_{α}^{(3)}, α ∈ A, such that the h_{α}^{(3)}’s satisfy (a) ∪ (b) when h′_{α} is replaced by h_{α}^{(3)}. In particular, each Image(h_{α}^{(3)}) ⫅ Ẑ ∖ L ⫅ Free(G, Z). Moreover, Ẑ \ L is 1-connected, because Z is 1-connected and dim Z − dim L ≧ 3. Hence, #_{g}(h′′_{α}) is defined for any (g, α) ∈ G × A. Furthermore, we can treat each index α by itself. For a fixed α, (c) is equivalent to the assertion that #_{G}(h_{α}^{(3)}, h_{α}^{(3)}) = 0 and #_{G}(h_{α}^{(3)}) = 0. It is clear that #_{G}(h_{α}^{(3)}, h_{α}^{(3)}) = #_{G}(h_{α}, h_{α}) = 0 by (iii), since #_{G}( , ) is invariant under homotopy. Thus it suffices to show #_{G}(h_{α}^{(3)}) = 0. By definition, #_{G}(h_{α}^{(3)}) = 0 ⟺ #_{g}(h_{α}^{(3)}) = 0 for all g ∈ G. If g ∈ G \ S(G, Z) then #_{g}(h_{α}^{(3)}) = ♮_{g}(h_{α}^{(3)}) = 0 by (iii). Suppose g ∈ S(G, Z). Then by Lemma 4.7, λw_{Z}(g) = 1. Thus #_{g}(h_{α}^{(3)}) ∈ ℤ_{g} = ℤ. Lemma 4.10 gives 2#_{g}(h_{α}^{(3)}) = #(h_{α}^{(3)}, g^{−1}h_{α}^{(3)}), and this is equal to 0 by (iii). Thus, #_{g}(h_{α}^{(3)}) = 0. □

Lemma 4.20. Let Z be as in Theorem 4.19 and let A be a closed, connected, k-dimensional, oriented, smooth manifold. If h : A → Ẑ is an immersion and

τ ∈ ⨁_{g∈Q(G,Z)} ℤ_{g}g

then there exists a regular homotopy h ∼ h′ in Z such that h′(A) ⫅ Ẑ and ˆ♮_{G}(h′) = ˆ♮_{G}(h) + τ.

We remark that ℤ_{g} = ℤ_{2} by Lemma 4.7.

Proof. For simplicity, we treat first the case when τ = g for some element g ∈ Q(G, Z). Let γ ∈ π_{0}(Z^{H}, k − 1) such that ρ_{G}(Z_{γ}) ∋ g. (It follows from Assumption 4.1 that Z_{γ} is a (k − 1)-dimensional connected component of Z^{g}.) Consider how ˆ♮_{G}(h) is altered by a regular homotopy h ∼ h′′ crossing γ once (i.e. the intersection number of Z_{γ} and the regular homotopy is equal to 1). The resulting immersion h′′ can be realized by a connected sum construction as follows. Fix a point p ∈ Z_{γ}^{=ρ_{G}(Z_{γ}^{H})} (hence G_{p} = ρ_{G}(Z_{γ})). Decompose the tangent space T_{p}(Z) in Z into a direct sum of the tangent space T_{p}(Z_{γ}) in Z_{γ} and the normal space ν_{p}(Z_{γ}) of Z_{γ} in Z. Clearly, dim ν_{p}(Z_{γ}) = k + 1. Take a tiny ball D^{k+1} in ν_{p}(Z_{γ}) centered at 0 so that the exponential map Exp maps D^{k+1} diffeomorphically into Z, Exp(D^{k+1}) ∩ Sing(G, Z) = {p}, and Image(h) ∩ G(Exp(D^{k+1})) = ∅. Set S^{k} = ∂D^{k+1} and h_{g} = Exp|_{S^{k}} : S^{k} → Free(G, Z). Then h′′ above may be realized as h#h_{g} (connected sum in Ẑ). Since Image(h_{g}) bounds the embedded disk Exp(D^{k+1}), h_{g} is null homotopic in Z. Moreover, Exp(D^{k+1}) ⊂ Z ∖ Z^{a} if a ∉ ρ_{G}(Z_{γ}). Recall that ρ_{G}(Z_{γ}) ∩ G(2) = {g}, by Proposition 4.18. Thus, for all a ∈ G(2) ∖ {g}, ♮_{a}(h_{g}) = #_{a}(h_{g}) = 0. It follows that ˆ♮_{G}(h_{g}) = #_{g}(h_{g})g. It is elementary to compute that #_{g}(h_{g}) = 1 ∈ ℤ_{2}. Since #_{G}(h, h_{g}) = 0, ˆ♮_{G}(h′′) = ˆ♮_{G}(h) + ˆ♮_{G}(h_{g}) = ˆ♮_{G}(h) + g by Theorem 4.16. We consider now the case of a general τ. Let g_{i}, i = 1, 2, ⋯, ℓ, denote all the elements such that g_{i} ∈ Q(G, Z) and the g_{i}-th coefficient of τ is nonzero. By induction on ℓ, one shows that there is a regular homotopy h ∼ h′ in Z such that ˆ♮_{G}(h′) = ˆ♮_{G}(h) + τ. □

Theorem 4.21. Let {A_{α} | α ∈ A} be a finite set of connected, closed, k-dimensional, oriented, smooth manifolds and for each α ∈ A, let h_{α} : A_{α} → Z be an immersion. Then (4.21.1) and (4.21.2) below are equivalent.

(4.21.1) (i) #_{G}(h_{α}, h_{α′}) = 0 for all α ≠ α′ ∈ A.

(ii) #(h_{α}, h_{β}) = 0 for all (α, β) ∈ A × B.

(iii) #_{G}(h_{α}, h_{α}) = 0 and ♮_{G}(h_{α}) = 0 for all α ∈ A. (4.21.2) There is a family of regular homotopies h_{α} ∼ h′_{α} in Z where α ranges through A

such that the following holds:

(a) Image(h′_{α}) ∩ gImage(h′_{α′}) = ∅ for all α ≠ α′ ∈ A and for all g ∈ G.

(b) Image(h′_{α}) ∩ B_{β} = ∅ for all (α, β) ∈ A × B.

(c) h′_{α} is an embedding such that χ(ν(h′_{α})) = 0 and Image(h′_{α}) ∩ gImage(h′_{α}) = ∅ for all α ∈ A and all g ≠ 1 ∈ G.

Proof. Since #_{G}( , ), #( , ) and ♮_{G}( ) are invariant under regular homotopy in Z, the assertion (4.21.2) ⟹ (4.21.1) is an obvious corollary of the assertion (4.19.2) ⟹ (4.19.1). We show that (4.21.1) ⟹ (4.21.2). It suffices to find regular homotopies h_{α} ∼ h′′_{α} such that h′′_{α} : A_{α} → Ẑ and (4.21.1) (i)–(iii) are satisfied when h_{α} is replaced by h′′_{α}. There are regular homotopies h_{α} ∼ h_{α}^{(3)}, A_{α} ∈ A, in Z such that h_{α}^{(3)} : A_{α} → Ẑ. Clearly (4.21.1) (i) and (ii) are valid after h_{α} is replaced by h_{α}^{(3)}. Furthermore #_{G}(h_{α}^{(3)}, h_{α}^{(3)}) = #_{G}(h_{α}, h_{α}) = 0 and ♮_{G}(h_{α}^{(3)}) = ♮_{G}(h_{α}) = 0 by assumption. It remains to find a regular homotopy h_{α}^{(3)} ∼ h′′_{α} in Z such that ˆ♮_{G}(h′′_{α}) = 0, for each α ∈ A. But, one can find such a regular homotopy by applying Lemma 4.20 with h = h_{α}^{(3)} and τ = −ˆ♮_{G}(h_{α}^{(3)}) = ˆ♮_{G}(h_{α}^{(3)}). □

5. G-surgery obstruction modules

Let X and Y denote compact, connected, oriented smooth G-manifolds of dimension n = 2k (n ≧ 6) such that dim Sing(G, X) ≦ k and Y is simply connected.

This section defines the doubly parametrized positioning module M M M_{fff} of a k-connected G-surgery map fff = (f, b). M M M_{fff} is called the G-surgery obstruction module of fff. It will be shown that a geometrically defined invariant ∇_{𝔹} vanishes on M M M_{fff} . The G-surgery obstruction σ(fff) is by Definition 5.11 the class of M M M_{fff} in the G-surgery obstruction group. Theorem 6.3 of the next section will show that the vanishing of σ(fff) is a sufficient condition for performing G-surgery. Theorem 7.8 will show that the vanishing of σ(fff) is a necessary condition for performing G-surgery.

The module M M M_{fff} will actually be defined with respect to any given G-set

L = L^{(k−1)} ∪ ⋃_{β∈B} B_{β} (⊂ X)

satisfying Assumption 4.1 in Section 4. By definition, L ⫆ Sing(G, X) and is properly thought of as a generalization of Sing(G, X). See Assumption 4.2. In this setting M M M_{fff} will be denoted by M M M_{fff,𝔹} and σ(fff) by σ(fff, 𝔹), where 𝔹 = {h_{β} : B_{β} → X | β ∈ B} denotes the set of canonical inclusion maps. Define

B_{+,L} = {β ∈ B | B_{β} is orientable}.

Recall that for each β ∈ B_{+,L}, B_{β} is assumed to be oriented. Its orientation class in H_{k}(B_{β}, ∂B_{β}; ℤ) is denoted by [β]. We introduce the notion of a G-singularity structure equipped with a free {±1}-action on B_{+,L} so that the G-action on the structure is compatible with that on the family of orientations {[β] | β ∈ B}.

Definition 5.1. The set 𝔹, or more precisely the pair (L, 𝔹), is called a G-singularity structure for X if the following conditions are satisfied. (5.1.1) B is a G × {±1}-set such that B^{{±1}} = B ∖ B_{+,L}. (5.1.2) For all g ∈ G and β ∈ B, B_{gβ} = gB_{β}. (Hence the map g : X → X, x ↦ gx, has

the well-defined restriction g : B_{β} → B_{gβ}.) (5.1.3) For all β ∈ B_{+}, B_{(−1)β} = B_{β} as subsets of X and [(−1)β] = −[β]. (Hence

h_{β} = h_{(−1)β}.) (5.1.4) For all g ∈ G and β ∈ B_{+,L}, [gβ] = g[β] (namely, g_{∗}[β]). (5.1.5) Let β and β′ be arbitrary elements of B such that B_{β} = B_{β′} as subsets of X. If

B_{β} is orientable then β = β′ or β = (−1)β′; if B_{β} is not orientable then β = β′.

For the rest of this paper, we let (L, 𝔹) be a G-singular structure for X. Define

Θ̃_{𝔹} = B_{+,L}, Θ_{2,𝔹} = B/{1, −1}

and let

p_{𝔹} : Θ̃_{𝔹} → Θ_{2,𝔹}

denote the canonical map. For each element t of Θ_{2,𝔹}, let [t] denote the nontrivial element of H_{k}(B_{β(t)}, ∂B_{β(t)}; ℤ_{2}). It follows that for β ∈ Θ̃_{𝔹}, the ℤ_{2}-reduction of [β] is equal to [p_{𝔹}(β)].

In the remainder of the paper, we often denote a lifting in B of t ∈ Θ_{2,𝔹} by β(t) which is uniquely determined up to {±1}-action. In this occasion, β is a section Θ_{2,𝔹} → B of the quotient map B → Θ_{2,𝔹}.

Let R denote a commutative ring with unit element.

Definition 5.2. Let

\begin{matrix}H_{k}(X; R) = K(X; R) ⊕ K(X; R)^{c} and \\ H_{k}(X, ∂X; R) = K(X, ∂X; R) ⊕ K(X, ∂X; R)^{c} \end{matrix}

be R[G]-direct sum decompositions such that

Int_{G,X,∂X;R}(K(X; R), K(X, ∂X; R)^{c}) = 0 and
Int_{G,X,∂X;R}(K(X; R)^{c}, K(X, ∂X; R)) = 0.

Let

\begin{matrix}H_{k}(X; R_{2}) = K(X; R_{2}) ⊕ K(X; R_{2})^{c} and \\ H_{k}(X, ∂X; R_{2}) = K(X, ∂X; R_{2}) ⊕ K(X, ∂X; R_{2})^{c} \end{matrix}

be R_{2}[G]-direct decompositions compatible with the decompositions of H_{k}(X; R) and H_{k}(X, ∂X; R) such that

Int_{G,X,∂X;R_{2}}(K(X; R_{2}), K(X, ∂X; R_{2})^{c}) = 0 and
Int_{G,X,∂X;R_{2}}(K(X; R_{2})^{c}, K(X, ∂X; R_{2})) = 0,

where R_{2} = R/2R. The term compatible above means that the canonical maps

\begin{matrix}π_{X} : H_{k}(X; R) → H_{k}(X; R_{2}) and \\ π_{X,∂X} : H_{k}(X, ∂X; R) → H_{k}(X, ∂X; R_{2}) \end{matrix}

preserve the decompositions, i.e.

π_{X}(K(X; R)) ⫅ K(X; R_{2}),
π_{X}(K(X; R)^{c}) ⫅ K(X; R_{2})^{c},
π_{X,∂X}(K(X, ∂X; R)) ⫅ K(X, ∂X; R_{2}), and
π_{X,∂X}(K(X, ∂X; R)^{c}) ⫅ K(X, ∂X; R_{2})^{c}.

If x ∈ H_{k}(X, ∂X; R) (resp. H_{k}(X, ∂X; R_{2})) then x has a decomposition

x = x_{K} + x_{K^{c}}

where x_{K} ∈ K(X, ∂X; R) (resp. K(X, ∂X; R_{2})) and x_{K^{c}} ∈ K(X, ∂X; R)^{c} (resp. K(X, ∂X; R_{2})^{c}). Define

\begin{matrix}θ_{K} : Θ̃_{𝔹} → K(X, ∂X; R); β ↦ (h_{β}_{∗}[β]_{R})_{K} \\ θ_{2,K} : Θ_{2,𝔹} → K(X, ∂X; R_{2}); t ↦ (h_{β(t)}_{∗}[t]_{R_{2}})_{K}. \end{matrix}

Clearly, the diagram

\xymatrix{Θ̃_{𝔹} \ar[d]_{p_{𝔹}} \ar[r]^{θ_{K}} & K(X, ∂X; R) \ar[d]^{π_{X,∂X}} \\ Θ_{2,𝔹} \ar[r]_{θ_{2,K}} & K(X, ∂X; R_{2})}

commutes. The 8-tuple

\begin{aligned}(K(X, R), K(X, ∂X; R), & Int_{G,X,∂X;R}, θ_{K}, \\ & K(X, ∂X; R_{2}), K(X; R_{2}), Int_{G,X,∂X;R_{2}}, θ_{2,K}) \end{aligned}

is called a geometric positioning module.

The 8-tuple is usually abbreviated by (K(X, R), K(X, ∂X; R), Int_{G,X,∂X;R}, θ_{K}, θ_{2,K}) if the canonical maps

\begin{matrix}π_{X} : K(X; R) → K(X; R_{2}) and \\ π_{X,∂X} : K(X, ∂X; R) → K(X, ∂X; R_{2}) \end{matrix}

are surjective.

Definition 5.3. We call the geometric positioning module above nonsingular, if (5.3.1) K(X; R) and K(X, ∂X; R) are finitely generated, projective R[G]-modules, (5.3.2) Int_{G,X,∂X;R} on K(X; R) × K(X, ∂X; R) is nonsingular, and (5.3.3) π_{X} : K(X; R) → K(X; R_{2}) and π_{X,∂} : K(X, ∂X; R) → K(X, ∂X; R_{2}) are surjec-

tive.

If the geometric positioning module is nonsingular, it follows of course that K(X; R_{2}) and K(X, ∂X; R_{2}) are finitely generated R_{2}[G]-modules and Int_{G,X,∂X;R_{2}} is nonsingular.

Definition 5.4. We shall call the geometric positioning module Hermitian if the canonical maps K(X; R) → K(X, ∂X; R) and K(X; R_{2}) → K(X, ∂X; R_{2}) are isomorphisms.

This make sense, since by Lemma 4.3 the diagram

\xymatrix{K(X; R) × K(X; R) \ar[r]^{^{Int^{G,X;R}}} \ar[d] & R[G] \\ K(X; R) × K(X, ∂X; R) \ar[ur]_{_{Int_{G,X,∂X;R}}}}

commutes and Int_{G,X;R} is λ-Hermitian, where λ = (−1)^{k}. Similar remarks hold when R is replaced by R_{2}. We shall abbreviate a Hermitian geometric positioning module by

(K(X; R), Int_{G,X;R}, θ_{K}, θ_{2,K}).

Theorem 5.5. Suppose that X is simply connected. Let {h_{α} : A_{α} → X | α ∈ A} be a finite set of immersions h_{α} of connected, closed, k-dimensional, oriented, smooth manifolds A_{α}. Let (K(X, ℤ), K(X, ∂X; ℤ), Int_{G,X,∂X;ℤ}, θ_{K}, θ_{2,K}) be a geometric positioning module in Definition 5.2 such that h_{α}_{∗}[α] ∈ K(X; ℤ) for each α ∈ A, where [α] is the orientation class of A_{α}. Let ♮_{G} be the G-self-intersection form in Definition 4.13. Then (5.5.1) and (5.5.2) below are equivalent. (5.5.1) (i) Int_{G,X,∂X;ℤ}(h_{α}_{∗}[α], h′_{α′}_{∗}[α′]) = 0 for all α ≠ α′ ∈ A.

(ii) Int_{G,X,∂X;ℤ}(h_{α}_{∗}[α], θ_{K}([β])) = 0 for all (α, β) ∈ A × Θ̃_{𝔹} and Int_{G,X,∂X;ℤ_{2}}(h_{α}_{∗}[α], θ_{2,K}(t)) = 0 for all (α, t) ∈ A × Θ_{2,𝔹}. (iii) Int_{G,X,∂X;ℤ}(h_{α}_{∗}[α], h_{α}_{∗}[α]) = 0 and ♮_{G}(h_{α}) = 0 for all α ∈ A.

(5.5.2) There is a family of regular homotopies h_{α} ∼ h′_{α} in X where α ranges through A such that the following holds:

(a) Image(h′_{α}) ∩ gImage(h′_{α′}) = ∅ for all α ≠ α′ ∈ A and g ∈ G.

(b) Image(h′_{α}) ∩ B_{β(t)} = ∅ for all (α, t) ∈ A × Θ_{2,𝔹}.

(c) h′_{α} is an embedding such that χ(ν(h′_{α})) = 0 and Image(h′_{α}) ∩ gImage(h′_{α}) = ∅ for all α ∈ A and g ≠ 1 ∈ G.

Proof. It follows from Lemma 4.6 and the observations

Int_{G,X,∂X;ℤ}((h_{α})_{∗}[α], θ_{K}(β)) = Int_{G,X,∂X;ℤ}(h_{α}_{∗}[α], h_{β}_{∗}([β]))

and

Int_{G,X,∂X;ℤ_{2}}(h_{α}_{∗}[α], θ_{2,K}(t)) = Int_{G,X,∂X;ℤ_{2}}(h_{α}_{∗}[α], h_{β(t)}_{∗}([t]))

that (5.5.1) ⟺ (4.21.1). Thus the assertion of the theorem follows from the assertion in Theorem 4.21. □

Let f : (X, ∂X) → (Y, ∂Y) be a G-map. The map f is called an R-boundary quasiequivalence if the induced map H_{∗}(∂X; R) → H_{∗}(∂Y; R) is an isomorphism. If f is an Rboundary quasiequivalence then f is necessarily an R_{2}-boundary quasiequivalence. The map f is called a (G, R)-singularity quasiequivalence if the induced map

H_{∗}(Sing(G, X); R) → H_{∗}(Sing(G, Y); R)

is an isomorphism. If f is a (G, R)-singularity quasiequivalence then f is necessarily a (G, R_{2})-singularity quasiequivalence.

Lemma 5.6. Let f : (X, ∂X) → (Y, ∂Y) be a degree-one G-map. Then

(K_{k}(X; R), K_{k}(X, ∂X; R), Int_{G,X,∂X;R}, θ_{K_{k}}, θ_{2,K_{k}})

is a geometric positioning module. If f is a k-connected, R-boundary quasiequivalence and if K_{k}(X; R) is R[G]-projective then (K_{k}(X; R), Int_{G,X;R}, θ_{K_{k}}, θ_{2,K_{k}}) is a nonsingular Hermitian geometric positioning module.

Proof. By (4.4.1)–(4.4.3), the decompositions H_{k}(X; R′) = K_{k}(X; R′) ⊕ f̂_{∗}(H_{k}(Y; R′)) and H_{k}(X, ∂X; R′) = K_{k}(X, ∂X; R′) ⊕ f̂_{∗}(H_{k}(Y, ∂Y; R′)) (for R′ = R and R_{2}) satisfy the conditions required in Definition 5.2. Thus,

\begin{matrix}(K_{k}(X; R), K_{k}(X, ∂X; R), & Int_{G,X,∂X;R}, θ_{K_{k}}, \\ K_{k}(X; R_{2}), K_{k}(X, ∂X; R_{2}), Int_{G,X,∂X;R}, θ_{2,K_{k}}) \end{matrix}

is a geometric positioning module.

By k-connectedness, K_{k−1}(X; ℤ) = 0. By the universal coefficient theorem, the canonical map

R′ ⊗_{ℤ} K_{k}(X; ℤ) → K_{k}(X; R′)

is an isomorphism for an arbitrary ℤ-module R′. Thus the canonical map R_{2}⊗_{R}K_{k}(X; R) → K_{k}(X; R_{2}) is an isomorphism.

Next suppose f is an R-boundary quasiequivalence. Then the maps K_{k}(X; R) → K_{k}(X, ∂X; R) and K_{k}(X; R_{2}) → K_{k}(X, ∂X; R_{2}) are isomorphisms. It follows that the map R_{2} ⊗_{R} K_{k}(X, ∂X; R) → K_{k}(X, ∂X; R_{2}) is also an isomorphism. The facts above establish that (K_{k}(X; R), Int_{G,X;R}, θ_{K_{k}}, θ_{2,K_{k}}) is a Hermitian geometric positioning module.

If K_{k}(X; R) is an R[G]-projective module then by Lemma 4.5, (K_{k}(X; R), Int_{G,X;R}) is a nonsingular, λ-Hermitian module. Thus (K_{k}(X; R), Int_{G,X;R}, θ_{K_{k}}, θ_{2,K_{k}}) is a nonsingular Hermitian geometric positioning module. □

Let f : (X, ∂X) → (Y, ∂Y) be a G-map of smooth G-manifolds X and Y. Let T (X) denote the tangent bundle of X. The smooth action of G on X gives T (X) a real Gvector bundle structure. Let ξ_{+} and ξ_{−} be real G-vector bundles on Y and let f^{∗}(ξ_{+}) and f^{∗}(ξ_{−}) denote the pullback bundles of ξ_{+} and ξ_{−}, respectively, to X. Let ξ = ξ_{+} − ξ_{−} denote the formal difference. A G-framing of f with respect to ξ is an isomorphism b : T (X) ⊕ f^{∗}(ξ_{−} ⊕ η) → f^{∗}(ξ_{+} ⊕ η) of real G-vector bundles where η is a real G-vector bundle over Y. Usually η = ε_{Y} (V) for a finite dimensional, real G-module (i.e. real G-representation space) V. Here ε_{Y} (V) is the product bundle on Y with fiber V ( i.e. trivial G-vector bundle) and f^{∗}ε_{Y} (V) = ε_{X}(V). A G-framed map is a pair fff = (f, b) : (X, ∂X, T (X)) → (Y, ∂Y, ξ).

Definition 5.7. Let fff = (f, b) : (X, ∂X, T (X)) → (Y, ∂Y, ξ) be a k-connected, degreeone G-framed map. Suppose that f is an R-boundary quasiequivalence and K_{k}(X; R) is R[G]-projective. Such a map fff is called a k-connected (R, G)-surgery map. Define

B_{f,R} = Int_{G,X;R|_{K_{k}(X;R)×K_{k}(X;R)}}, θ_{f,𝔹} = θ_{K_{k}}, θ_{2,f,𝔹} = θ_{2,K_{k}}.

By Lemma 5.6, (K_{k}(X; R), B_{f,R}, θ_{f,𝔹}, θ_{2,f,𝔹}) is a nonsingular Hermitian geometric positioning module on the projective R[G]-module K_{k}(X; R). Let

Q(G, X) = G(X, k − 1) ∩ G(2)
S(G, X) = G(X, k) ∩ G(2)
Λ(Q(G, X)) = min_{λ}(ℤ[G]) + ℤ[Q(G, X)]
Γ(S(G, X)) = min_{−λ}(ℤ[G]) + ℤ[S(G, X)]

(cf. Section 3). Since f is k-connected and of degree one, we deduce that the relative Hurewicz map π_{k}(f) → K_{k}(X; ℤ) is an isomorphism. Thus, each element of K_{k}(X; ℤ) can be represented by an immersion h : S^{k} → X of the oriented k-sphere S^{k}. Since fff is G-framed, we can in fact choose h such that its normal bundle ν(h) in X is trivial, cf. [44, Part 0, p.10, Theorem 1.1 and Proposition]. Such an h is unique up to regular homotopy in X, cf. [44, Part 0, Theorem 1.1] or the proposition subsequent to it. For each x ∈ K_{k}(X; ℤ), let h_{x} : S^{k} → X be an immersion of the k-sphere S^{k} with trivial normal bundle such that h_{x}_{∗}[S^{k}] = x. By Lemma 4.6,

B_{f,ℤ}(x, y) = #_{G}(h_{x}, h_{y}).

Let

♮_{G} : Immersion(S^{k}, Free(G, X)) → ℤ[G]/(Λ(Q(G, X)) + ℤ[S(G, X)])

denote the generalized, G-equivariant, self-intersection form in Definition 4.13 and define

q_{fff} : K_{k}(X; ℤ) → ℤ[G]/(Λ(Q(G, X)) + ℤ[S(G, X)]); x ↦ ♮_{G}(h_{x}).

By Theorem 4.14, we have the equality

B_{f,ℤ}(x, y) = q̃_{fff,ℤ}(x) + q̃_{fff,ℤ}(y) in ℤ[G]/ℤ[S(G, ℤ)]

where q̃_{fff,ℤ}(x) and q̃_{fff,ℤ}(y) denote liftings of q_{fff,ℤ}(y) and q_{fff,ℤ}(y), respectively, to ℤ[G]/ℤ[S(G, ℤ)]. By Theorem 4.16, we have the equality

q_{fff}(x + y) = q_{fff}(x) + q_{fff}(y) + B_{f,ℤ}(x, y) in ℤ[G]/(Λ(Q(G, X)) + ℤ[S(G, X)]).

Furthermore, one shows routinely that for any element a ∈ ℤ[G], q_{fff}(ax) = aq_{fff}(x)\overline{a}. It follows that the pair (B_{f,ℤ}, q_{fff}) is a doubly parametrized form, in the sense of Definition 2.6, on K_{k}(X; ℤ) over the parameter algebra

(ℤ[G], (−, λ), Γ(S(G, X)), G, ℤ[S(G, X)], Λ(Q(G, X)) + ℤ[S(G, X)]).

By applying the exact functor R ⊗ −, we extend B_{f,ℤ} and q_{fff,ℤ}, respectively, to maps

B_{f,R} : K_{k}(X, R) × K_{k}(X, R) → R[G]

and

q_{fff,R} : K_{k}(X; R) → R[G]/(RΛ(Q(G, X) + R[S(G, X)]))

and obtain straightforward that (B_{f,R}, q_{fff,R}) is a doubly parametrized form on K_{k}(X; R) over the parameter algebra

(R[G], (−, λ), RΓ(S(G, X)), G, R[S(G, X)], RΛ(Q(G, X)) + R[S(G, X)])).

The 5-tuple

M M M_{fff,𝔹} = (K_{k}(X; R), B_{f,R}, q_{fff,R}, θ_{f,𝔹}, θ_{2,f,𝔹})

is called the R[G]-surgery obstruction module of (fff, 𝔹).

Proposition 5.8. Let fff : (X, ∂X; T (X)) → (Y, ∂Y; ξ) be a k-connected (R, G)-surgery map. Then its surgery obstruction module M M M_{fff,𝔹} lies in the category Q Q Q(A A A,Θ Θ Θ_{𝔹}), where

A A A = (R[G], (−, λ), RΓ(S(G, X)), G, R[S(G, X)], RΛ(Q(G, X)) + R[S(G, X)])

is as in Section 3 and Θ Θ Θ_{𝔹} = (Θ̃_{𝔹}, p_{𝔹}, Θ_{2,𝔹}). Moreover on Image[K_{k}(X; ℤ) → K_{k}(X; R)], B_{f,R} = #_{G} and q_{fff,R} = ♮_{G}.

Proof. The nonsingularity of B_{f,R} was proved in Lemma 5.6. Everything else was shown directly above. □

Our next goal is to show that the surgery obstruction module M M M_{fff,𝔹} above lies in a certain subcategory

∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹})

of Q Q Q(A A A,Θ Θ Θ_{𝔹}) where ∇_{𝔹} is an invariant on the latter category, in the sense of Definition 3.1. To describe ∇_{𝔹}, it is convenient to use the notation established in Section 3. By definition A = R[G], Γ = RΓ(S(G, X)), A_{s} = R[S(G, X)], Λ = RΛ(Q(G, X)) + R[S(G, X)], A_{2} = R_{2}[G]. Let Γ′_{2} = min_{−λ}(A_{2}) (= min_{λ}(A_{2}), because 2 = 0 in A_{2}), ε_{2} : A_{2} → R_{2}; ∑_{g∈G} r_{g}g ↦ r_{1}, Γ′_{2,g} = ε_{2}(Γ′_{2}g^{−1}), where g ∈ G, and

ρ_{2} = ρ_{G} : Θ_{2,𝔹} → S(G); t ↦ ⋂_{x∈B_{β(t)}} G_{x}.

If (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ_{𝔹}) and g ∈ S(G, X), let

Σ_{ρ_{2},g}^{θ_{2,𝔹}} = ∑_{t∈Θ_{2,𝔹}|_{g}} θ_{2}(t) ∈ M_{2},

where Θ_{2,𝔹}|_{g} = {t ∈ Θ_{2,𝔹} | ρ_{2}(t) ∋ g} and M_{2} = M/2M. Define Θ_{2}(G, X) to be the set of all submanifolds of X obtained as k-dimensional connected components of X^{H} for subgroups H of G. By Assumption 4.1, Θ_{2}(G, X) is a subset of Θ_{2,𝔹}.

Lemma 5.9. Let g ∈ S(G, X). Then the following hold. (5.9.1) Θ_{2,𝔹}|_{g} = Θ_{2}(G, X)|_{g}, thus

Σ_{ρ_{2},g}^{θ_{2}} = ∑_{t∈Θ_{2}(G,X)|_{g}} θ_{2}(t).

(5.9.2) If A is a connected closed manifold and h : A → X a continuous map then

#_{2}(h, gh) = ∑_{t∈Θ_{2}(G,X)|_{g}} #_{2}(h, h_{β(t)}) in ℤ_{2},

where #_{2}(h, gh) is the mod 2 geometric intersection number of h and gh.

Proof. (5.9.1): Let t ∈ Θ_{2,𝔹} such that ρ_{2}(t) ∋ g. Then B_{β(t)} ⫅ X^{g}. Since dim X^{g} ≦ k, B_{β(t)} is a k-dimensional connected component of X^{g}. Thus t belongs to Θ_{2}(G, X).

(5.9.2): For the computation of the intersection number, we may assume that h : A → X is an immersion and that h, gh and X^{g} intersect transversely with one another. If h and gh meet each other at a point x in X ∖X^{g} then h and gh meet at gx ∈ X ∖X^{g}, too. Thus the number of points in X \ X^{g} at which h and gh meet each other is an even integer. Thus #_{2}(h, gh) is equal (mod 2) to the number of points in X^{g} at which h and gh meet each other. This number is equal (mod 2) to #_{2}(h, X^{g}). On the other hand

#_{2}(h, X^{g}) = ∑_{t∈Θ_{2}(G,X)|_{g}} #_{2}(h, h_{β(t)}).

□

If (M, B, q, θ, θ_{2}) ∈ Q Q Q(A A A,Θ Θ Θ_{𝔹}) and s ∈ S(G, X), let

α_{Γ′_{α,s}}(M, B, q, θ, θ_{2}) = Hom_{ℤ}(M_{2}, R_{2}Γ′_{2,s}) = Hom_{ℤ}(M_{2}, R_{2})

and

∇_{Γ′_{2,s}}(M, B, q, θ, θ_{2}) : M_{2} → R_{2}; m ↦ ε_{2}(B_{f,R_{2}}(sm, Σ_{ρ_{2},s}^{θ_{2}} − m)).

Let

\begin{aligned}∇_{Γ′_{2}} & = ⨁_{s∈S(G,X)} ∇_{Γ′_{2,s}} \\ α_{Γ′_{2}} & = ⨁_{s∈S(G,X)} α_{Γ′_{2,s}} \\ ∇_{𝔹} & = (∇_{Γ′_{2}}, α_{Γ′_{2}}). \end{aligned}

By Lemma 3.25, ∇_{𝔹} defines an invariant on Q Q Q(A A A,Θ Θ Θ_{𝔹}). It is called the geometric invariant on Q(A A A,Θ Θ Θ_{𝔹}).

Lemma 5.10. Suppose R_{2} = ℤ/2ℤ or 0. If fff : (X, ∂X, T (X)) → (Y, ∂Y, ξ) is a k-connected (R, G)-surgery map then the surgery obstruction module M M M_{fff,𝔹} belongs to ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}), namely

ε_{2}(B_{f,R_{2}}(sm, Σ_{ρ_{2},s}^{θ_{2,f,𝔹}} − m))) = 0

for all m ∈ K_{k}(X; R_{2}) = R_{2} ⊗_{R} K_{k}(X; R).

Proof. Unwinding definitions, one obtains that

ε_{2}(B_{f,R_{2}}(sm, Σ_{ρ_{2},s}^{θ_{2,f,𝔹}} − m)) = Int_{X;R_{2}}(sm, Σ_{ρ_{2},s}^{θ_{2,f,𝔹}} − m)
= Int_{X,∂X;R_{2}}(m, Σ_{ρ_{2},s}^{θ_{2,f,𝔹}} − sm).

Choose an immersion h : S^{k} → X such that the normal bundle ν(h) of h in X is trivial and h_{∗}[S^{k}] = m. By definition and (5.9.1),

Σ_{ρ_{2},s}^{θ_{2,f,𝔹}} = ∑_{t∈Θ_{2}(G,X)|_{s}} θ_{2,f,𝔹}(t).

Since h_{β(t)} : B_{β(t)} → X is the canonical embedding, it follows over R_{2} that

Int_{X,∂X;R_{2}}(m, Σ_{ρ_{2},s}^{θ_{2,f,𝔹}} − sm) = ∑_{t∈Θ_{2}(G,X)|_{s}} Int_{X,∂X;R_{2}}(h_{∗}[S^{k}], s(h_{β(t)})_{∗}[t])
− Int_{X,∂X;R_{2}}(h_{∗}[S^{k}], sh_{∗}[S^{k}])

= ∑_{t∈Θ_{2}(G,X)|_{s}} #_{2}(h, sh_{β(t)}) − #_{2}(h, sh) (by Lemma 4.6)

= ∑_{t∈Θ_{2}(G,X)|_{s}} #_{2}(h, h_{β(t)}) − #_{2}(h, sh)

= 0 (by (5.9.2)).

□

Definition 5.11. Suppose R_{2} = ℤ/2ℤ or 0. If fff : (X, ∂X, T (X)) → (Y, ∂Y, ξ) is a k-connected (R, G)-surgery map, define

σ(fff, 𝔹) = [M M M_{fff,𝔹}] ∈ W_{2k}(R, G, Q(G, X), S(G, X),Θ Θ Θ_{𝔹})_{proj}.

If K_{k}(X; R) is stably free over R[G] (cf. Remark 3.35) then we can regard

σ(fff, 𝔹) ∈ W_{2k}(R, G, Q(G, X), S(G, X),Θ Θ Θ_{𝔹})_{free}.

In the remainder of this paper we suppose ℤ ⊆ R ⊆ ℚ.

Lemma 5.12. A finitely generated R-free, R[G]-module M is R[G]-projective if and only if M_{(p)} = M ⊗_{ℤ} ℤ_{(p)} is ℤ_{(p)}[G]-projective for all (rational) primes p that are not invertible in R.

Proof. The proof is similar to that of Lemma 2 of Nakayama [34]. □

Let P(G) denote the set of all subgroups of G of prime power order. A G-map f : X → Y (or (X, ∂X) → (Y, ∂Y)) is called a P(G)_{R}-singularity quasiequivalence if the induced map H_{∗}(X^{P}; ℤ_{(p)}) → H_{∗}(Y^{P}; ℤ_{(p)}) is an isomorphism for any prime p not invertible in R and any nontrivial p-subgroup P of G.

Lemma 5.13. Let f : (X, ∂X) → (Y, ∂Y) be a degree-one R-boundary and P(G)_{R}- singularity quasiequivalence such that K_{ℓ}(X; R) = 0 for all ℓ ≦ k −1. Then K_{ℓ}(X; R) = 0 if ℓ ≠ k and K_{k}(X; R) is a finitely generated, projective R[G]-module. If moreover f is a (G, R)-singularity quasiequivalence then K_{k}(X; R) is a stably free R[G]-module.

Proof. Let C_{f} denote the mapping cone of f. Then K_{∗}(X; M) ≅ \overline{H}_{∗+1}(C_{f}; M) for any ℤ-module M. Thus the universal coefficient theorem holds for the functors K_{∗}(−; −) and K^{∗}(−; −). Since K_{ℓ}(X; R) = 0 for ℓ ≦ k − 1, we obtain K^{ℓ}(X; R) = 0 for ℓ ≦ k − 1 by the universal coefficient theorem. By Poincaré-Lefschetz duality (cf. [8, I.2.8]), K_{ℓ}(X, ∂X; R) = 0 for ℓ ≧ k + 1. From the exact sequence of the pair (X, ∂X) with coefficient ring R, it follows that K_{ℓ}(X; R) = 0 for ℓ ≧ k + 1. Thus K_{ℓ}(X; R) = 0 for ℓ ≠ k. Since

K_{k}(X; R) = K_{k}(X, ∂X; R) = K^{k}(X; R) = Hom(K_{k}(X; R), R)

and K_{k}(X; R) = K_{k}(X; ℤ) ⊗ R, it follows that K_{k}(X; R) is a free R-module.

Next we prove that K_{k}(X; R) is R[G]-projective. By Lemma 5.12, it suffices to show that K_{k}(X; ℤ_{(p)}) is ℤ_{(p)}[G]-projective for each prime p not invertible in R. It is enough to show that K_{k}(X; ℤ_{(p)}) is ℤ_{(p)}[P]-projective for any prime p as above and any Sylow psubgroup P of G. If P is the trivial group then it is obviously the case. Suppose P ≠ {1}. By hypothesis, f|_{Sing(P,X)} : Sing(P, X) → Sing(P, Y) is a ℤ_{(p)}-homology equivalence. This amounts to K_{∗}(X; ℤ_{(p)}) = K_{∗}(X, Sing(P, X); ℤ_{(p)}). Set K = K_{k}(X, Sing(G, X); ℤ_{(p)}). By observation of the cellular chain complex with coefficient ring ℤ_{(p)} of the mapping cone of f : (X, Sing(P, X)) → (Y, Sing(P, Y)) using [42][Proposition 3.5 (i) and (v)], we conclude that K/pK is ℤ_{p}[P]-free. Let ψ : ℤ_{p}[P]^{m} → K/pK be a ℤ_{p}[P]-isomorphism and ψ̃: ℤ_{(p)}[P] → K be a ℤ_{(p)}[P]-homomorphism covering ψ. Let 𝔄 = {ga_{i} | g ∈ P, i = 1, …, m} be the canonical ℤ_{(p)}-basis of ℤ_{(p)}[P]. Let

𝔅 = {b_{g,i} | g ∈ P, i = 1, …, m}

be a ℤ_{(p)}-basis of K. Take the matrix representation τ of ψ̃ with respect to the bases 𝔄 and 𝔅. Since ψ is an isomorphism, [det τ] is an invertible element in ℤ_{p}, which implies that det τ is invertible in ℤ_{(p)}. Thus ψ̃ is an isomorphism. This implies K_{k}(X; ℤ_{(p)}) is ℤ_{(p)}[P]-free. Consequently, K_{k}(X; R) is R[G]-projective.

If f is a (G, R)-singularity quasiequivalence then, by inspecting the cellular chain complex with coefficient ring R of the mapping cone of f : (X, Sing(G, X)) → (Y, Sing(G, Y)), one can deduce that K_{k}(X; R) = K_{k}(X, Sing(G, X); R) is a stably free R[G]-module. □

Lemma 5.14. If a degree-one G-map f : (X, ∂X) → (Y, ∂Y) is a k-connected, Rboundary, P(G)_{R}-singularity quasiequivalence then

K_{k}(X; R) = Ker[f_{∗} : H_{k}(X; R) → H_{k}(Y; R)]

is a finitely generated, projective R[G]-module. If moreover, f is a (G, R)-singularity quasiequivalence then K_{k}(X; R) is a stable free R[G]-module.

Proof. This follows immediately from Lemma 5.13. □

6. G-surgery obstructions and theorems

The present section has two main results Theorems 6.2 and 6.3. The surgery obstruction σ(fff, 𝔹) is defined in Theorem 6.3 as the class of surgery module M M M_{fff,𝔹} in the surgery obstruction group W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ_{𝔹})_{proj}. The results are for any coefficient ring R such that ℤ ⫅ R ⫅ ℚ. Let M M M′ denote the orthogonal sum M M M′ = M M M_{fff,𝔹} ⊕ 𝕄 of a G-surgery obstruction module M M M_{fff,𝔹} and a metabolic plane 𝕄 which vanishes under the geometric invariant ∇_{𝔹}. Theorem 6.2 says that one can convert by G-surgery the Gframed map fff to a G-framed map fff′ such that M M M′ = M M M_{fff′,𝔹}. Since 𝕄 vanishes by definition in the G-surgery obstruction group, it will follow that σ(fff, 𝔹) = σ(fff′, 𝔹). Theorem 6.3 says that the vanishing of σ(fff, 𝔹) is a sufficient condition for converting fff by G-surgery into a G-framed homotopy equivalence fff′.

Let I denote the closed unit interval [0, 1] and p_{X} : I ×X → X the canonical projection on the second factor X. Recall that a G-cobordism W between X and X′ is a compact oriented smooth G-manifold of dimension n + 1 such that the boundary ∂W of W is a union of two G-submanifolds ∂_{+}W and ∂_{−}W satisfying the properties

∂_{+}W = −X ⨿ X′ (−X denotes X with the opposite orientation) ∂_{−}W ≅_{G} I × ∂X′ (≅_{G} reads G-diffeomorphic to) ∂_{+}W ∩ ∂_{−}W = ∂(∂_{+}W) = ∂(∂_{−}W) = ∂(−X) ⨿ ∂(X′).

T W |_{∂_{+}W} is identified with ε_{∂W} (ℝ) ⊕ T (∂_{+}W)

in the standard way, namely the inward normal bundle ν(−X, W) of −X in W is identified with ε_{−X}(ℝ) and the outward normal bundle ν(X′, W) of X′ in W is identified with ε_{X′}(ℝ). If η_{i} (i = 1, 2) are bundles over X, if b : η_{1} → η_{2} is a bundle map covering the identity map on X, and if C is a subspace of X, then b|_{C} denotes the bundle map b|_{η_{1}|_{C}} : η_{1}|_{C} → η_{2}|_{C}. Let fff = (f, b) : (X, ∂X, T X) → (Y, ∂Y, f^{∗}ξ) and fff′ = (f′, b′) : (X′, ∂X′, T X′) → (Y, ∂Y, f′^{∗}ξ) denote G-framed maps. Then a G-framed cobordism between fff and fff′ consists of a G-cobordism W from X to X′ and a G-framed map

(F, B) : (W, ∂W, T W) → (I × Y, ∂(I × Y), (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ)),

where

B : T W ⊕ (p_{Y} ∘ F)^{∗}(ξ_{−} ⊕ η) → ε_{W} (ℝ) ⊕ (p_{Y} ∘ F)^{∗}(ξ_{+} ⊕ η),

such that F (X) ⫅ {0} × Y, F (X′) ⫅ {1} × Y, F |_{X} = f, F |_{X′} = f′, B|_{X} = id_{ε_{X}(ℝ)}⊕ b and B|_{X′} = id_{ε_{X′}(ℝ)} ⊕ b′. Unless specifically mentioned otherwise, we assume that the stable term η has the form η = ε_{Y} (U) for a real G-module U such that dim U^{G} ≧ n.

Let C denote a G-simplicial subcomplex of X with respect to some smooth equivariant triangulation of X. The G-framed cobordism (F, B) above is said to be relative to C, if there exists a manifold G-neighborhood N = N(C, X) of C in X such that the canonical inclusion N → X extends to a (neat) G-embedding I × N → W with the property that N = X ∩ (I × N) = {0} × N (the left equality holds in X, and the right equality holds in I×N), X′∩(I×N) = {1}×N, and for each t ∈ I, F |_{{t}×C} = f|_{C} (i.e. F ({t}×C) ⫅ {t}×Y and the composite map

C ⟶ {t} × C ⟶^{F} I × Y ⟶^{p_{Y}} Y

is the map f|_{C}), B|_{{t}×C} = (id_{ε_{X}(ℝ)} ⊕ b)|_{C} (i.e. the composite map

(ε_{X}(ℝ) ⊕ T X ⊕ f^{∗}(ξ_{−} ⊕ η))|_{C} ⟶ (ε_{N}(ℝ) ⊕ T N ⊕ (f|_{N})^{∗}(ξ_{−} ⊕ η))|_{C}

⟶ (T (I × N) ⊕ ((p_{Y} ∘ F)|_{I×N})^{∗}(ξ_{−} ⊕ η))|_{{t}×C} ⟶ (T W ⊕ (p_{Y} ∘ F)^{∗}(ξ_{−} ⊕ η))|_{{t}×C}

⟶^{B} ((p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ_{+} ⊕ η))|_{{t}×C}

= (ε_{X}(ℝ) ⊕ f^{∗}(ξ_{+} ⊕ η))|_{C}

is the map (id_{ε_{X}(ℝ)} ⊕ b)|_{C}). If (F, B) is relative to C then one says that fff is G-framed cobordant rel C to fff′.

Performing G-surgery on a G-framed map fff = (f, b), f : X → Y, defines a pair (F, B) of maps called the trace of G-surgery, which satisfies the conditions above for a G-framed cobordism, except possibly the condition that B|_{X′} = id_{ε_{X′}(ℝ)} ⊕ b′. The next lemma guarantees that we can perform the G-surgery in a way that its trace (F, B) is a G-framed cobordism in the sense above.

Lemma 6.1. Let (M, M_{0}) denote an n-dimensional, finite G-CW-pair, i.e. M_{0} ⫅ M and n = dim(M ∖ M_{0}). Let ω and ω′ denote real G-vector bundles supplied with a Ginvariant Riemannian metric over M. Let ⊕ denote the operation of orthogonal sum on G-vector bundles with a G-invariant Riemannian metric. If ω ⫆ ε_{M}(ℝ^{n′}) where n′ = max(n, 1) then any G-vector bundle isomorphism b : ε_{M}(ℝ) ⊕ ω → ε_{M}(ℝ) ⊕ ω′ such that b|_{M_{0}} = id_{ε_{M_{0}}(ℝ)} ⊕ b′_{0}, for some G-vector bundle isomorphism b′_{0} : ω_{M_{0}} → ω′_{M_{0}}, is regularly G-homotopic rel M_{0} to an orthogonal sum id_{ε_{M} (ℝ)} ⊕ b′ of G-vector bundle isomorphisms id_{ε_{M} (ℝ)} and b′ : ω → ω′ such that b′|_{M_{0}} = b′_{0}.

Proof. It is well known that b is regularly G-homotopic to a metric-preserving isomorphism. (This follows from the equivariant covering homotopy property and from the fact that if ⟨ , ⟩ and ⟨ , ⟩′ are G-invariant Riemannian metrics on the same underlying Gvector bundle ξ, then (1 − t)⟨ , ⟩ + t⟨ , ⟩′ (t ∈ I) is a G-invariant Riemannian metric on ξ.) Thus we may assume that b is metric preserving.

We prove Lemma 6.1 by double induction on n and the number of isotropy types of n-dimensional cells in M ∖ M_{0}. Suppose M_{0} ⫅ M′ and M = M′ ∪ ⋃_{j}(G/H × D_{j}^{n}), where D_{j}^{n} = D^{n}. Invoke the inductive hypothesis that b|_{M′} has the form id_{ε_{M′}(ℝ)} ⊕ b′′ where b′′ : ω|_{M′} → ω′|_{M′}. Under this hypothesis, we will find b′ as in Lemma 6.1. For fixed j, set E = H/H × D_{j}^{n}. Then b(ε_{M}(ℝ)|_{\overline{E}∖E^{∘}}) = ε_{M}(ℝ)|_{\overline{E}∖E^{∘}} where \overline{E} = Closure(E) and

E^{∘} = Interior(E), but it is not necessary that

(6.1.1)  b(ε_{M}(ℝ)|_{E}) = ε_{M}(ℝ)|_{E}.

Let b^{H} : ε_{M^{H}}(ℝ) ⊕ ω^{H} → ε_{M^{H}}(ℝ) ⊕ ω′^{H} be the restriction of b to the H-fixed point set. Then b|_{M^{H}} decomposes as a sum b|_{M^{H}} = b^{H} ⊕ b_{H} (N_{G}(H)-orthogonal sum). We deform b keeping b|_{M′} and b_{H} fixed. The obstruction σ to modifying b to satisfy (6.1.1) lies in π_{n−1}(S^{m−1}), where m = fiber-dim(ω^{H}) + 1. Since fiber-dim(ω^{H}) ≧ fiber-dim(ω^{G}) ≧ n, the obstruction group π_{n−1}(S^{m−1}) is trivial. Hence the obstruction σ vanishes. If (6.1.1) is satisfied for all j then b(ε_{M}(ℝ)) = ε_{M}(ℝ). Since b is metric preserving, we get b(ω) ⫅ ω′. Moreover, we can arrange b so that b|_{ε_{M} (ℝ)} = id_{ε_{M} (ℝ)}, because fiber-dim(ε_{M}(ℝ) ⊕ ε_{M}(ℝ^{n′})) ≧ 2. □

Recall that ℤ ⫅ R ⫅ ℚ, λ = (−1)^{k},

A A A = (R[G], (−, λ), RΓ(S(G, X)), G, R[S(G, X)], RΛ(Q(G, X) + R[S(G, X)]),

and Γ′_{2} = min_{−λ}(R_{2}[G]).

The formulation of the next theorem makes use of the classification in Lemma 3.32 of metabolic planes in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}).

Theorem 6.2. Let fff = (f, b) : (X, ∂X, T X) → (Y, ∂Y, f^{∗}ξ) denote a G-framed map such that f is a k-connected, P(G)_{R}-singularity, R-boundary quasiequivalence. Let (L, 𝔹) denote a G-singularity structure of X (cf. Definition 5.1). Let

M M M_{fff,𝔹} = (K_{k}(X; R), B_{f,R}, q_{fff,R}, θ_{f,𝔹}, θ_{2,f,𝔹})

denote the R[G]-surgery obstruction module (Definition 5.7) in Q Q Q(A A A,Θ Θ Θ_{𝔹}) of (fff, 𝔹). By Lemma 5.10, M M M_{fff,𝔹} lies in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}). Let 𝕄(R[G],ccc) (ccc : Θ Θ Θ_{𝔹} → (R, R_{2})) denote a metabolic plane in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}). Then one can perform G-surgery on fff along (k − 1)- dimensional spheres in Interior(X∖L) to obtain a k-connected G-framed map fff′ = (f′, b′) : (X′, ∂X′, T X′) → (Y, ∂Y, f′^{∗}ξ) such that M M M_{fff′,𝔹} ≅ M M M_{fff,𝔹} ⊕ 𝕄(R[G],ccc) in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}).

Note that in the theorem above, the resulting map f′ is a P(G)_{R}-singularity and Rboundary quasiequivalence, because the surgery is performed on Interior(X).

The proof of Theorem 6.2 will be given after the following application.

Theorem 6.3. Let fff = (f, b) : (X, ∂X, T X) → (Y, ∂Y, f^{∗}ξ) and (L, 𝔹) denote a G-framed map and a G-singularity structure of X, respectively, in the previous theorem. Let

σ(fff, 𝔹) ∈ W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ_{𝔹})_{proj}

denote the element afforded by M M M_{fff,𝔹}. If σ(fff, 𝔹) = 0 then one can perform G-surgery on fff along (k − 1)- and k-dimensional spheres in Interior(X ∖ L) to obtain a k-connected G-framed map fff′ = (f′, b′) : (X′, ∂X′, T X′) → (Y, ∂Y, f′^{∗}ξ) such that f′ : X′ → Y is an R-homology equivalence.

Proof. Suppose σ(fff, 𝔹) = 0. Then there exists modules M M M and N N N ∈ ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}) such that N N N is null in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹})_{F F F(A)}, and M M M_{fff,𝔹} ⊕ M M M ≅ N N N ⊕ M M M. By Lemma 3.11, the family of metabolic planes in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹})_{F F F(A)} is cofinal in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹}). Thus we can assume without loss of generality that M M M is isomorphic to a direct sum of metabolic planes in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹})_{F F F(A)}. By Lemma 3.32, we may assume that

M M M ≅ 𝕄(R[G],ccc_{1}) ⊕ ⋯ ⊕ 𝕄(R[G],ccc_{ℓ})

for some ccc_{i} = (c_{i}, c_{2,i}), where c_{i} : Θ̃_{𝔹} → R and c_{2,i} : Θ_{2,𝔹} → R_{2}, i = 1, ⋯, ℓ. By Theorem 6.2, one can perform G-surgery on fff along (k − 1)-dimensional spheres in Interior(X ∖ L) to get a G-surgery map fff′′ such that

M M M_{fff′′,𝔹} ≅ M M M_{fff,𝔹} ⊕ 𝕄(R[G],ccc_{1}) ⊕ ⋯ ⊕ 𝕄(R[G],ccc_{ℓ}).

Since this module is isomorphic to N N N ⊕ M M M, it follows that M M M_{fff′′,𝔹} is a null module in ∇_{𝔹}Q Q Q(A A A,Θ Θ Θ_{𝔹})_{F F F(A)}. Therefore we may assume that M M M_{fff,𝔹} is null in the free category. In this case there is a free Lagrangian U of M M M_{fff,𝔹} where U ⊂ K_{k}(X; R). Let {x_{1}, ⋯, x_{ℓ}} be a basis of U. We can assume without loss of generality that each x_{i} is represented by an immersion h_{i} : S^{k} → Interior(X ∖L) with trivial normal bundle. Note that #_{G}(h_{i}, h_{j}) = B_{f,R}(x_{i}, x_{j}), #_{G}(h_{i}, h_{β}) = B_{f,R}(x_{i}, θ(β)) ∈ ℤ for β ∈ Θ̃_{𝔹}, #_{G}(h_{i}, h_{β(t)}) = B_{2,f,R}(x_{i}, θ_{2}(t)) ∈ ℤ_{2} for t ∈ Θ_{2,𝔹} where β(t) ∈ is a lifting of t, and ♮_{G}(h_{i}) = q(x_{i}). Thus all these elements are trivial. By Theorem 5.5, we may assume that gh_{i} and g′h_{j} are disjoint unless g = g′ ∈ G and 1 ≦ i = j ≦ ℓ, that Image(h_{i}) ∩ B_{β} = ∅ for all 1 ≦ i ≦ ℓ and all β ∈ B, and that h_{i} is an embedding with trivial normal bundle such that Image(h_{i}) ∩ gImage(h_{i}) = ∅ for all 1 ≦ i ≦ ℓ and all g ≠ 1 ∈ G. Now we can perform G-surgery on fff along h_{1}, …, h_{ℓ}. The resulting G-framed map fff′ = (f′; b′) : (X′, ∂X′; T (X′)) → (Y, ∂Y; f′^{∗}ξ) is a Gsurgery map in the sense of Definition 5.7 (in particular X′ is 1-connected). Moreover K_{k}(X′; R) = 0 by the argument in [44, p.51, lines 12–(−7)] (cf. Proof of Theorem 7.3 Step 4 of [6, p.292]). Thus, f′ : X′ → Y is an R-homology equivalence. □

Proof of Theorem 6.2. The proof has 3 steps. Step 1 constructs a smooth G-embedding φ : G × S^{k−1} × D^{k+1} → Int(X ∖ L) satisfying certain properties. After identifying X with {1} × X, we obtain a smooth G-embedding φ : G × S^{k−1} × D^{k+1} → I × X. Step 2 begins by attaching the G-handle G × D^{k} × D^{k+1} to I × X, using φ as the attaching map. Let

W = (I × X)⋃_{φ}(G × D^{k} × D^{k+1})

denote the resulting attaching space and let

X′ = (({1} × X) ∖ φ(G × S^{k−1} × Int(D^{k+1})))⋃_{φ|_{G×S^{k−1}×S^{k}}} (G × D^{k} × S^{k}).

Clearly ∂W = X′ ∪ ({0} × X) ∪ (I × ∂X). Using the properties of φ and Lemma 6.1, we show that the G-framed map (f, b), f : X → Y, extends to a G-framed cobordism (F, B), F : W → I × Y, rel (∂X ∪ L), which satisfies the usual conditions plus an extra condition set out at the beginning of the section. Let (f′, b′), f′ : X′ → Y, denote the G-framed map obtained by restricting (F, B) to X′ and by identifying canonically Y with {1} × Y. Step 3 shows that f′ is k-connected and that 𝕄_{fff′,𝔹} = 𝕄_{fff,𝔹} ⊕ 𝕄(R[G],ccc).

Step 1. In order to construct an embedding φ which can be used to complete Step 3, we need to fix a model 𝕄(R[G],ccc, c̃) in the isomorphism class of 𝕄(R[G],ccc). This is done as follows. The module 𝕄(R[G],ccc) is defined in Definition 3.30 and by definition, ccc is a pair (c, c_{2}) of maps such that the diagram

\xymatrix{Θ̃_{𝔹} \ar[r]^{c} \ar[d]_{p_{𝔹}} & R \ar[d] \\ Θ_{2,𝔹} \ar[r]_{c_{2}} & R_{2}}

commutes. By Lemma 3.31, the isomorphism class of 𝕄(R[G],ccc) is not changed, if ccc is replaced by rccc = (rc, rc_{2}) for any invertible element r ∈ R. Let ℤ′_{2} = Image[ℤ → R_{2}]. Since ℤ ⫅ R ⫅ ℚ, there is an invertible element r ∈ R such that Image(rc) ⫅ ℤ and Image(rc_{2}) ⫅ ℤ′_{2}. After replacing ccc by rccc, we may assume Image(c) ⫅ ℤ and Image(c_{2}) ⫅ ℤ′_{2}. Let β : Θ_{2,𝔹} → B denote a section of the quotient map B → Θ_{2,𝔹} (cf. the paragraph subsequent to Definition 5.1). Now let c̃: Θ_{2,𝔹} → ℤ be any map such that the diagram

\xymatrix{Θ_{2,𝔹} \ar[r]^{c̃} \ar[dr]_{c_{2}} & ℤ \ar[d] \\  & ℤ′_{2}}

commutes and c̃(t) = c(β(t)) for all t ∈ p_{𝔹}(Θ̃_{𝔹}). This gives us the model 𝕄(R[G],ccc, c̃) we want.

Using c̃ above, we shall first construct for each t ∈ p_{𝔹}(Θ̃_{𝔹}) and each 1 ≦ i ≦ |c̃(t)|, an embedded oriented k-dimensional disk D_{t,i} in X such that the ordinary intersection number

B_{β(t)} · D_{t,i} = \begin{cases}1, & if t ∈ p_{𝔹}(Θ̃_{𝔹}), c̃(t) > 0 \\ −1, & if t ∈ p_{𝔹}(Θ̃_{𝔹}), c̃(t) < 0. \end{cases}

Then we shall construct for each t ∈ Θ_{2,𝔹} ∖ p_{𝔹}(Θ̃_{𝔹}) and each 1 ≦ i ≦ |c̃(t)|, an embedded k-dimensional disk D_{t,i} with arbitrary orientation such that

B_{β(t)} · D_{t,i} = ±1.

Finally we shall choose a certain embedded k-dimensional disk D_{0} and connect each ∂(D_{t,i}) to ∂(D_{0}) by a solid k-dimensional tube T_{t,i} ≅ I × D^{k−1}. This leads quickly to a G-embedding φ : G × S^{k−1} × D^{k+1} → X.

We fill in the details of the outline in the paragraph above.

Let t ∈ p_{𝔹}(Θ̃_{𝔹}) such that c̃(t) ≠ 0. Let N(t) denote a ρ_{G}(t)-tubular neighborhood of B_{β(t)}. Take a closed, k-dimensional disk D(t) in B_{β(t)} ∖L^{(k−2)} such that D(t)∩gD(t) = ∅ for all g ∈ G ∖ ρ_{G}(t). Since D(t) is ρ_{G}(t)-contractible, N(t)|_{D(t)} ≅ D(t) × M(t) for some ρ_{G}(t)-module M(t) of dimension k. Let D′_{t,1}, ⋯, D′_{t,|c̃(t)|} denote disjoint, embedded, closed, k-dimensional disks in the interior of D(t). Let x_{t,i}, 1 ≦ i ≦ |c̃(t)|, denote the center of D′_{t,i}. Clearly x_{t,i} ∉ B_{β(t′)} for any t′ ∈ p_{𝔹}(Θ̃_{𝔹}) such that t′ ≠ t. Since N(t)|_{D′_{t,i}} ≅ D′_{t,i} × M(t) for each i such that 1 ≦ i ≦ |c̃(t)|, there is a smooth section s′_{t,i} : D′_{t,i} → N(t)|_{D′_{t,i}} such that s′_{t,i} intersects transversally D′_{t,i} and Image(s′_{t,i}) ∩ D′_{t,i} = {x_{t,i}}. Note that ρ_{G}(t) acts freely on M(t) ∖ {0}. Thus we can take a δ-approximation s_{t,i} of s′_{t,i} in N(t)|_{D′_{t,i}} such that if y ∈ D′_{t,i} is close to x_{t,i}, i.e. for example, the distance between y and x_{t,i} is less than the half of the radius of D′_{t,i}, then s_{t,i}(y) = s′_{t,i}(y), and such that if gs_{t,i}(y) = g′s_{t,i}(y′), where g, g′ ∈ G and y, y′ ∈ ∂D′_{t,i}, then g = g′ and y = y′. Note that s_{t,i} is not necessarily a section. If we choose δ sufficiently small (in the sense of C^{1}- topology) then s_{t,i} is an embedding. Let D_{t,i} = Image(s_{t,i}). Clearly, GD_{t,i} ∩ GD_{t,j} = ∅ whenever i ≠ j. Furthermore, ∂D_{t,i} ∩ g∂D_{t′,j} = ∅ whenever t ≠ t′, i ≠ j, or g ≠ 1. Orient each D_{t,i} such that when c̃(t) > 0 (resp. c̃(t) < 0), the orientation at x_{t,i} given by the orientation on X agrees (resp. disagrees) with the orientation at x_{t,i} defined by the orientations of B_{β(t)} and D_{t,i}. The constructions above guarantee that the intersection number

B_{β(t)} · D_{t,i} = \begin{cases}1, & if t ∈ p_{𝔹}(Θ̃_{𝔹}), c̃(t) > 0 \\ −1, & if t ∈ p_{𝔹}(Θ̃_{𝔹}), c̃(t) < 0. \end{cases}

Now let t ∈ Θ_{2,𝔹} ∖ p_{𝔹}(Θ̃_{𝔹}) such that c̃(t) ≠ 0. Let N(t) denote a ρ_{G}(t)-tubular neighborhood of B_{β(t)}. As above, we take a k-dimensional closed disk D(t), k-dimensional closed disks D′_{t,i}, where 1 ≦ i ≦ |c̃(t)|, with centers x_{t,i}, and k-dimensional closed disks D_{t,i} (⊂ N(t)|_{D′_{t,i}}), except now we orient each D_{t,i} arbitrarily. Thus

B_{β(t)} · D_{t,i} = ±1.

Set

B B B = {(t, i) | t ∈ Θ_{2,𝔹}, c̃(t) ≠ 0, and 1 ≦ i ≦ |c̃(t)|}.

We put all D_{t,i}’s, (t, i) ∈ B B B, together as follows. Let N(L) denote a G-regular neighborhood of L. Let

D_{0} ⊂ Interior (X ∖ N(L))

denote an embedded, closed, k-dimensional disk such that D_{0}∩gD_{0} = ∅ for all g ≠ 1 ∈ G. Let {T_{t,i} | (t, i) ∈ B B B} be a set of disjoint, closed, k-dimensional, solid tubes (≅ I × D^{k−1}) in

(Interior(X) ∖ L) ∖ G (Interior(D_{0}) ∪ ⋃_{(t,i)∈B B B} Interior(D_{t,i}))

such that T_{t,i} connects ∂D_{0} to ∂D_{t,i} and such that T_{t,i} ∩ gT_{t′,j} = ∅ whenever one of the inequalities g ≠ 1, t ≠ t′, or i ≠ j holds. Let

D = D_{0} ∪ ⋃_{(t,i)∈B B B}(T_{t,i} ∪ D_{t,i}).

Then D is an embedded, closed, k-dimensional disk such that for each t ∈ Θ_{2,𝔹},

D ∩ B_{β(t)} = {x_{t,1}, ⋯, x_{t,|c̃(t)|}}.

By choosing the T_{t,i}’s appropriately, we can assume that D is the union of D_{0}, T_{t,i} and D_{t,i} as oriented manifolds. Thus, ∂D is a (k−1)-dimensional sphere embedded in Interior(X ∖ L) and ∂D ∩ g∂D = ∅ for g ≠ 1 ∈ G. Let

(6.2.1)  h_{ccc} : S^{k−1} = ∂D → Interior(X ∖ L)

denote the canonical inclusion. Since ∂D is the boundary of the embedded disk D in X and D is contractible, ∂D has a closed, tubular neighborhood \overline{N}(∂D) in X diffeomorphic to S^{k−1} × D^{k+1}. Moreover, we may assume that \overline{N}(∂D) lies in Interior(X ∖ L), and \overline{N}(∂D) ∩ g\overline{N}(∂D) = ∅ whenever g ≠ 1. Let H : S^{k−1} × D^{k+1} → \overline{N}(∂D) denote a coordinate diffeomorphism. Define φ = Ind_{{1}}^{G}H : G × S^{k−1} × D^{k+1} → I × X where Ind_{H}^{G}(g, u) = (1, gH(u)) for g ∈ G and u ∈ S^{k−1} × D^{k+1}.

Step 2. Form the attaching space

W = (I × X)⋃_{φ}(G × D^{k} × D^{k+1}),

and define the subset X′ ⊂ ∂W by

X′ = {({1} × X) ∖ Ind_{{1}}^{G}H(G × S^{k−1} × Interior(D^{k+1}))}⋃_{φ|_{G×S^{k−1}×S^{k}}} (G × D^{k} × S^{k}).

Clearly ∂W = X′ ∪ ({0} × X) ∪ (I × ∂X). Let ψ : D^{k} → D denote a diffeomorphism extending h_{ccc} : ∂D^{k} = S^{k−1} → ∂D. The G-map id_{I} × f : I × X → I × Y extends to a Gmap F : W → I × Y such that F (X′) ⫅ {1} × Y and F (g, u, 0) = (1, g(f(ψ(u)))) for any g ∈ G and u ∈ D^{k}. The obstruction σ_{H} to extending the G-vector bundle isomorphism

id_{ε_{I}(ℝ)} × b : ε_{I}(ℝ) × (T (X) ⊕ f^{∗}ξ_{−} ⊕ ε_{X}(U)) → ε_{I}(ℝ) × (f^{∗}ξ_{+} ⊕ ε_{X}(U))

over I × X to a G-vector bundle isomorphism

B′ : T (W) ⊕ (p_{Y} ∘ F)^{∗}ξ_{−} ⊕ ε_{W} (U) → (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ_{+}) ⊕ ε_{W} (U)

over W lies in π_{k−1}(SO(k + 1 + m)), where m is a certain nonnegative integer. The flexibility we have for choosing the coordinate diffeomorphism H : S^{k−1} ×D^{k+1} → \overline{N}(∂D) above, allows us to control the element σ_{H}. Let ω : S^{k−1} → SO(k + 1) be a smooth map. Let H^{ω} denote the diffeomorphism S^{k−1} × D^{k+1} → \overline{N}(∂D) defined by H^{ω}(x, y) = H(x, ω(x)y), where x ∈ S^{k−1} and y ∈ D^{k+1}. Then we have the formula σ_{H^{ω}} = σ_{H} +[ω] for the obstructions, where [ω] ∈ π_{k−1}(SO(k+1+m)) is the homotopy class of the composition of ω : S^{k−1} → SO(k + 1) and the canonical map SO(k + 1) → SO(k + 1 + m). Since the homomorphism π_{k−1}(SO(k + 1)) → π_{k−1}(SO(k + 1 + m)) induced by the canonical map is an isomorphism, we can choose H such that σ_{H} = 0. In other words, we can choose H such that id_{ε_{I}(ℝ)} × b extends to B′ (cf. the proof of Corollary to Theorem 1.1 in [44, Part 0, p.10]). Define f′ : X′ → Y by F (u) = (1, f′(u)) (u ∈ X′). By Lemma 6.1, B′ is regularly G-homotopic rel (I × (∂X ∪ L)) ∪ ({0} × X) to B : T (W) ⊕ (p_{Y} ∘ F)^{∗}ξ_{−} ⊕ ε_{W} (U) → (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ_{+}) ⊕ ε_{W} (U) such that B|_{X′} = id_{ε_{X′}(ℝ)} ⊕ b′ for some b′ : T (X′) ⊕ f′^{∗}ξ_{−} ⊕ ε_{X′}(U) → f′^{∗}ξ_{+} ⊕ ε_{X′}(U). Therefore (F, B) is a G-framed cobordism rel ∂X ∪ L from fff = (f, b) to fff′ = (f′, b′). The map fff′ is called the G-framed map obtained from fff by G-surgery along h_{ccc} (or more precisely along H).

Step 3. The step takes place in 3 parts. Part 1 constructs a pair (e_{1}, f_{1}) of elements in K_{k}(X′; R) such that B_{f′,R}(e_{1}, e_{1}) = 0, B_{f′,R}(f_{1}, e_{1}) = 1 and q_{fff′,R}(e_{1}) = q_{fff′,R}(f_{1}) = 0. Suppose this has been done. Let M(e_{1}, f_{1}) denote the R[G]-submodule of K_{k}(X′; R) generated by e_{1} and f_{1}. M(e_{1}, f_{1}) is an R[G]-free module with basis {e_{1}, f_{1}} and the canonically induced sequence

0 → M(e_{1}, f_{1})^{⊥} → K_{k}(X′; R) ⟶^{B_{f′,R}(•,−)} Hom_{R[G]}(M(e_{1}, f_{1}), R[G]) → 0

of R[G]-modules is split exact, where

M(e_{1}, f_{1})^{⊥} = {x ∈ K_{k}(X′; R) | B_{f′,R}(x, y) = 0 for all y ∈ M(e_{1}, f_{1})}.

Clearly K_{k}(X′; R) = M(e_{1}, f_{1})^{⊥} ⊕ M(e_{1}, f_{1}) and the restriction of B_{f′,R} to M(e_{1}, f_{1}) is nonsingular. Let N N N(e_{1}, f_{1}) denote the doubly parametrized module

(M(e_{1}, f_{1}), B_{f′,R}|_{M(e_{1},f_{1})}, q_{fff′,R}|_{M(e_{1},f_{1})}).

(K_{k}(X′; R), B_{f′,R}, q_{fff′,R}) obviously splits as the orthogonal sum (K_{k}(X′; R), B_{f′,R}, q_{fff′,R}) = N N N(e_{1}, f_{1})^{⊥}⊕N N N(e_{1}, f_{1}), where N N N(e_{1}, f_{1})^{⊥} = (M(e_{1}, f_{1})^{⊥}, B_{f′,R}|_{M(e_{1},f_{1})^{⊥}}, q_{fff′,R}|_{M(e_{1},f_{1})^{⊥}}). Accordingly the maps θ_{f′,𝔹} : Θ̃_{𝔹} → K_{k}(X′; R) and θ_{2,f′,𝔹} : Θ_{2,𝔹} → K_{k}(X′; R_{2}) split canonically as direct sums θ_{f′,𝔹} = θ_{M^{⊥}} ⊕ θ_{M} and θ_{2,f′,𝔹} = θ_{2,M^{⊥}} ⊕ θ_{2,M} where θ_{M^{⊥}} : Θ̃_{𝔹} → M(e_{1}, f_{1})^{⊥}, θ_{M} : Θ̃_{𝔹} → M(e_{1}, f_{1}), θ_{2,M^{⊥}} : Θ_{2,𝔹} → M(e_{1}, f_{1})_{2}^{⊥}, and θ_{2,M} : Θ_{2,𝔹} → M(e_{1}, f_{1})_{2}. Let M M M(e_{1}, f_{1}) = (N N N(e_{1}, f_{1}), θ_{M}, θ_{2,M}), and let M M M(e_{1}, f_{1})^{⊥} = (N N N(e_{1}, f_{1})^{⊥}, θ_{M}^{⊥}, θ_{2,M}^{⊥}). By construction, M M M_{fff′,𝔹} = M M M(e_{1}, f_{1})^{⊥}⊕M M M(e_{1}, f_{1}). Part 2 shows that M M M(e_{1}, f_{1}) ≅ 𝕄(R[G],ccc, c̃). Part 3 shows that f′ is k-connected. Part 4 shows that M M M(e_{1}, f_{1})^{⊥} ≅ M M M_{fff,𝔹}.

We supply now details of the above.

Part 1. Let

S_{1} = {1} × {x_{0}} × S^{k} (⊂ {1} × S^{k−1} × S^{k} ⊂ G × D^{k} × D^{k+1} ⊂ X′) .

Using a parallel translation of {1} × D (⊂ {1} × X), we can find a k-dimensional closed disk D′ embedded in

{1} × (X ∖ G · Interior(Image(H)))

such that

∂D′ = {1} × S^{k−1} × {x′_{0}} (⊂ G × D^{k} × D^{k+1} ⊂ X′)

for some x′_{0} ∈ S^{k}. Let

S_{2} = {1} × D^{k} × {x′_{0}} ∪_{{1}×S^{k−1}×{x′_{0}}} D′ (⊂ X′) .

Let h_{S_{i}} : S_{i} → X′, i = 1, 2, denote the inclusion maps. We shall use the h_{S_{i}}, i = 1, 2, to construct e_{1} and f_{1}. Give S_{2} the orientation which agrees with that on D′ and S_{1} the orientation such that #(h_{S_{2}}, h_{S_{1}}) = 1, i.e. #(h_{S_{1}}, h_{S_{2}}) = (−1)^{k}. Obviously, #_{G}(h_{S_{1}}, h_{S_{1}}) = 0, #_{G}(h_{S_{2}}, h_{S_{1}}) = 1, and ♮_{G}(h_{S_{1}}) = 0. Since S_{2} is an embedded sphere with trivial normal bundle, ♮_{1}(h_{S_{2}}) = 0. If

g ∈ G ∖ ⋃_{t∈Θ_{2,𝔹}: c̃(t)≠0} ρ_{G}(t),

then ♮_{g}(h_{S_{2}}) = 0. Moreover using Lemma 9.1 of [6] (as in the proof of Theorem 8.1 of [6, §9]), we can choose D so that for g ∈ S(G, X),

#(h_{S_{2}}, gh_{S_{2}}) = ∑_{t∈Θ_{2,𝔹}|_{g}}(−1)^{k}c̃(t).

Let [S_{i}] ∈ H_{k}(S_{i}; ℤ) denote the orientation class of S_{i}. Let e_{1} = h_{S_{1}}_{∗}[S_{1}] and h_{2} = h_{S_{2}}_{∗}[S_{2}] ∈ H_{k}(X′; ℤ). Then e_{1}, h_{2} ∈ K_{k}(X′; ℤ) (⫅ K_{k}(X′; R)). It follows from Lemma 4.6 and the values #_{G}(h_{S_{1}}, h_{S_{1}}) = 0 and #_{G}(h_{S_{2}}, h_{S_{1}}) = 1 above that B_{f′,R}(e_{1}, e_{1}) = 0 and B_{f′,R}(h_{2}, e_{1}) = 1. We will show that f′ : X′ → Y is k-connected. Once this has been shown, we obtain a quadratic map q_{fff′,R} : K_{k}(X′; R) → R[G]/RΛ(Q(G, X)+R[S(G, X)])) by Definition 5.7. By definition, q_{fff′,R}(e_{1}) = ♮_{G}(h_{S_{1}}) and q_{fff′,R}(h_{2}) = ♮_{G}(h_{S_{2}}). Thus from the values ♮_{G}(h_{S_{1}}) = 0 and ♮_{g}(h_{S_{2}}) = 0 above, it follows that q_{fff′,R}(e_{1}) = 0 and q_{fff′,R}(h_{2})_{g} = 0 (the g-th coefficient of q_{fff′,R}(h_{2})) if g ∈ {1} ∪ (G(2) ∖ (Q(G, X) ∪ S(G, X))) (see Definition 5.7). Choose v ∈ ℤ[G ∖ ({1} ∪ G(2))] such that after setting f_{1} = h_{2} + ve_{1}, we get q_{fff′,R}(f_{1}) = 0. We maintain that B_{f′,R}(f_{1}, e_{1}) = 1 and obtain additionally that

B_{f′,R}(f_{1}, f_{1}) = ∑_{g∈S(G,X)} ∑_{t∈Θ_{2,𝔹}|_{g}}(−1)^{k}c̃(t)g.

Part 2. By Lemma 3.31, 𝕄(R[G],ccc, c̃) ≅ 𝕄(R[G], (−1)^{k}ccc, (−1)^{k}c̃). We shall show

M M M(e_{1}, f_{1}) = (M(e_{1}, f_{1}), B_{f′,R}|_{M(e_{1},f_{1})}, q_{fff′,R}|_{M(e_{1},f_{1})}, θ_{M}, θ_{2,M})

is isomorphic to 𝕄(R[G], (−1)^{k}ccc, (−1)^{k}c̃). According to Definition 3.28, we must show that

(6.2.2)  \begin{matrix}B_{f′,R}(e_{1}, e_{1}) = 0, B_{f′,R}(f_{1}, e_{1}) = 1, \\ B_{f′,R}(f_{1}, f_{1}) = ∑_{g∈S(G,X)} ∑_{t∈Θ_{2,𝔹}|_{g}} (−1)^{k}c̃(t)g, \\ q_{fff′,R}(e_{1}) = q_{fff′,R}(f_{1}) = 0, \end{matrix}

and

(6.2.3)  θ_{M} = sr((−1)^{k}c), θ_{2,M} = sr((−1)^{k}c_{2}).

The property (6.2.2) has been demonstrated already in Part 1.

Clearly θ_{M}(β) = B_{fff′,R}(f_{1}, (h_{β}_{∗}[β])_{K_{k}(X′;R)}) for β ∈ Θ̃_{𝔹} (cf. Definition 5.2). But

B_{f′,R}(f_{1}, (h_{β}_{∗}[β])_{K_{k}(X′;R)}) = B_{f′,R}(h_{2}, (h_{β}_{∗}[β])_{K_{k}(X′;R)})

= ∑_{g∈G} Int(h_{S_{2}}, g^{−1}h_{β})g

= ∑_{g∈G} Int(h_{S_{2}}, h_{g^{−1}β})g

= ∑_{g∈G}(−1)^{k}c(g^{−1}β)g

= sr((−1)^{k}c)(β).

Similarly, one shows that θ_{M_{2}} = sr((−1)^{k}c_{2}).

Part 3. We show first that f′ : X′ → Y is k-connected. We begin by simplifying the Mayer-Vietoris exact sequences (6.2.4)

\xymatrix{⋯ \ar[r] & K_{i+1}(W, X; ℤ) \ar[r]^{∂} & K_{i}(X; ℤ) \ar[r]^{^{i_{X}^{∗}}} & K_{i}(W; ℤ) \ar[r]^{τ} & K_{i}(W, X; ℤ) \ar[r] & ⋯}

and (6.2.5)

\xymatrix{⋯ \ar[r] & K_{i+1}(W, X′; ℤ) \ar[r]^{∂′} & K_{i}(X′; ℤ) \ar[r]^{^{i_{X′}^{∗}}} & K_{i}(W; ℤ) \ar[r]^{τ′} & K_{i}(W, X′; ℤ) \ar[r] & ⋯}

for the pairs (W, X) and (W, X′), respectively.

By definition, K_{i}(W, X; ℤ) = Ker[F_{∗} : H_{i}(W, X; ℤ) → H_{i}(I × Y, Y; ℤ)]. Since H_{i}(I × Y, Y; ℤ) = 0, it follows that K_{i}(W, X; ℤ) = H_{i}(W, X; ℤ). Since the quotient space W/X is G-homotopy equivalent to (G × S^{k})/(G × {pt}), we obtain

(6.2.6)  K_{i}(W, X; ℤ) ≅ \begin{cases}ℤ[G], & if i = k \\ 0, & if i ≠ k. \end{cases}

Let h_{2} ∈ K_{k}(X; ℤ) be as in Part 1. Set h′_{2} = i_{X′}_{∗}(h_{2}) and h′′_{2} = τ(h′_{2}). From the definition of h_{2}, we obtain that

(6.2.7)  K_{k}(W, X; ℤ) = ℤ[G]h′′_{2}, a free ℤ[G]-module generated by h′′_{2}.

Since f is k-connected, it follows that K_{i}(X; ℤ) = 0 if i < k. Thus the exact sequence for (W, X) above simplifies to a split exact sequence

(6.2.8)  \xymatrix{0 \ar[r] & K_{k}(X; ℤ) \ar[r]^{^{i_{X}^{∗}}} & K_{k}(W; ℤ) \ar[r]^{τ} & K_{k}(W, X; ℤ) \ar[r] & 0}

and canonical identifications

(6.2.9)  K_{j}(X; ℤ) = K_{j}(W; ℤ) for j ≠ k.

As above, K_{i}(W, X′; ℤ) = H_{i}(W, X′; ℤ). Since W/X′ is G-homotopy equivalent to (G × S^{k+1})/(G × {pt}), we obtain

(6.2.10)  K_{i}(W, X′; ℤ) ≅ \begin{cases}ℤ[G], & if i = k + 1 \\ 0, & if i ≠ k + 1. \end{cases}

Thus the exact sequence for (W, X′) above simplifies to an exact sequence

(6.2.11)  \xymatrix{⋯ \ar[r] & K_{k+1}(W, X′; ℤ) \ar[r]^{∂′} & K_{k}(X′; ℤ) \ar[r]^{^{i_{X′}^{∗}}} & K_{k}(W; ℤ) \ar[r]^{τ′} & 0}

and

(6.2.12)  K_{j}(X′; ℤ) = K_{j}(W; ℤ) = 0 j < k.

From this fact and the fact that X′ and Y are simply connected, it follows that f′ : X′ → Y is k-connected. Additionally, we obtain

(6.2.13)  K_{k+1}(W, X′; ℤ) = ℤ[G]e′_{1}, a ℤ[G]-free module generated by e′_{1},

for an unique element e′_{1} ∈ K_{k+1}(W, X′; ℤ) such that ∂′(e′_{1}) = e_{1}.

Part 4. Let φ = Ind_{{1}}^{G}H : G × S^{k−1} × D^{k+1} → X be as in Part 1. Let

X_{0} = X ∖ Interior(Image(φ)).

Let i_{0} : X_{0} → X and i_{1} : X_{0} (= {1} × X_{0}) → X′ denote the canonical inclusions. Let i_{X} : X → W and i_{X′} : X′ → W denote also the canonical inclusions. Let

K_{k}(X_{0}; ℤ) = Ker[(f ∘ i_{0})_{∗} : H_{k}(X_{0}; ℤ) → H(Y; ℤ)].

Clearly i_{X} ∘ i_{0} is homotopic (in fact, G-isotopic) to i_{X′} ∘ i_{1} and hence f ∘ i_{0} is homotopic to f′ ∘ i_{1}. Thus K_{k}(X_{0}; ℤ) = Ker[(f′ ∘ i_{1})_{∗} : H_{k}(X_{0}; ℤ) → H(Y; ℤ)] and the diagram

(6.2.14)  \xymatrix{K_{k}(X_{0}; ℤ) \ar[d]_{_{i_{1}_{∗}}} \ar[r]^{^{i_{0}^{∗}}} & K_{k}(X; ℤ) \ar[d]^{_{i_{X}_{∗}}} \\ K_{k}(X′; ℤ) \ar[r]_{_{i_{X′}_{∗}}} & K_{k}(W; ℤ)}

of ℤ[G]-modules commutes. Moreover the map i_{0}_{∗} is surjective, because i_{0} is k-connected.

Since ℤ ⫅ R ⫅ ℚ, R is a ring of fractions of ℤ. Thus the functor M ↦ R ⊗ M, where M is a ℤ-module, is exact. Thus we can canonically identify the modules R ⊗ H_{j}(V; ℤ) = H_{j}(V; R) for any topological space V.

By applying the functor R ⊗ − to (6.2.14), we obtain a commutative diagram

\xymatrix{K_{k}(X_{0}; R) \ar[d]_{_{i_{1}_{∗}}} \ar[r]^{^{i_{0}^{∗}}} & K_{k}(X; R) \ar[d]^{_{i_{X}_{∗}}} \\ K_{k}(X′; R) \ar[r]_{_{i_{X′}_{∗}}} & K_{k}(W; R)}

of R[G]-modules such that i_{0}_{∗} is surjective. By Lemma 5.13, K_{k}(X; R) is R[G]-projective. Thus there is an R[G]-submodule K_{R} ⫅ K_{k}(X_{0}; R), which maps isomorphically under i_{0}_{∗} onto K_{k}(X; R). Let j_{0} = i_{0}_{∗}|_{K_{R}} : K_{R} → K_{k}(X; R), j_{1} = i_{1}_{∗}|_{K_{R}} : K_{R} → K_{k}(X′; R), and

σ = j_{1} ∘ j_{0}^{−1} : K_{k}(X; R) → K_{k}(X′; R).

From Part 1, we know that K_{k}(X′; R) has an orthogonal decomposition

(K_{k}(X′; R), B_{f′,R}, q_{fff′,R}) = N N N(e_{1}, f_{1})^{⊥} ⊕ N N N(e_{1}, f_{1}).

Let p^{⊥} denote the projection of K_{k}(X′; R) on M(e_{1}, f_{1}). We shall show that the R[G]- homomorphism p^{⊥} ∘ σ : K_{k}(X; R) → M(e_{1}, f_{1})^{⊥} is an isomorphism M M M_{fff,𝔹} → M M M(e_{1}, f_{1})^{⊥} of doubly parametrized modules with positioning data.

First we show that σ : K_{k}(X; R) → K_{k}(X′; R) is a morphism (K_{k}(X; R), B_{f,R}, q_{fff,R}) → (K_{k}(X′; R), B_{f′,R}, q_{fff′,R}) of doubly parametrized modules.

Let K′_{ℤ} = K_{R} ∩ Image[K_{k}(X_{0}; ℤ) → K_{k}(X_{0}; R)]. K′_{ℤ} is a ℤ[G]-submodule of K_{k}(X_{0}; R) such that R · K′_{ℤ} = K_{R}. It suffices to show that the map σ|_{j_{0}(K′_{ℤ})} preserves Hermitian and quadratic forms. Let K_{ℤ} denote the preimage in K_{k}(X_{0}; ℤ) of K′_{ℤ}. The groups and maps above fit into a commutative diagram

(6.2.15) [figure: K_{k}(X_{0}; ℤ); 1⊗−; loc. epi.; K_{k}(X_{0}; R); _{i_{1}_{∗}}; _{i_{0}_{∗}}; K_{ℤ}; 1⊗−; K′_{ℤ}; loc. epi.; K_{R}; j_{0}; ≅; j_{1}; K_{k}(X; R); _{i_{X}_{∗}}; σ; K_{k}(X′; R); _{i_{X′}_{∗}}; K_{k}(W; R).]

Let x ∈ K_{ℤ}. Then x is represented by an immersion h_{x} : S^{k} → Interior(X_{0}) with trivial normal bundle. Suppose y ∈ K_{ℤ}. By definition, x′ = 1 ⊗ x, y′ = 1 ⊗ y ∈ K′_{ℤ}. Clearly h_{x} (resp. h_{y}) represents also j_{0}(x′) and j_{1}(x′) (resp. j_{0}(y′) and j_{1}(y′)). By Proposition 5.8, B_{f,R}(j_{0}(x′), j_{0}(y′)) = #_{G}(h_{x}, h_{y}) = B_{f′,R}(j_{1}(x′), j_{1}(y′)) = B_{f′,R}(σ(j_{0}(x′)), σ(j_{0}(y′))) and q_{fff,R}(j_{0}(x′)) = [♮_{G}(h_{x})] = q_{fff′,R}(τ_{1}(j_{1}(x′))) = q_{fff′,R}(σ(j_{0}(x′))).

Next we show that σ(K_{k}(X; R)) ⫅ M(e_{1}, f_{1}) ⊕ R[G]e_{1}. Recall that e_{1} itself is totally isotropic, i.e. B_{f,R}(e_{1}, e_{1}) = 0 and q_{fff,R}(e_{1}) = 0, and B_{f,R}(f_{1}, e_{1}) = 0. Obviously

(R[G]e_{1})^{⊥} = {u ∈ K_{k}(X′; R) | B_{f′,R}(u, e_{1}) = 0}

coincides with M(e_{1}, f_{1})^{⊥} ⊕ R[G]e_{1}. By Diagram (6.2.15), it suffices to show that i_{1}((1 ⊗ −)(K_{k}(X_{0}, Z))) ⫅ (R[G]e_{1})^{⊥}. Suppose this has been done. The map p^{⊥}|_{M(e_{1},f_{1})^{⊥}⊕R[G]e_{1}} : M(e_{1}, f_{1})^{⊥} ⊕ R[G]e_{1} → M(e_{1}, f_{1})^{⊥} preserves Hermitian and quadratic forms, because e_{1} is orthogonal to M(e_{1}, f_{1})^{⊥} by definition and e_{1} itself is totally isotropic. Thus the map p^{⊥} ∘ σ preserves Hermitian and quadrqatic forms and therefore p^{⊥} ∘ σ is a morphism (K_{k}(X; R), B_{f,R}, q_{fff,R}) → N N N(e_{1}, f_{1})^{⊥} of doubly parametrized modules. Suppose x ∈ K_{k}(X_{0}; ℤ). Then x is represented by an immersion h_{x} : S^{k} → Interior(X_{0}) with trivial normal bundle. On the other hand e_{1} is represented by the embedding h_{S_{1}} appearing in Part 1. By Proposition 5.8, B_{f′,R}(i_{1}_{∗}(1 ⊗ x), e_{1}) = #_{G}(h_{x}, h_{S_{1}}). But #_{G}(h_{x}, h_{S_{1}}) = 0 because Image(h_{x}) ∩ GImage(h_{S_{1}}) = ∅. Thus i_{1}_{∗}(1 ⊗ x) ∈ (R[G]e_{1})^{⊥}.

Next we show that p^{⊥} ∘ σ : (K_{k}(X; R), B_{f,R}, q_{fff,R}) → N(e_{1}, f_{1})^{⊥} is an isomorphism of doubly parametrized modules. Since B_{f,R} is nonsingular and p^{⊥} ∘ σ preserves Hermitian forms, it follows from [2, Lemma 2.3] that p^{⊥} ∘ σ is injective and M(e_{1}, f_{1})^{⊥} splits as an orthogonal sum Image(p^{⊥} ∘ σ) ⊕ M′. Thus it suffices to show that rank_{R}(K_{k}(X; R)) = rank_{R}(M_{(}e_{1}, f_{1})^{⊥}). We know K_{k}(X′; R) = M(e_{1}, f_{1})^{⊥} ⊕ M(e_{1}, f_{1}). Therefore it suffices to show

(6.2.16)  K_{k}(X′; R) ≅ K_{k}(X; R) ⊕ R[G] ⊕ R[G].

Consider the diagram (6.2.17)

[figure: ⋯; K_{k}(X; R); σ; ^{i_{X}^{∗}}; K_{k}(W; R); τ; K_{k}(W, X; R); ⋯; ⋯; K_{k+1}(W, X′; R); ∂′; K_{k}(X′; R); _{i_{X′}_{∗}}; K_{k}(W; R); ⋯]

afforded by (6.2.4) and (6.2.5). We compute first the top row and then the bottom row. This will establish (6.2.14) above.

The following is a continuation of the computation in Part 3. It follows from (6.2.6) and (6.2.7) that

(6.2.18)  K_{j}(W, X; R) = \begin{cases}R[G]h′′_{2}, & if j = k \\ 0, & if j ≠ k. \end{cases}

By (6.2.10)–(6.2.13) and Poincaré-Lefschetz duality, we obtain

(6.2.19)  K_{j}(W, X′; R) = \begin{cases}R[G]e′_{1}, & if j = k + 1 \\ 0, & if j ≠ k + 1, \end{cases}

where e′_{1} is an element in K_{k+1}(W, X′; R) such that ∂′(e′_{1}) = e_{1}. Thus Diagram (6.2.17) simplifies to a commutative diagram

(6.2.20)  [figure: K_{k}(X; R); σ; ^{i_{X}^{∗}}; K_{k}(W; R); τ; R[G]h′′_{2}; R[G]e_{1}; K_{k}(X′; R); _{i_{X′}_{∗}}; K_{k}(W; R)]

Since K_{k}(X; R) is R[G]-projective, so is K_{k}(W; R). Thus the bottom row is split exact as well as the top row. We obtain K_{k}(X′; R) ≅ K_{k}(X; R) ⊕ R[G] ⊕ R[G], in fact

(6.2.21)  K_{k}(X′; R) = σ(K_{k}(X; R)) ⊕ R[G]e_{1} ⊕ R[G]h_{2}.

We prove next that p^{⊥} ∘ σ preserves positioning functions. This will complete the proof of the theorem.

Let β ∈ Θ̃. We must show p^{⊥}(σ(θ_{f,𝔹}(β))) = θ_{M^{⊥}}(β). By definition, θ_{M^{⊥}}(β) = p^{⊥}(θ_{f′,𝔹}(β)). Thus it suffices to show p^{⊥}(σ(θ_{f,𝔹}(β))) = p^{⊥}(θ_{f′,𝔹}(β)). Let p_{1} denote the projection M(e_{1}, f_{1})^{⊥} ⊕ M(e_{1}, f_{1}) → M(e_{1}, f_{1})^{⊥} ⊕ R[G]f_{1} and p_{2} denote the projection M(e_{1}, f_{1})^{⊥} ⊕ R[G]f_{1} → M(e_{1}, f_{1})^{⊥}. Clearly p^{⊥} = p_{2} ∘ p_{1}. It suffices to show p_{1}(σ(θ_{f,𝔹}(β))) = p_{1}(θ_{f′,𝔹}(β)). By definition, θ_{f,𝔹}(β) = (h_{β}_{∗}[β]_{R})_{K}. Thus i_{X}_{∗}(θ_{f,𝔹}(β)) = i_{X′}_{∗}(θ_{f′,𝔹}(β)). From i_{X′}_{∗} ∘ σ = i_{X}_{∗} (cf. (6.2.15)) and the exactness of the bottom row in (6.2.20), it follows that p_{1}(σ(θ_{f,𝔹}(β))) = p_{1}(θ_{f′,𝔹}(β)).

Replacing Θ̃_{f,𝔹} above by Θ_{2,f,𝔹} and applying ℤ_{2} ⊗ − to the argument above, we obtain that p^{⊥} ∘ σ preserves mod 2 positioning functions. (Note that R_{2} := ℤ_{2} ⊗ R = ℤ_{2} if 1/2 ∉ R and R_{2} = 0 if 1/2 ∈ R.) Thus p^{⊥} ∘ σ preserves positioning data. □

7. G-framed Θ Θ Θ-cobordism invariance

The surgery obstruction in Wall’s surgery theory is a framed-cobordism invariant. Similar results are expected in equivariant surgery theory. We prove in this section that our surgery obstruction is a G-framed Θ Θ Θ-cobordism invariant under hypotheses on the singular set and the boundary. Theorem 1.1 in the introduction follows immediately from the last result in the section, Theorem 7.8. The proof of this theorem provides the construction of the surgery obstruction element σ(fff) required in the statement of Theorem 1.1.

We introduce first the notion G-framed Θ Θ Θ-cobordism. Let fff = (f, b) : (X, ∂X, T X) → (Y, ∂Y, f^{∗}ξ) and fff′ = (f′, b′) : (X′, ∂X′, T X′) → (Y, ∂Y, f′^{∗}ξ) denote G-framed maps, and

F F F = (F, B) : (W, ∂_{+}W, ∂_{−}W, T W) → (I × Y, ∂I × Y, I × ∂Y, (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ))

a G-framed cobordism between fff and fff′. Let (L, 𝔹), 𝔹 = {h_{β} : B_{β} → X | β ∈ B}, and (L′, 𝔹′), 𝔹′ = {h′_{β} : B′_{β} → X′ | β ∈ B′}, denote G-singularity structures for X and X′, respectively. We shall assume throughout B = B′ and B_{+,L} = B_{+,L′} as G × {±1}-sets (see Definition 5.1). To simplify notation, we shall use the abbreviations B_{+} = B_{+,L}(= B_{+,L′}), Θ̃ = Θ̃_{𝔹}(= Θ̃_{𝔹′}), Θ_{2} = Θ_{2,𝔹}(= Θ_{2,𝔹′}), and Θ Θ Θ = Θ Θ Θ_{𝔹}(= Θ Θ Θ_{𝔹′}). We remind the reader that the G-framed cobordism (F, B) above is called trivial, if W is just a cylinder, i.e. W = I ×X. If (F, B) is trivial then f is G-homotopic to f′ and b is regularly G-homotopic to b′.

Definition 7.1. Let W be a G-cobordism between X and X′. A Θ Θ Θ-cobordism on W between (L, 𝔹) and (L′, 𝔹′) is a pair (L̂, 𝔹̂) consisting of a G-simplicial subcomplex L̂ of W and a set

𝔹̂ ={inclusion maps H_{β} : (B̂_{β}, ∂_{+}B̂_{β}, ∂_{−}B̂_{β}) → (W, ∂_{+}W, ∂_{−}W)
of submanifolds B̂_{β} ⊂ W | β ∈ B}

satisfying the following conditions. (7.1.1) L̂ ⫆ Sing(G, W). (7.1.2) L̂ = L̂^{(k)} ∪ ⋃_{β∈B} B̂_{β}, where L̂^{(k)} is the k-skeleton of L̂.

(7.1.3) For each β ∈ B, B̂_{β} is a (k + 1)-dimensional connected submanifold of W. (7.1.4) If β ∈ B ∖ B_{+} (resp. B_{+}) then B̂_{β} is a cobordism (resp. oriented cobordism)

between B_{β} and B′_{β} (hence ∂B̂_{β} = ∂_{+}B̂_{β} ∪∂_{−}B̂_{β}, ∂_{+}B̂_{β} = B_{β} ⨿B′_{β}, ∂_{+}B̂_{β} ∩∂_{−}B̂_{β} = ∂B_{β} ⨿ ∂B′_{β}).

(7.1.5) H_{β}|_{∂_{+}B̂_{β}} = h_{β} ∪ h′_{β}. (7.1.6) For each β ∈ B, L̂^{(k)} ∩ B̂_{β} ⫅ L̂^{(k−1)}. (7.1.7) For every β, β′ ∈ B, B̂_{β} ∩ B̂_{β′} is a submanifold of W. (7.1.8) For every β, β′ ∈ B, if B̂_{β} ≠ B̂_{β′} as subsets of W then B̂_{β} ∩ B̂_{β′} ⫅ L̂^{(k−1)}. (7.1.9) For all g ∈ G and β ∈ B, B̂_{gβ} = gB̂_{β}. (Hence the map g : W → W has the

well-defined restriction g : B̂_{β} → B̂_{gβ}.) (7.1.10) For all β ∈ B_{+}, B̂_{(−1)β} = B̂_{β} as subsets of W (hence H_{β} = H_{(−1)β}) and [B̂_{(−1)β}, ∂B̂_{(−1)β}] =

−[B̂_{β}, ∂B̂_{β}]. (7.1.11) For all g ∈ G and β ∈ B_{+}, [B̂_{gβ}, ∂B̂_{gβ}] = g[B̂_{β}, ∂B̂_{β}].

A pair (F F F , (L̂, 𝔹̂)) is called a G-framed Θ Θ Θ-cobordism between (fff, (L, 𝔹)) and (fff′, (L′, 𝔹′)) if F F F = (F, B) is a G-framed cobordism between fff = (f, b) and fff′ = (f′, b′), and (L̂, 𝔹̂) is a Θ Θ Θ-cobordism on W between (L, 𝔹) and (L′, 𝔹′).

For the remainder of the section, let (F F F , (L̂, 𝔹̂)) denote a G-framed Θ Θ Θ-cobordism between (fff, (L, 𝔹)) and (fff′, (L′, 𝔹′)). In addition, let A = R[G], λ = (−1)^{k},

A A A = (A, (−, λ), RΓ(S(G, X)), G, R[S(G, X)], RΛ(Q(G, X)) + R[S(G, X)]),

and Γ′_{2} = min_{−λ}(R_{2}[G]). Γ′_{2} defines ∇ = ∇_{𝔹} (cf. Definition 3.18, Definition 3.33 and the notation prior to Lemma 5.10).

Lemma 7.2. One can perform G-surgery on F F F along spheres of dimension ≦ k − 1 in Interior(W ∖ L̂) to obtain a G-framed Θ Θ Θ-cobordism (F F F′, (L̂, 𝔹̂)), F F F′ = (F′, B′), between (fff, (L, 𝔹)) and (fff′, (L′, 𝔹′)) such that the resulting map F′ : W′ → I × Y is k-connected.

Proof. This is clear from the observation

dim L̂ + (k − 1) < dim W (for any β ∈ B).
□

Recall the assumption ℤ ⫅ R ⫅ ℚ. Suppose fff and fff′ are k-connected, R-boundary quasiequivalences, F is k-connected, and ∂_{−}F : ∂_{−}W → I × ∂Y is an R-homology equivalence. Then, the sequence

0 → K_{k+1}(W) → K_{k+1}(W, ∂W) → K_{k}(∂W) → K_{k}(W) → K_{k}(W, ∂W) → 0

is exact, where K_{ℓ}(W, ∂W) = Ker[F_{∗} : H_{ℓ}(W, ∂W; R) → H_{ℓ}(I × Y, ∂(I × Y); R)]

K_{ℓ}(W) = Ker[F_{∗} : H_{ℓ}(W; R) → H_{ℓ}(I × Y; R)], etc.

Lemma 7.3. Suppose f and f′ are k-connected, P(G)_{R}-singularity, R-boundary quasiequivalences, ∂_{−}F = F |_{∂_{−}W} : ∂_{−}W → I × ∂Y is an R-homology equivalence, and F is kconnected. Then F F F is G-framed cobordant rel X ∪ ∂_{−}W ∪ L̂ to F F F′ = (F′, B′), where F′ : (W′, ∂_{+}W′, ∂_{−}W′) → (I × Y, ∂I × Y, I × ∂Y), satisfying the following properties.

(7.3.1) (F F F′, (L̂, 𝔹̂)) is a G-framed Θ Θ Θ-cobordism between (fff, (L, 𝔹)) and (fff′′, (L′, 𝔹′)) such that fff′′ = (f′′, b′′) : (X′′, ∂X′′, T X′′) → (Y, ∂Y, f′′^{∗}ξ) is a k-connected G-framed map.

(7.3.2) F′ : W′ → I × Y is k-connected and K_{k}(W′, ∂W′; R) = 0 = K_{k+1}(W′; R). (7.3.3) M M M_{fff′′,𝔹′} ≅ M M M_{fff′,𝔹′} ⊕ 𝕄(R[G],ccc_{1}) ⊕ ⋯ ⊕ 𝕄(R[G],ccc_{ℓ}) in ∇Q Q Q(A A A,Θ Θ Θ) for some ccc_{i} : Θ Θ Θ →

(ℤ, ℤ_{2}), i = 1, ⋯, ℓ, (cf. Definitions 3.28 and 3.30).

Proof. Fix a small, closed, n-dimensional disk D_{0} in general position in Interior(X′ ∖ L′). Thus D_{0} ∩ gD_{0} = ∅ if g ∈ G ∖ {1}. Let t ∈ Θ_{2}. Take a point x_{t} ∈ B′_{β(t)} such that G_{x_{t}} = ρ_{G}(t). Further choose small, closed, n-dimensional, G_{x_{t}}-invariant disks D_{t} with center x_{t} in X′ ∖ GD_{0} such that gD_{t} ∩ g′D_{t′} = ∅ unless t = t′ and gG_{x_{t}}g^{−1} = g′G_{x_{t′}}g′^{−1}, where t, t′ ∈ Θ_{2}, and g, g′ ∈ G. Thus each GD_{t} is a G-equivariant tubular neighborhood of Gx_{t}. For each t, take a connecting tube T_{t} from ∂D_{0} to ∂D_{t} in general position in (X′ ∖ L′) ∖ ⋃_{t∈Θ_{2}}G(Interior(D_{t})). Then,

D = D_{0} ∪ ⋃_{t∈Θ_{2}} (D_{t} ∪ T_{t})

is homeomorphic to the n-dimensional disk D^{n}. Since K_{k}(W) → K_{k}(W, ∂W) is surjective, we can take finitely many embeddings with trivial normal bundle, h_{i} : S^{k} → Interior(W), i = 1, ⋯, ℓ say, which together generate K_{k}(W, ∂W). Without loss of generality, we can assume gh_{i} and g′h_{j} are disjoint unless i = j. For each i, take a path p_{i} : [0, 1] → W ∖ L̂ in general position starting in Interior(D_{0}) and ending in Im(h_{i}) (i.e., p_{i}(0) ∈ Interior(D_{0}) and p_{i}(1) ∈ Im(h_{i})). Choose a thin band u_{i} : [0, 1] × D^{k} ↪ X along the path p_{i} (i.e., u_{i}(t, 0) = p_{i}(t) for all t ∈ [0, 1]) in general position such that u_{i}(0, D^{k}) ⊂ Interior(D_{0}) and u_{i}(1, D^{k}) ⊂ Image(h_{i}). Then taking the union of u_{i}([0, 1] × S^{k−1}) and Image(h_{i}) ∖ u_{i}(1, Interior(D^{k})), we obtain an embedding h′_{i} : (D^{k}, S^{k−1}) → (W, X′ ∖ L′). Here the restriction ∂h′_{i} : S^{k−1} → X′ ∖ L′ of h′_{i} is a trivial embedding close to p_{i}(0). We may suppose gh_{i} and g′h_{j} are disjoint unless i = j. For β ∈ Θ̃, let c_{i}(β) denote the ordinary intersection number in ℤ afforded by W_{β} and Image(h_{i}). Set c̃_{i}(t) = c_{i}(β(t)) and c_{2,i}(t) = [c_{i}(β(t))] ∈ ℤ_{2} for t ∈ p_{𝔹}(Θ̃), where β(t) ∈ Θ̃ denotes as usual a lifting of t. For t ∈ Θ_{2} ∖ p_{𝔹}(Θ̃), let c̃_{i}(t) denote the number (∈ ℤ) of intersection points of B̂_{t} with Image(h_{i}). Set c_{2,i}(t) = [c̃_{i}(t)] ∈ ℤ_{2}. Thus ccc_{i} = (c_{i}, c_{2,i}) : Θ Θ Θ → (ℤ, ℤ_{2}) and c̃_{i} : Θ_{2} → ℤ. Without loss of generality, we can assume that for t ∈ Θ_{2}, h_{i} meets with B̂_{β(t)} at m_{i,t}-points A(i, t, 1), ⋯, A(i, t, m_{i,t}), each having isotropy subgroup G_{x_{t}}, where m_{i,t} = |c̃_{i}(t)|. Take points B(i, t, 1), ⋯, B(i, t, m_{i,t}) in h′_{i}(S^{k−1}) and C(i, t, 1), ⋯, C(i, t, m_{i,t}) in D_{t} ∩ B′_{β(t)}. Let △(i, t, j) = △A(i, t, j)B(i, t, j)C(i, t, j) denote embedded triangles in W (each of which is homeomorphic to D^{2}) such that

\overline{A(i, t, j)B(i, t, j)} ⊂ Im(h′_{i}),
\overline{A(i, t, j)C(i, t, j)} ⊂ B̂_{β(t)}^{=ρ_{G}(t)}, and
\overline{B(i, t, j)C(i, t, j)} ⊂ Interior(D).

We may assume g(Interior(△(i, t, j))) ∩ g′(Interior(△(i′, t′, j′))) = ∅ unless i = i′, t = t′, j = j′ and g = g′. Moreover we may assume △(i, t, j) is perpendicular to B̂_{β(t)}, X′ and Image(h′_{i}). For each i, delete the intersection points A(i, t, j) of h′_{i} with B′_{β(t)} along the triangle △(i, t, j) (see Lemma 1.2 of [30]), where t ∈ Θ_{2} and j = 1, ⋯, m, and obtain an embedding h′′_{i} : (D^{k}, S^{k−1}) → (W ∖ L̂, X′ ∖ L′). It is remarkable that the restriction ∂h′′_{i} : S^{k−1} → X′ ∖ L′ of h′′_{i} is the connected sum of ∂h′_{i} and h_{ccc_{i}} (cf. (6.2.1)). Thickening h′′_{i}, we obtain embeddings

H′′_{i} : (D^{k} × D^{k+1}, S^{k−1} × D^{k+1}) → (W ∖ L̂, X′ ∖ L′),

i = 1, ⋯, ℓ. Now we may suppose gH′′_{i} and g′H′′_{j} are disjoint unless g = g′ and i = j. Further we can suppose F (Image(H′′_{i})) is a point in {1} × Y. Set

V = ⋃_{i=1}^{ℓ} G(Image(H′′_{i})),

∂_{+}V = ⋃_{i=1}^{ℓ} G(H′′_{i}(S^{k−1} × D^{k+1})),

W′ = Closure(W ∖ V),

X′′ = (X′ ∪ V) ∖ {⋃_{i=1}^{ℓ} G(H′′_{i}(D^{k} × (Interior(D^{k+1}))))} ,

F′ = F |_{W′} : W′ → I × Y,

B′ = B|_{W} : T (W′) → (p_{Y} ∘ F′)^{∗}(ε_{Y} (ℝ) ⊕ ξ),

f′′ = F |_{X′′} : X′′ → {1} × Y = Y.

Deforming B by a regular G-homotopy (cf. Lemma 6.1) if necessary, we can assume without loss of generality that B′|_{X′′} has the form

id_{X′′} ⊕ b′′ : ε_{X′′}(ℝ) ⊕ T (X′′) → ε_{X′′}(ℝ) ⊕ f′′^{∗}ξ.

We check that F F F′ = (F′, B′) and fff′′ = (f′′, b′′) satisfy (7.3.1)–(7.3.3). Clearly, by construction, f′′ and F′ are k-connected. By excision, K_{k}(W′, ∂W′; R) = K_{k}(W, ∂W ∪ V; R). There is an exact sequence

\xymatrix{K_{k}(∂W ∪ V, ∂W; R) \ar[r] & K_{k}(W, ∂W; R) \ar[r] & K_{k}(W, ∂W ∪ V; R) \ar[r] & 0.}

Since the first arrow is surjective, K_{k}(W, ∂W ∪ V; R) = 0. Thus we get K_{k}(W′, ∂W′; R) = 0. By the universal coefficient theorem, K^{k}(W′, ∂W′; R) = 0. The Poincaré-Lefschetz duality implies K_{k+1}(W′; R) = 0. Note that fff′′ can be obtained from fff′ by G-surgery along the embeddings ∂h′′_{1}, ⋯, ∂h′′_{ℓ} : S^{k−1} → X′. These embeddings are isotopic in Interior(X′ ∖ L′) to h_{ccc_{1}}, ⋯, h_{ccc_{ℓ}}, respectively (see (6.2.1)). Thus we obtain

M M M_{fff′′,𝔹} ≅ M M M_{fff′,𝔹′} ⊕ 𝕄(R[G],ccc_{1}) ⊕ ⋯ ⊕ 𝕄(R[G],ccc_{ℓ}).
□

Suppose fff and fff′ are k-connected, P(G)_{R}-singularity, R-boundary quasiequivalences. Suppose F is k-connected, ∂_{−}F : ∂_{−}W → I × ∂Y is an R-homology equivalence, and furthermore K_{k+1}(W; R) = 0 = K_{k}(W, ∂W; R). Note that ∂_{+}F F F is the disjoint union of two G-framed maps −fff and fff′, where −fff is a copy of fff whose underlying manifold has orientation opposite to the original one. Let \overline{θ}_{∂(F,𝔹̂)} (resp., \overline{θ}_{2,∂(F,𝔹̂)}) denote the composition:

Θ̃ ⟶^{id∪id} Θ̃_{𝔹} ⨿ Θ̃_{𝔹′} ⟶^{θ_{∂_{+}F,𝔹∪𝔹′}} K_{k}(∂_{+}W; R) = K_{k}(∂W; R)

(resp., Θ_{2} ⟶^{id∪id} Θ_{2,𝔹} ⨿ Θ_{2,𝔹′} ⟶^{θ_{2,∂_{+}F,𝔹∪𝔹′}} K_{k}(∂_{+}W; R_{2}) = K_{k}(∂W; R_{2})).

Set

\overline{M M M}_{∂(F F F ,𝔹̂)} = (K_{k}(∂W; R), B_{∂F,R}, q_{∂F F F ,R}, \overline{θ}_{∂(F,𝔹̂)}, \overline{θ}_{2,∂(F,𝔹̂)}).

Clearly this module is isomorphic to M M M_{−(fff,𝔹)} ⊕ M M M_{fff′,𝔹′}.

Lemma 7.4. Suppose fff and fff′ are k-connected, P(G)_{R}-singularity, R-boundary quasiequivalences, ∂_{−}F : ∂_{−}W → I × ∂Y is an R-homology equivalence, F : W → I × Y is k-connected, and K_{k+1}(W; R) = 0 = K_{k}(W, ∂W; R). Then the following hold. (7.4.1) \overline{M M M}_{∂(F F F ,𝔹̂)} = −M M M_{fff,𝔹} ⊕ M M M_{fff′,𝔹′} (hence \overline{M M M}_{∂(F F F ,𝔹̂)} ∈ ∇Q Q Q(A A A,Θ Θ Θ)), where

−M M M_{fff,𝔹} = (K_{k}(X, R), −q_{fff,R}, −B_{f,R}, −θ_{f,𝔹}, −θ_{2,f,𝔹}).

(7.4.2) The submodule ∂K_{k+1}(W, ∂W; R) of K_{k}(∂W; R) contains Im(\overline{θ}_{∂(F,𝔹̂)}); the submod-

ule ∂K_{k+1}(W, ∂W; R_{2}) of K_{k}(∂W; R_{2}) contains Im(\overline{θ}_{2,∂(F,𝔹̂)}). (7.4.3) ∂K_{k+1}(W, ∂W; R) is totally isotropic in \overline{M M M}_{∂(F F F ,𝔹̂)}.

Proof. Property (7.4.1) follows from M M M_{−(fff,𝔹)} ≅ −M M M_{fff,𝔹}.

Property (7.4.2) is obtained by chasing the following commutative diagram

\xymatrix{H_{k+1}(B̂_{β}, ∂B̂_{β}) \ar[r] \ar[d]_{∂} & H_{k+1}(W, ∂W) \ar[r] \ar[d]^{∂} & K_{k+1}(W, ∂W) \ar[d]^{∂} \\ H_{k}(∂B̂_{β}) \ar[r] \ar[d] & H_{k}(∂W) \ar[r] \ar[d] & K_{k}(∂W) \ar[d]^{=} \\ H_{k}(∂B̂_{β}, ∂_{−}B̂_{β}) \ar[r] & H_{k}(∂W, ∂_{−}W) \ar[r] & K_{k}(∂W, ∂_{−}W) \\ H_{k}(−B_{β}, ∂(−B_{β})) ⊕ H_{k}(B′_{β}, ∂B′_{β}). \ar[u]^{=}}

for the coefficient rings R and R_{2}.

We prove (7.4.3). Here the coefficient ring of homology groups is R. Regard K_{k+1}(W, ∂W) as a submodule of K_{k}(∂W) via the connecting homomorphism ∂. It is well known that the ordinary intersection form Int_{∂W;R} : K_{k}(∂W) × K_{k}(∂W) → R satisfies

Int_{∂W;R}(K_{k+1}(W, ∂W), K_{k+1}(W, ∂W)) = 0.

Since K_{k+1}(W, ∂W) is G-invariant,

B_{∂F,R}(K_{k+1}(W, ∂W), K_{k+1}(W, ∂W)) = 0.

We conclude the proof by showing q_{∂F} vanishes on ∂K_{k+1}(W, ∂W). By Definition 5.7, q_{∂F F F} is a map

K_{k}(∂W) ⟶ (R/(1 − λ))[{1}] ⊕ R_{2}[G(2)_{λ} ∖ Q] ⊕ R[G(2)_{−λ} ∖ S] ⊕ R[G(≥ 3)_{h}],

where G(2)_{λ} = {a ∈ G(2) | w(a) = −λ}, G(2)_{−λ} = {a ∈ G(2) | w(a) = λ}, Q = Q(G, X), S = S(G, X), and G(≧ 3)_{h} is a subset of G such that

G = {1} ⨿ G(2) ⨿ G(≧ 3)_{h} ⨿ G(≧ 3)_{h}^{−1}.

Let x ∈ K_{k+1}(W, ∂W). For g ∈ G(≧ 3)_{h}, the g-th coefficient q_{∂F F F ,R}(x)_{g} of q_{∂F F F ,R}(x) is equal to 0, because

q_{∂F F F ,R}(x)_{g} = ε(Int_{∂W;R}(x, g^{−1}x)).

For g ∈ G(2)_{−λ} ∖ S, q_{∂F F F ,R}(x)_{g} = 0, because

ε(Int_{∂W;R}(x, g^{−1}x)) = 2q_{∂F F F ,R}(x)_{g}.

Thus we have reduced to the case g ∈ {1} ∪ (G(2)_{λ} ∖ Q). Let ⟨g⟩ denote the subgroup generated by g in G. By definition,

q_{∂F F F ,R}(x)_{g} = q_{∂(Res_{⟨g⟩}^{G}F F F),R}(x)_{g}

for any x ∈ K_{k}(∂W) = K_{k}(∂(Res_{⟨g⟩}^{G}W)). But

dim(Sing(⟨g⟩, Res_{⟨g⟩}^{G}W)) + (k + 1) ≤ {(k − 2) + 1} + (k + 1)
= 2k
< dim(Res_{⟨g⟩}^{G}W).

In other words, the strong gap hypothesis holds for the group ⟨g⟩. In this case, an argument similar to [44, p.53, lines 7–11] proves that q_{∂(Res_{⟨g⟩}^{G}F F F),R}(x) = 0 for all x ∈ K_{k+1}(Res_{⟨g⟩}^{G}W, ∂(Res_{⟨g⟩}^{G}W)). Consequently q_{∂F F F ,R}(x) = 0 for all x ∈ K_{k+1}(W, ∂W). □

Corollary 7.5. Suppose fff, fff′ and F F F are as in Lemma 7.4. Let σ(fff, 𝔹) and σ(fff′, 𝔹′) ∈ W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ)_{proj} denote their surgery obstructions defined in Theorem 6.3. If K_{k+1}(W, ∂W; R) is a stably free R[G]-module then

σ(−(fff, 𝔹)) + σ(fff′, 𝔹′) = 0 ∈ W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ)_{proj}.

Proof. First note that K_{k}(∂W; R) = K_{k}(X; R) ⊕ K_{k}(X; R) is a projective R[G]-module and has the nonsinglar Hermitian form B_{∂F,R}.

Since K_{k+1}(W; R) = 0 = K_{k}(W, ∂W; R), we obtain from the Mayer-Vietoris exact sequence for the pair (W, ∂W), a short exact sequence

(7.5.1)  \xymatrix{∂K_{k+1}(W, ∂W; R) \ar@{^{(}->}[r] & K_{k}(∂W; R) \ar[r]^{^{i_{∂W}^{∗}}} & K_{k}(W; R) \ar[r] & 0.}

of R[G]-modules. By Poincaré-Lefschetz duality, K_{k}(W, R) ≅ K^{k+1}(W, ∂W; R) as Rmodules. By the universal coefficient theorem, K^{k+1}(W, ∂W; R) ≅ Hom(K_{k+1}(W, ∂W; ℤ), R) as R-modules. Thus K_{k}(W; R) is a free R-module. In addition, we get rank_{R}K_{k}(W; R) = rank_{R}(∂K_{k+1}(W, ∂W; R)). The exact sequence (7.5.1) splits over R and

(7.5.2)  K_{k}(∂W; R) ≅ ∂K_{k+1}(W, ∂W; R) ⊕ K_{k}(W; R)

as R-modules. Let ε_{1} denote the projection map R[G] → R on the coefficient of 1 ∈ G. The ordinary intersection pairing, i.e. ε_{1} ∘ B_{∂f,R}, yields an R[G]-homomorphism

(7.5.3)  ψ : K_{k}(∂W; R) → Hom_{R}(∂K_{k+1}(W, ∂W; R), R), x ↦ ε_{1}(B_{∂f,R}(x, −)).

Since ε_{1} ∘ B_{∂f,R} is nonsingular, ψ is epic. Thus K_{k}(∂W; R) possesses an R[G]-direct sum decomposition

(7.5.4)  K_{k}(∂W; R) = Ker(ψ) ⊕ M

where M ≅ Hom_{R}(∂K_{k+1}(W, ∂W; R), R) via ψ|_{M}. It follows from (7.5.2) and (7.5.4) that

rank_{R}(Ker(ψ)) = rank_{R}K_{k}(W; R) = rank_{R}(∂K_{k+1}(W, ∂W; R)).

By Lemma 7.4, ∂K_{k+1}(W, ∂W; R) ⫅ Ker(ψ). Since Ker(ψ) and ∂K_{k+1}(W, ∂W; R) are both R-free modules having the same R-rank, we obtain Ker(ψ) = ∂K_{k+1}(W, ∂W; R). Set

∂K_{k+1}(W, ∂W; R)^{⊥} = {x ∈ K_{k}(∂W; R) | B_{∂F,R}(x, y) = 0 for all y ∈ ∂K_{k+1}(W, ∂W; R)}.

Then the equality ∂K_{k+1}(W, ∂W; R)^{⊥} = Ker(ψ) follows from the formula

B_{∂F,R}(x, y) = ∑_{g∈G} ε_{1}(B_{∂F,R}(x, g^{−1}y))g (x, y ∈ K_{k}(∂W; R)).

Thus we obtain

(7.5.5)  ∂K_{k+1}(W, ∂W; R) = Ker(ψ) = ∂K_{k+1}(W, ∂W; R)^{⊥}.

By (7.5.4) and (7.5.5), ∂K_{k+1}(W, ∂W; R) is an R[G]-direct summand of K_{k}(∂W; R). Hence it follows from Lemma 7.4 that K_{k}(W, ∂W; R) is an R[G]-stably free Lagrangian of \overline{M M M}_{∂(F F F ,𝔹̂)}. Thus \overline{M M M}_{∂(F F F ,𝔹̂)} vanishes in W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ)_{free}, and consequently σ(−(fff, 𝔹)) + σ(fff′, 𝔹′) = 0. □

Remark 7.6. Let K_{k}(W; R), K_{k}(∂W; R) and K_{k+1}(W, ∂W; R) be as in Corollary 7.5. Then K_{k}(W; R) ≅ Hom_{R}(∂K_{k+1}(W, ∂W; R), R) as R[G]-modules. Moreover K_{k}(W; R) and K_{k}(∂W; R) are both stably R[G]-free modules.

Proof. Let M be as in (7.5.4). Then i_{∂W}_{∗} maps M isomorphically onto K_{k}(W; R). Thus K_{k}(W; R) ≅ Hom_{R}(∂K_{k+1}(W, ∂W; R), R) as R[G]-modules. Since ∂K_{k+1}(W, ∂W; R) is stably free over R[G], so is Hom_{R}(∂K_{k+1}(W, ∂W; R), R). Thus K_{k}(W; R) and K_{k}(∂W; R) are both stably free over R[G]. □

Let H(G) denote the set of all hyperelementary subgroups of G. The map f : (X, ∂X) → (Y, ∂Y) is called an (H(G), R)-singularity quasiequivalence if f^{H} : X^{H} → Y^{H} is an Rhomology equivalence for all nontrivial H ∈ H(G), i.e. H ≠ {1}.

Lemma 7.7. Suppose f and f′ are k-connected, R-boundary, (H(G), R)-singularity equivalences such that ∂X = ∂X′, ∂f = ∂f′ and ∂b = ∂b′. If (F F F , 𝔹̂) is a G-framed Θ Θ Θ-cobordism rel ∂X between fff and fff′ and F is an (H(G), R)-singularity quasiequivalence then the Gsurgery obstructions

σ(fff, 𝔹) and σ(fff′, 𝔹′) lie in W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ)_{free}

(⫅ W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ)_{proj}) and furthermore

σ(fff, 𝔹) = σ(fff′, 𝔹′).

Proof. For each hyperelementary subgroup H of G, K_{k}(X; R) is stably free over R[H], by Lemma 5.14. Thus by Swan’s induction theorem, K_{k}(X; R) is stably free over R[G]. Hence the G-surgery obstruction σ(fff, 𝔹) lies in W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ)_{free}. By Lemmas 2.20 and 2.37, σ(−(fff, 𝔹)) = −σ(fff, 𝔹). Similarly σ(fff′, 𝔹′) lies in the same abelian group. Since F is a cobordism rel ∂X and f is an R-boundary quasiequivalence, ∂_{−}F is also an R-homology equivalence. Thus K_{k}(∂W; R) ≅ K_{k}(X; R) ⊕ K_{k}(X′; R) as R[G]- modules and K_{k}(∂W; R) is stably free over R[G]. Moreover, by Lemma 7.3 we can assume F satisfies the hypotheses of Lemma 7.4. Then K_{j}(W; R) = 0 except for j = k, and K_{k}(W; R) is free over R. For each hyperelementary subgroup H of G, F : W → I ×Y is an (H, R)-singularity quasiequivalence in the sense prior to Lemma 5.6. One can adapt now the proof of Lemma 5.14 to show that K_{k}(W; R) is stably free over R[H]. Thus by Swan’s induction theorem, K_{k}(W; R) is stably free over R[G]. By the short exact sequence (7.5.1), K_{k+1}(W, ∂W; R) is also stably free over R[G]. Thus by Corollary 7.5, σ(fff, 𝔹) = σ(fff′, 𝔹′). □

We remind the reader once again of the assumption ℤ ⫅ R ⫅ ℚ.

Theorem 7.8. Let X and Y denote compact, connected, oriented, smooth G-manifolds of dimension n = 2k ≧ 6. Let (L, 𝔹) denote a G-singularity structure for X. Let fff = (f, b) : (X, ∂X, T X) → (Y, ∂Y, f^{∗}ξ) denote a degree one G-framed map. Suppose Y is simply connected and f : X → Y is an R-boundary, (H(G), R)-singularity quasiequivalence. Then σ(fff, 𝔹) possesses a complete G-surgery obstruction

σ(fff, 𝔹) ∈ W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ_{𝔹})_{free},

i.e., σ(fff, 𝔹) = 0 if and only if one can perform G-surgery on fff along spheres of dimension ≦ k in Interior(X ∖ L) to obtain a k-connected G-framed map fff′ = (f′, b′) : (X′, ∂X, T X′) → (Y, ∂Y, f′^{∗}ξ) such that f′ : X′ → Y is an R-homology equivalence.

The definition of σ(fff, 𝔹) above is given in the proof below. Note that σ(fff, 𝔹) has been defined previously in Theorem 6.3, but only for k-connected maps fff.

Proof. Since dim B_{β} ≦ k, we can perform G-surgery on fff along spheres of dimension ≦ k − 1 in Interior(X ∖ L) to obtain a k-connected (R, G)-surgery map fff_{1} = (f_{1}, b_{1}), f_{1} : (X_{1}, ∂X_{1}) → (Y, ∂Y). When f itself is k-connected, we allow surgery along an empty set of spheres and get fff_{1} = fff. Note that ∂X_{1} = ∂X and Sing(G, X_{1}) = Sing(G, X). Thus, f_{1} is also an R-boundary, (H(G), R)-singularity quasiequivalence. Using Swan’s induction theorem (cf. proof of Lemma 7.7), we get

σ(fff_{1}, 𝔹) ∈ W_{n}(R, G, Q(G, X), S(G, X),Θ Θ Θ_{𝔹})_{free}.

We define σ(fff, 𝔹) to be the element σ(fff_{1}, 𝔹). We check that the element σ(fff, 𝔹) is independent of the choice of fff_{1}. Let fff_{2} = (f_{2}, b_{2}) be another k-connected (R, G)-surgery map obtained in the same way as fff_{1}. Then there exists a G-framed Θ Θ Θ_{𝔹}-cobordism (F F F , 𝔹̂) rel ∂X between (fff_{1}, (L, 𝔹)) and (fff_{2}, (L, 𝔹)) such that L̂ = I × L and 𝔹̂ = I × 𝔹. In this situation, F : W → I × Y is automatically an (H(G), R)-singularity quasiequivalence and ∂_{−}(F, B) = I × ∂(f, b). Thus by Lemma 7.7, σ(fff_{1}, 𝔹) = σ(fff_{2}, 𝔹).

Suppose σ(fff, 𝔹) = 0. By definition, this means that σ(fff_{1}, 𝔹) = 0. By Theorem 6.3, we can perform G-surgery on fff_{1} along spheres of dimension ≦ k in Interior(X ∖ L) to obtain a G-framed map fff′ as described in the theorem.

On the other hand, if we can obtain by a G-surgery in Interior(X ∖L), a G-framed map fff′ as described in the theorem then by independence of the choice of fff_{1} in constructing σ(fff, 𝔹), we have σ(fff, 𝔹) = σ(fff′, 𝔹). But σ(fff′, 𝔹) = 0, because now K_{k}(X′; R) = 0. □

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Department of Mathematics University of Bielefeld 33615 Bielefeld, Germany

E-mail address: bak@mathematik.uni-bielefeld.de

Graduate School of Natural Science and Technology Okayama University Okayama, 700-8530 Japan

E-mail address: morimoto@ems.okayama-u.ac.jp