Forum Math 8 (1996), 267 - 302

EQUIVARIANT SURGERY WITH

MIDDLE-DIMENSIONAL SINGULAR SETS. I

Anthony Bak^{∗} and Masaharu Morimoto^{∗∗}

University of Bielefeld^{∗} and Okayama University^{∗∗}

Abstract. Let G be a finite group. Let f : X → Y be a k-connected, degree 1, G-framed map of simply connected, closed, oriented, smooth manifolds X and Y of dimension 2k ≧ 6. Under the assumption that the dimension of the singular set of the action of G on X is at most k, we construct an abelian group W (G, Y) and an element σ(f) ∈ W (G, Y), called the surgery obstruction of f such that the vanishing of σ(f) in W (G, Y) guarantees that f can converted by G-surgery to a homotopy equivalence.

Acknowledgement. We wish to express our gratitude to I. Madsen for stressing the importance of the problem above. We are also grateful to E. Laitinen and K. Pawałowski for their suggestions concerning brushing up the manuscript. The second author gratefully acknowledges support by the Japan Association for Mathematical Sciences and a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan.

1991 Mathematics Subject Classification: 57R67, 57R91, 19G24

1. Introduction

Let G be a finite group and X a smooth G-manifold. In the current article, the term G-equivariant surgery or simply G-surgery will be used in a restricted sense. Namely, it will refer to G-surgery on that part of X where each nontrivial element of G acts without fixed points. Thus, G-surgery on X will not change the G-singular set

Sing(G, X) = ⋃_{g∈G\{1}}X^{g},

Key words and phrases. surgery, singular set, quadratic form.

Typeset by AMS-TEX Typeset by AMS-TEX

2 ANTHONY BAK^{∗} AND MASAHARU MORIMOTO^{∗∗}

where X^{g} = {x ∈ X | gx = x}.

Equivariant surgery theory in the sense above has been developed by several authors, beginning in the early early 70-ies. For references, see [18], [11], and [13]. C. T. C. Wall’s nonsimply connected surgery for compact manifolds X can be viewed as G-surgery on the universal covering space X̃ of X where G is the fundamental group of X. Except for [7], equivariant surgery theory has proceeded under the gap hypothesis: 2 dim Sing(G, X) < dim X. Under this hypothesis, the surgery obstruction group is either an L-group of Wall or a quotient of such involving form parameters, cf. [13], [14]. However, it turns out that there are interesting geometric problems for G-manifolds X, which require using G-surgery, where 2 dim Sing(G, X) = dim X. In order to handle such problems, we develop in this paper an equivariant surgery obstruction theory under the assumptions that dim X = 2k ≧ 6 and dim Sing(G, X) ≦ k.

Applications of our G-surgery will appear in subsequent papers. They include the following. Buchdahl, Kwasik, and Schultz [6] proved that if a standard n-sphere S^{n} admits a one fixed point, smooth (or locally linear) G-action for some finite group G then n ≧ 6. We shall prove a converse to this result, namely that the alternating group A_{5} on 5 letters has a one fixed point, smooth action on each S^{n} for n ≧ 6 ([3]). Another application is the following. Recall that an Oliver group is a finite group G which does not possess a series of subgroups P ◁ H ◁ G such that P and G/H are of prime power order and H/P is cyclic. According to [16], if a finite group G acts smoothly on a standard sphere, with precisely one fixed point then G is Oliver. The converse of this result, namely that each Oliver group has a one fixed point, smooth action on some standard sphere, is proved for odd order abelian groups in [17], for nontrivial perfect groups in [10], and in full generality in [9].

The equivariant surgery obstruction theory which is presented in the current article is sufficient for the applications in [3], [9], and [10] above, but is not best possible, because extra assumptions will be imposed on middle dimensional fixed point sets. The advantage of making these assumptions is that new constructions needed for surgery with middle dimensional fixed point sets can be introduced, while at the same the details of the proofs can be considerably simplified over the general situation. The general situation will be treated in a paper under preparation and will show that vanishing of the surgery obstruction invariant of a G-framed map is equivalent to the map being G-framed cobordant (relatively to the G-singular set) to a homotopy equivalence.

We describe now our main result in the current article. Let X and Y be oriented, smooth G-manifolds. Let T (X) denote the tangent bundle of X. Recall that a G-framed map fff = (f, b) : X → Y is a pair consisting of a smooth map f : X → Y and a real G-vector bundle isomorphism b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η) covering the identity map on X, for some real G-vector bundles ξ and η on Y. A G-framed map fff = (f, b) is said to be of degree 1 (resp. k-connected) if f is of degree 1 (resp. k-connected). As usual, ℤ will denote the ring of integers, ℤ_{(p)} the localization of ℤ at a prime number p, and ℚ the ring of rational numbers. We regard the set

G(2) = {g ∈ G | g^{2} = 1 and g ≠ 1}

as a G-set by letting G act by conjugation on the elements of G(2).

Theorem 1.1. Let G be a finite group and Y a closed, 1-connected, oriented, smooth G-manifold of even dimension n = 2k ≧ 6. Suppose that (1.1.1)–(1.1.3) below hold.

(1.1.1) dim Y^{g} ≦ k for any g ∈ G \ {1}. (This is equivalent to dim Sing(G, Y) ≦ k.)

(1.1.2) If dim Y^{H} = k for some subgroup H ≦ G then |H| = 2 and Y^{H} is connected and oriented such that each g : Y^{H} → Y^{gHg^{−1}} (g ∈ G) is orientation preserving.

(1.1.3) dim(Y^{H} ∩ Y^{K}) ≦ k − 2 whenever dim Y^{H} = k and dim Y^{K} = k − 1 (H, K ≦ G).

Let R be one of ℤ, ℤ_{(p)} (p a prime), or ℚ. Then there is an abelian group W (G, Y; R) having the properties (DP) and (SP) below.

(DP) W (G, Y; R) is determined solely by the data (R, G, Q, S, λ, w_{Y}^{G}), where Q = Q(G, Y) = {g ∈ G(2) | dim Y^{g} = k − 1}, S = S(G, Y) = {g ∈ G(2) | dim Y^{g} = k}, λ = (−1)^{k}, and w_{Y}^{G} : G → {±1} is the orientation homomorphism associated to Y.

(SP) Let fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}ξ ⊕ η) be a degree 1, k-connected, G-framed map where X also satisfies (1.1.1)–(1.1.3). Suppose that Q(G, X) = Q(G, Y), S(G, X) = S(G, Y), and K_{k}(f; R) = Ker[f_{∗} : H_{k}(X; R) → H_{k}(Y, R)] is stably free over R[G]. Then there is an element σ(fff) ∈ W (G, Y; R) depending on fff, such that if σ(fff) = 0 then fff can be converted by G-surgery to a degree 1, k-connected, G-framed map fff′ = (f′ : X′ → Y, b′ : T (X′)⊕f′^{∗}η → f′^{∗}ξ) with the property that f′ is an R-homology equivalence.

Theorem 1.1 will be deduced in the main body of the paper from a slight generalization Theorem 7.3 of it.

Remark 1.2. Let

fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}ξ ⊕ η)

be as in Theorem 1.1. Let X_{0} be a G-simplicial subcomplex of X with respect to some equivariant smooth triangulation of X such that dim X_{0} ≦ k −1. Suppose σ(fff) = 0. Then in the proof of Theorem 7.3, the G-surgery used to convert f to an R-homology equivalence will be along embeddings h : S^{ℓ} → X such that ℓ ≦ k. Since dim S^{ℓ} + dim X_{0} < dim X, we can modify these embeddings so that h(S^{ℓ}) ∩ X_{0} = ∅. Thus, the G-surgery required in Theorem 1.1 (SP) and also in Theorem 7.3 can be performed in the free part of X \ X_{0}.

A special case of equivariant surgery theory on manifolds having middle-dimensional singular sets was treated by K. H. Dovermann [7], namely the case |G| = 2. His surgery obstruction was expressed in terms of several invariants in classical surgery theory. The approach in the current paper is very different from that in [7], in that we construct a new surgery group over a ring with form parameters, housing a single invariant to detect the obstruction to performing G-surgery.

We describe now this surgery invariant. Recall that the usual surgery invariant, under the gap hypothesis, is obtained by equipping K_{k}(f; ℤ) with the restriction B_{f} of the equivariant intersection form on H_{k}(X; ℤ) and then showing that the self-intersection form on X defines a quadratic form q_{f f f} on K_{k}(F; ℤ), whose associated Hermitian form is B_{f}. If the triple (K_{k}(f; ℤ), B_{f}, q_{f f f}) has (stably) a ℤ[G]-free Lagrangian L then one can realize geometrically any ℤ[G]-basis for L by equivariantly embedded, disjoint k-spheres and then remove these by performing G-surgery. Thus, the class of (K_{k}(f; ℤ), B_{f}, q_{f f f}) in the Grothendieck group of all such algebraically defined triples modulo all triples having a ℤ[G]-free Lagrangian is a sufficient invariant for performing equivariant surgery. We want to modify this procedure so that it works under the hypothesis dim Sing(G, X) ≦ k. It turns out that K_{k}(f; ℤ) is still a finitely generated, projective ℤ[G]-module, that B_{f} is still a nonsingular Hermitian form, but that it is not necessarily even. The first ingredient we develop is a new notion of quadratic form q whose associated Hermitian form B is not necessarily even. This involves two parameters instead of one as above and we call a triple (M, B, q) a doubly parametrized quadratic module. In the geometric situation, we construct a quadratic form q_{f f f} in the new sense above, on K_{k}(f; ℤ), which incorporates selfintersection information needed later and whose associated Hermitian form is the intersection form B_{f} above. The notion of Lagrangian L for (M, B, q) is the usual one, but it is not necessary that a Lagrangian in the generality we are working has a direct sum complement which is a Lagrangian. We are still not finished building our surgery invariant. The subset S(G, X) of G(2) is G-invariant under the action of G via conjugation. We replace now the triples (M, B, q) above by quadruples M M M = (M, B, q, α) where α : S(G, X) → K is a G-map and define a Lagrangian L for (M, B, q, α) to be one for (M, B, q) such that Im(α) ⊆ L. The G-map α is called the positioning data of (M, B, q, α). In the geometric situation, α_{f} is the G-map which assigns to each s ∈ S(G, X) the image in K_{k}(f; ℤ) of the orientation class of X^{s}. ¿From the geometric standpoint, our main result is the following: If L is a ℤ[G]-free Lagrangian for (K_{k}(f; ℤ), B_{f}, q_{f f f}, α_{f}) then any ℤ[G]-basis of L can be realized geometrically by equivariantly embedded, disjoint kspheres which do not meet Sing(G, X). This being the case, we can perform G-surgery on the embedded spheres and convert f to a homology equivalence and therefore, to a homotopy equivalence. One would like now to form the Grothendieck group of all algebraically defined quadruples (M, B, q, α) modulo the subgroup of all quadruples having a ℤ[G]-free Lagrangian and claim that the class of (K_{k}(f; ℤ), B_{f}, q_{f f f}, α_{f}) in this group is a sufficient obstruction to performing equivariant surgery. But, this doesn’t work, because stabilization with respect to this group is too strong. It turns out that the quadruples (K_{k}(f; ℤ), B_{f}, q_{f f f}, α_{f}) vanish under a certain invariant ∇. This is a crucial observation. The right group W (G, Y) for housing (K_{k}(f; ℤ), B_{f}, q_{f f f}, α_{f}) is the Grothendieck group of all algebraically defined quadruples M M M = (M, B, q, α) with trivial ∇_{M M M} modulo the subgroup generated by all such quadruples having a Lagrangian. A few words concerning ∇ are in order. ¿From the geometric point of view, the definition of ∇ is motivated by the obstruction that if x ∈ K_{k}(f; ℤ) is realized by an immersion h : S^{k} → X and if s ∈ S(G, X) then the intersection number of h and sh is congruent mod 2 to that of X^{s} and sh. The observation shows that if we define for an arbitrary algebraic object M M M = (M, B, q, α), ∇_{M M M} : M → Map(S(G, X), ℤ/2ℤ) by ∇_{M M M}(x)(s) = [ε(B(α(s) − x, sx))] (x ∈ M and s ∈ S), where ε : ℤ[G] → ℤ is the map defined by ε(∑_{g∈G} a_{g}g) = a_{1} (a_{g} ∈ ℤ), then for any geometric

object M M M_{f f f} = (K_{k}(f; ℤ), B_{f}, q_{f f f}, α_{f}), ∇_{M M M_{_{f}_{f}_{f}}} = 0. In other words, ∇ vanishes on all geometric objects. Now our result that the family of all metabolic planes with trivial ∇-invariant is cofinal in the category of all algebraic objects with trivial ∇-invariant and our result that any metabolic plane with trivial ∇-invariant can be added to a geometric object M M M_{f f f} by performing G-surgery on fff shows that the group W (G, Y; ℤ) is the correct one for housing our G-surgery invariant.

The rest of the article is organized as follows. In Section 2, we recall certain foundations of equivariant surgery including the equivariant intersection form and equivariant selfintersection form. We construct a doubly parametrized selfintersection form which is used later to define q_{f f f}. In Section 3, we prove the geometric result that a k-dimensional immersion h into a 2k-dimensional G-manifold X, which does not meet the G-singular set Sing(G, X) and vanishes under our doubly parametrized selfintersection form, is regularly homotopic to an equivariant embedding h′, i.e. an embedding h′ such that Imh′ ∩ gImh′ = ∅ for all g ∈ G\{1}. Section 4 is purely algebraic. It defines doubly parametrized quadratic modules with positioning data and the invariant ∇ of such modules. Various Grothendieck-Witt groups relevant to studying surgery groups are constructed. One of these groups, namely that defined in (4.4), is the surgery group. Section 5 studies special metabolic planes whose ∇-invariant is trivial. The main result is Theorem 5.6: A doubly parametrized quadratic module which has a free Lagrangian and trivial ∇-invariant, decomposes as an orthogonal sum of special metabolic planes with trivial ∇-invariant. A corollary of this result is that the family of metabolic planes with trivial ∇-invariant is cofinal in the category of all doubly parametrized quadratic modules with trivial ∇-invariant. Section 6 is devoted to the proof of Theorem 5.6. Section 7 begins by constructing the geometric module M M M_{f f f} = (K_{k}(f; ℤ), B_{f}, q_{f f f}, α_{f}), where fff = (f, b), and showing that its ∇-invariant is 0. Let σ(fff) denote the class of M M M_{f f f} in the surgery group W (G, Y; ℤ). The main result of the paper is Theorem 7.3 asserting that if σ(fff) = 0 then fff is G-framed cobordant to fff′ = (f′, b′) such that f′ : X′ → Y is a homology equivalence. Theorem 7.3 is proved in Section 8 on the basis of Theorem 8.1: Any metabolic plane with trivial ∇-invariant can be added to M M M_{f f f} by performing G-surgery on fff; i.e., given a metabolic plane M M M such that ∇_{M M M} = 0, there is a G-framed map fff′′ = (f′′, b′′) obtained from fff by G-surgery such that M M M_{f f f}′′ ≅ M M M_{f f f} ⊕ M M M. Theorem 8.1 is proved in Section 9.

2. Geometric preliminaries

In this section, we develop notation to be used in the following sections.

Let X be a G-space. For a point x ∈ X and for a subgroups H of G, let H_{x} denote the isotropy subgroup at x in the H-space res_{H}^{G}X. Let

X^{H} = Fix(H, X) = {x ∈ X | G_{x} ⫆ H}
Fix_{G}(>H, X) = {x ∈ X | G_{x} ⫌ H},
Fix_{G}(=H, X) = {x ∈ X | G_{x} = H}
Free(H, X) = {x ∈ X | H_{x} = {1}}, and
Sing(H, X) = {x ∈ X | H_{x} ≠ {1}}.

If g ∈ G, let X^{g} = X^{⟨g⟩}, Free(g, X) = Free(⟨g⟩, X), etc. For a subspace Z ⫅ X, define

ρ_{X}^{G}(Z) = ⋂_{x∈Z} G_{x}.

Let Mnf Mnf Mnf^{n}(G) denote the family of all paracompact, 1-connected (i.e. connected and simply connected), oriented, smooth G-manifolds of dimension n. Let Mnf Mnf Mnf_{cp}^{n}(G) denote the family of all compact G-manifolds in Mnf Mnf Mnf^{n}(G).

For X ∈ Mnf Mnf Mnf^{n}(G), the orientation homomorphism w_{X}^{G} : G → {±1} is defined by w_{X}^{G}(g) = 1 if g : X → X is orientation preserving and w_{X}^{G}(g) = −1 if otherwise. For any commutative ring R with the unity, let R[G] denote the group ring of G with coefficients in R. For any set U, the set Map(U, R) consisting of all maps U → R is regarded as an R-module in the canonical way. As R-modules, R[G] = Map(G, R). For a subset S of G, let R[S] denote the R-submodule of R[G] generated by S; thus R[S] = Map(S, R). We shall always give R[G] the antiinvolution a ↦ \overline{a} defined by w = w_{X}^{G}; thus

\overline{∑_{g∈G} r_{g}g} = ∑_{g∈G} r_{g}w(g)g^{−1} (r_{g} ∈ R).

Let H be a subgroup G, which will be indicated by H ≦ G. Since X has an equivariant smooth triangulation (cf. [8]), the H-fixed-point set X^{H} is an N_{G}(H)-simplicial complex of dimension ≦ n where N_{G}(H) = {g ∈ G | gHg^{−1} = H}.

Let π_{0}(X^{H}) denote the set of all connected components γ of X^{H}. The underlying space of γ will be denoted by X_{γ} (or X_{γ}^{H} when we want to emphasize the group H). For a nonnegative integer ℓ, let π_{0}(X^{H}, ℓ) denote the subset of π_{0}(X^{H}) consisting of all γ such that dim X_{γ} = ℓ. Set

\begin{matrix}Π(G, X) = ∐_{H≦G} π_{0}(X^{H}), and \\ Π(G, X, ℓ) = ∐_{H≦G} π_{0}(X^{H}, ℓ). \end{matrix}

For γ ∈ Π = Π(G, X) such that γ ∈ π_{0}(X^{H}), define ρ(γ) = ρ_{Π}(γ) := H (Π = Π(G, X)). One should note that if H = ρ_{Π}(γ) then H ≦ ρ_{X}^{G}(X_{γ}), but H is not necessarily equal to ρ_{X}^{G}(X_{γ}). For g ∈ G and γ ∈ Π(G, X), let gγ be the connected component γ′ ∈ π_{0}(X^{gHg^{−1}}) such that X_{γ′}^{gHg^{−1}} = gX_{γ}^{H}. The assignment g ↦ gγ defines an action of G on Π(G, X). Obviously, Π(G, X, ℓ) is G-invariant. Let Φ : Π(G, X) → Π(G, X) denote the map γ ↦ β such that ρ_{Π}(β) = ρ_{X}^{G}(X_{γ}) and X_{β} = X_{γ} as subsets of X. The map Φ is a G-map. The property ρ_{Π}(Φ(γ)) = ρ_{X}^{G}(X_{Φ(γ)}) should be kept in mind. Generally speaking, the subsets below are more useful than Π(G, X) and Π(G, X, ℓ) for handling problems arising from Sing(G, X). Define

Θ(G, X) = Im(Φ) and Θ(G, X, ℓ) = Θ(G, X) ∩ Π(G, X, ℓ).

Let Mnf Mnf Mnf_{sg}^{2k}(G) denote the family of all X ∈ Mnf Mnf Mnf^{2k}(G) satisfying the following hypotheses:

(2.1.1) dim Sing(G, X) ≦ k (namely, dim X^{g} ≦ k for all g ∈ G \ {1}).

(2.1.2) |π_{0}(X^{H}, k)| ≦ 1 for any H ≦ G.

(2.1.3) If γ ∈ Θ(G, X, k) then |ρ_{Π}(γ) ∩ G(2)| = 1.

(2.1.4) If γ ∈ Θ(G, X, k) and δ ∈ Θ(G, X, k − 1) then dim(X_{γ} ∩ X_{δ}) ≦ k − 2.

(2.1.5) All submanifolds X_{γ}^{H} (γ ∈ Θ(G, X, k)) are oriented in such a way that each g ∈ G acts as an orientation preserving diffeomorphism X_{γ} → X_{gγ}.

Set Mnf Mnf Mnf_{cp,sg}^{2k}(G) = Mnf Mnf Mnf_{cp}^{2k}(G) ∩ Mnf Mnf Mnf_{sg}^{2k}(G).

Lemma 2.2. Let X ∈ Mnf Mnf Mnf_{sg}^{2k}(G) where k ≧ 2. Then for every γ ∈ Θ(G, X, k),

dim Fix_{G}(>ρ_{Π}(γ), X_{γ}) ≦ k − 2.

In particular, Fix_{G}(=ρ(γ), X_{γ}) is connected and open dense in X_{γ}.

Proof. Let γ ∈ Θ(G, X, k). If X_{γ} ⫌ X_{δ} for some δ ∈ Π(G, X) then by (2.1.4), dim X_{δ} ≦ k − 2. Thus

dim Fix_{G}(>ρ_{Π}(γ), X_{γ}) ≦ k − 2. Q.E.D.

Let Y be a closed, connected, oriented, smooth manifold of dimension k. Let X ∈ Mnf Mnf Mnf^{n}(G) where n = 2k and let λ = (−1)^{k}. Let Map(Y, X) denote the set of all continuous maps Y → X. Let Immer(Y, X) denote the set of all smooth immersions Y → X and let Immer^{t}(Y, X) denote the subset of Immer(Y, X) consisting of all immersions Y → X with trivial normal bundle. Let Int(X) denote the interior of X. For f_{1}, f_{2} ∈ Map(Y, Int(X)), let intsec(f_{1}, f_{2}) denote the geometric intersection number of f_{1} and f_{2}. This number is determined as follows. Approximate f_{1} and f_{2} by f′_{1} and f′_{2} ∈ Immer(Y, Int(X)) such that Imf′_{1} ∩ Imf′_{2} = {a_{1}, ⋯, a_{m}}, f′_{1}^{−1}(a_{i}) = {b_{i}}, f′_{2}^{−1}(a_{i}) = {c_{i}}, and each a_{i} is a transversal-intersection point of f′_{1} and f′_{2}. For each point a_{i}, define intsec(f′_{1}, f′_{2}; a_{i}) = 1 (resp. −1) if the ordered direct sum df′_{1}(T_{b_{i}}(Y)) ⊕ df′_{2}(T_{c_{i}}(Y)) has the same (resp.

opposite) orientation as T_{a_{i}}(X). Then intsec(f_{1}, f_{2}) = ∑^{m} ; i=1intsec(f′_{1}, f′_{2}; a_{i}). The G-

intersection number intsec_{G}(f_{1}, f_{2}) of f_{1} and f_{2} is defined by

intsec_{G}(f_{1}, f_{2}) = ∑_{g∈G} intsec(f_{1}, g^{−1}f_{2})g ∈ ℤ[G].

intsec_{G}(f_{1}, f_{2}) is well-defined and invariant under homotopies of f_{1} and f_{2} in Int(X).

If dim X > dim Sing(H, X) + 2 then Free(H, X) is 1-connected. Hence if f ∈ Immer(Y, Int(Free(H, X))), the composition π_{H} ∘ f determines the selfintersection number selfintsec_{H}(f) ∈ ℤ[H]/min^{λ}(ℤ[H]) (cf. [19, Part I §5]), where π_{H} : Int(Free(H, X)) → Int(Free(H, X))/H is the canonical projection, and min^{λ}(ℤ[H]) = \{x−λ\overline{x} | x ∈ ℤ[H]\}. The number selfintsec_{H}(f) is invariant under regular homotopies of f in Int(Free(H, X)). Let T ⫅ G be a subset closed under taking inverses. For a commutative ring R with 1, we define a coefficient quasibundle B_{T} (R) over T as follows. For g ∈ G set R_{g} = R/(1 − λw(g))R if g^{2} = 1, and R_{g} = R otherwise. Define B_{T} (R) = ∐_{g∈T}R_{g}. The map

p_{B} : B_{T} (R) → T such that p_{B}(R_{g}) = {g} is called the projection. A map s : T → B_{T} (R) is called a section if p_{B}∘s = id_{T}. Define Γ^{λ,w}(T; R) to be the set of all sections s : T → B_{T} (R) such that s(g^{−1}) = λw(g)s(g). Define

Q(G, X) = {g ∈ G(2) | dim X^{g} = k − 1}, and
S(G, X) = {g ∈ G(2) | dim X^{g} = k}.

It is easy to show that w(g)(= w_{X}^{G}(g)) = (−1)^{k+1} (resp. (−1)^{k}) for all g ∈ Q(G, X) (resp. S(G, X)). Thus, Q(G, X) ⫅ \{g ∈ G(2) | g = −(−1)^{k}\overline{g}\} and S(G, X) ⫅ \{g ∈ G(2) | g = (−1)^{k}\overline{g}\}. Letting Q = Q(G, X), S = S(G, X) and defining

Λ(G, Q; R) = ⟨x − λ\overline{x} | x ∈ A⟩_{R} + R[Q],

we see that there is a canonical identification R[G \ S]/Λ(G, Q; R) = Γ^{λ,w}(G \ (Q ∪ S); R). Thus we can regard selfintsec_{H}(f) ∈ Γ^{λ,w}(H; ℤ) for f ∈ Immer(Y, Int(Free(H, X))). If H, K ≦ G and f ∈ Immer(Y, Int(Free(H, X))) ∩ Immer(Y, Int(Free(K, X))) then it follows that

(2.3)  selfintsec_{H}(f)(g) = selfintsec_{K}(f)(g) ∈ ℤ_{g} for any g ∈ H ∩ K.

Furthermore if f ∈ Immer^{t}(Y, Int(Free(H, X))) then

(2.4)  intsec_{H}(f, f) = selfintsec_{H}(f) + λ\overline{ selfintsec_{H}(f)} in ℤ[H]

where selfintsec_{H}(f) is a lifting of selfintsec_{H}(f) ∈ ℤ[H]/min^{λ}(ℤ[H]) (cf. [19, Part I Theorem 5.2 (iii)]).

The following lemma is well-known.

Lemma 2.5. Let k be an integer ≧ 3, n = 2k, X ∈ Mnf Mnf Mnf^{n}(G) such that dim X > dim Sing(G, X)+2. Let Y be a closed, connected, oriented, k-dimensional smooth manifold. Let f ∈ Immer(Y, Int(Free(G, X))). If intsec_{G}(f, f) = 0 ∈ ℤ[G] and selfintsec_{G}(f) = 0 ∈ ℤ[G]/min^{λ}(ℤ[G]) then f is regularly homotopic in Int(Free(G, X)) to a smooth embedding f′ : Y → Int(Free(G, X)) such that Im(f′) ∩ gIm(f′) = ∅ for all g ∈ G \ {1}.

Lemma 2.6. If X ∈ Mnf Mnf Mnf_{sg}^{2k}(G) then there is a canonical bijection

S(G, X) → Θ(G, X, k); s ↦ γ(s) such that ρ_{Π}(γ(s)) ∋ s.

Proof. This follows from (2.1.2)–(2.1.3). Q.E.D.

G acts on S(G, X) by conjugation and the bijection above is a G-map. In this paper we identify S(G, X) with Θ(G, X, k) via this bijection, whenever X ∈ Mnf Mnf Mnf_{sg}^{2k}(G).

Definition 2.7. Let X ∈ Mnf Mnf Mnf_{sg}^{n}(G) (n = 2k ≧ 6) and let Y be as above. Then for f ∈ Immer(Y, Int(X)) define

μ_{X}(f) ∈ Γ^{λ,w}(G(μ), ℤ) (where G(μ) = G \ (Q(G, X) ∪ S(G, X)))

by

μ_{X}(f)(g) = \{ \begin{matrix}selfintsec_{⟨g⟩}(f̂_{g})(g) (g ∈ (\{1\} ∪ G(2)) ∩ G(μ)) \\ intsec(f, g^{−1}f) (g ∈ G \ (\{1\} ∪ G(2))), \end{matrix}

where f̂_{g} ∈ Immer(Y, Int(Free(g, X))) is an approximation regularly homotopic to f. We can regard

μ_{X}(f) ∈ ℤ[G \ S]/Λ(G, Q; ℤ) = ℤ[G]/(Λ(G, Q; ℤ) + ℤ[S])

in a canonical way, where Q = Q(G, X), S = S(G, X) and

Λ(G, Q; ℤ) = min^{λ}(ℤ[G]) + {∑_{g∈Q} a_{g}g | a_{g} ∈ ℤ}.

The well-definedness of μ_{X}(f) is easily checked because dim X^{g} ≦ k − 2 for all g ∈ ({1} ∪ G(2)) ∩ G(μ).

Theorem 2.8. Let k be an integer ≧ 3, n = 2k, X ∈ Mnf Mnf Mnf_{sg}^{n}(G), and X̂ = X \ (⋃_{γ} X_{γ}) where γ runs over Θ(G, X, k). Let Y be a closed, connected, oriented, k-dimensional, smooth manifold. If f ∈ Immer(Y, Int(X̂)) satisfies intsec_{G}(f, f) = 0 and μX̂(f) = 0 ∈ ℤ[G \ S]/Λ(G, Q; ℤ) then f is regularly homotopic in X̂ to a smooth embedding f′ : Y → Int(Free(G, X)) such that Im(f′) ∩ gIm(f′) = ∅ for all g ∈ G \ {1}, where Q = Q(G, X) and S = S(G, X).

The result above is proved below.

3. Regular homotopies of immersions to embeddings

The present section is devoted to the proof of Theorem 2.8.

Let k be an integer ≧ 3, n = 2k, and X ∈ Mnf Mnf Mnf_{sg}^{n}(G).

Lemma 3.1. If γ ∈ Θ(G, X, k − 1) then Fix_{G}(=H, X_{γ}) (where H = ρ(γ)) is open dense in X_{γ}.

Proof. The conclusion follows from the observation dim Fix_{G}(>H, X_{γ}) ≦ k − 2. Q.E.D.

Lemma 3.2. If γ ∈ Θ(G, X, k − 1) then |ρ_{Π}(γ) ∩ G(2)| ≦ 1.

Proof. Set H = ρ_{Π}(γ). By Lemma 3.1, we can take a point z in Int(Fix_{G}(= H, X_{γ})). By definition, G_{z} = H. Let T_{z}(X) be the tangential H-representation at z in X. Then T_{z}(X) is the direct sum T_{z}(X^{H}) ⊕ ν_{z}(X^{H}, X) of H-representations. Set V = ν_{z}(X^{K}, X). By (2.1.1) and (2.1.4), H acts freely on V \ {0} and dim_{ℝ} V = k + 1. In particular, L = H ∩ G(2) acts freely on V \ {0}. Thus each g ∈ L acts on V like scalar multiplication by −1. Since V is a faithful H-representation, we get |L| ≦ 1. Q.E.D.

For the remainder of the current section, let Y be a closed, connected, oriented, kdimensional, smooth manifold, and set X̂ = X \ (⋃_{γ} X_{γ}) where γ runs over Θ(G, X, k).

Lemma 3.3. Let f : Y → Int(Free(G, X)) be a smooth immersion. If τ ∈ (ℤ/2ℤ)[Q(G, X)] then there exists a regular homotopy f_{t} : f ∼ f_{1} (f_{0} = f) in Int(X̂) such that Im(f_{1}) ⊂ Int(Free(G, X)) and selfintsec_{G}(f_{1}) = selfintsec_{G}(f) + τ in ℤ[G]/min^{λ}(ℤ[G]) (λ = (−1)^{k}).

We shall assume for the moment that the lemma has been proved and deduce Theorem 2.8 from the lemma.

Proof that Lemma 3.3 ⟹ Theorem 2.8. Let f : Y → Int(X̂) be an immersion satisfying the hypotheses in Theorem 2.8. Since dim Sing(G, X̂) ≦ k − 1, f is regularly homotopic to an immersion in Int(Free(G, X)) = Int(Free(G, X̂)). Thus we suppose Im(f) ⊂ Int(Free(G, X)). Since intsec_{G}(f, f) = 0 and μX̂(f) = 0, we get selfintsec_{G}(f) ∈ (ℤ/2ℤ)[Q(G, X)]. By Lemma 3.3, f is regularly homotopic to f′′ in X̂ such that Im(f′′) ⊂ Int(Free(X)) and selfintsec_{G}(f′′) = 0 in ℤ[G]/min^{λ}(ℤ[G]). As the intersection form is invariant under homotopies, intsec_{G}(f′′, f′′) = intsec_{G}(f, f) = 0. By Lemma 2.5, f′′ is regularly homotopic in Int(Free(G, X)) to a smooth embedding f′ such that Im(f′)∩gIm(f′) = ∅ for all g ∈ G \ {1}. Q.E.D.

Proof of Lemma 3.3. It suffices to prove the lemma in the case τ = g (g ∈ Q(G, X)).

Set H = ⟨g⟩. Since dim X^{H} = k − 1, there is a connected component X_{β}^{H} of dimension k − 1. Let δ = Φ(β) ∈ Θ(G, X, k − 1). Set K = ρ_{Π}(δ) (= ρ_{X}^{G}(X_{β}^{H})). Fix a point z ∈ Int(Fix_{G}(= K, X_{δ}^{K})). Let ν = ν(X_{δ}^{K}, X) be the N_{G}(K)-normal bundle of X_{δ}^{K} in X. This normal bundle is often identified with an N_{G}(K)-tubular neighborhood of X_{δ}^{K}. Let D_{r}(ν) (resp. S_{r}(ν)) be the radius r closed-disk (resp. sphere) bundle over X_{δ}^{K} associated with ν. Regard each S(ν) ⊂ D_{r}(ν) as a submanifold of ν ⊂ X. Thus D_{r} = D_{r}(ν|_{z}) (resp. S_{r} = S_{r}(ν|_{z})) is a (k + 1)-dimensional closed disk (resp. k-dimensional sphere) centered at z. Take r > 0 so small that

(3.4)  D_{r} ∩ aD_{r} ≠ ∅ (a ∈ G) ⟹ a ∈ K,

and that GD_{r} ∩ GIm(f) = ∅. Then K acts freely on D_{r} \ {z}. Let h′ : D^{k+1} = D_{r} → X be the canonical inclusion (hence a smooth embedding). Set h = h′|_{∂D^{k+1}} : S^{k} → Int(Free(G, X)). We regard selfintsec_{G}(h) ∈ Γ^{λ,w}(G; ℤ). Clearly selfintsec_{G}(h)(1) = 0 in ℤ/(1 − λ)ℤ. Since h bounds a disk,

(3.5)  intsec_{G}(h, h) = 0.

By (3.4), selfintsec_{G}(h)(a) = 0 for all a ∈ G \ K. Set J = \{a ∈ G | a = −λ\overline{a}\}. By (3.5), selfintsec_{G}(h)(a) = 0 for all a ∈ G \ J. Since K ∩ G(2) = g, selfintsec_{G}(h)(a) is possibly nontrivial only when a = g. Note that selfintsec_{G}(h)(g) = selfintsec_{⟨g⟩}(h)(g). It is elementary to check that selfintsec_{⟨g⟩}(h)(g) = 1 in ℤ/2ℤ. Thus, we get selfintsec_{G}(h) = g in ℤ[G]/min^{λ}(ℤ[G]). Take a (k +1)-dimensional connecting band ψ(I ×D^{k}) from Im(f) to S_{r} in Int(Free(G, X)) as follows. Let B = I × D^{k} (I = [0, 1]). Take a smooth embedding ψ : B → (Int(X) \ Int(D_{r}(ν))) such that Im(ψ) ∩ aIm(ψ) ≠ ∅ (a ∈ G) ⟹ a = 1, such that ψ^{−1}(Im(f)) = {0} × D^{k} and ψ^{−1}(S_{r}) = {1} × D^{k}, and such that f^{−1}(Im(ψ)) ≅ D^{k}. Set U = f^{−1}(Im(ψ)) and V = h^{−1}(Im(ψ)) (≅ D^{k}). Construct the connected sum Y′ of Y = Domain(f) with S_{h}^{k} = Domain(h) by

Y′ = {Y \ Int(U)} ∪ (I × S^{k−1}) ∪ {S_{h}^{k} \ Int(V)} .

Since S_{h}^{k} = S^{k}, Y′ is diffeomorphic to Y. Define f_{1} : Y′ → Int(Free(G, X)) by gluing f|_{Y \Int(U)}, ψ|_{I×S^{k−1}}, and h_{S^{k}\Int(V)}. By construction, f is regularly homotopic to f_{1} in Int(X̂). In addition, one has that selfintsec_{G}(f_{1}) = selfintsec_{G}(f) + g in ℤ[G]/min^{λ}(ℤ[G]). Q.E.D.

4. Doubly parametrized quadratic modules

Let R denote a commutative ring with the unity, such that a ≡ a^{2} mod 2R for all a ∈ R. For applications in surgery, the ring ℤ of integers and the ring U^{−1}ℤ of U-fractions of ℤ, where U is a multiplicative set in ℤ, will be of primary interest. Let λ = 1 or −1 and let w : G → {±1} be a homomorphism. In the following, the ring A = R[G] is equipped with the antiinvolution − defined by (∑_{g∈G} a_{g}g)^{−} = ∑_{g∈G} a_{g}w(g)g^{−1} (a_{g} ∈ R). Let

G(2) = {g ∈ G | g^{2} = 1, and g ≠ 1}.

G acts on G(2) by conjugation f ↦ gfg^{−1} (g ∈ G(2), g ∈ G). Let Q and S be conjugationinvariant subsets of G(2) satisfying

(QC)  Q ⫅ \{g ∈ G(2) | g = −λ\overline{g}\}, and

(SC)  S ⫅ \{g ∈ G(2) | g = λ\overline{g}\}.

We define three R-submodules A_{q}, A_{s}, and Λ of A as follows:

A_{q} = A_{q}(G, S) := R[G \ S], A_{s} = A_{s}(G, S) := R[S], and

Λ = Λ(G, Q; R) := min^{λ}(R[G]) + R[Q] (the form parameter generated by Q),

where min^{λ}(R[G]) is the minimal form parameter of R[G], i.e.

min^{λ}(R[G]) = ⟨x − λ\overline{x} | x ∈ A⟩_{R}.

Clearly A_{q}(G, S) = Map(G \ S, R) and A_{s}(G, S) = Map(S, R) as R-modules. In the following, let

A A A = (R, G, Q, S, λ, w).

Definition 4.1. A map B : M × M → A (where M is a finitely generated A-module) is called a λ-Hermitian form on M if (4.1.1)–(4.1.3) are satisfied:

(4.1.1) B is biadditive,

(4.1.2) B(ax, by) = bB(x, y)\overline{a},

(4.1.3) B(x, y) = λ\overline{B(y, x)},

for all x, y ∈ M, a, b ∈ A. A map q : M → A_{q}/Λ is called an A A A-quadratic form (or simply quadratic form) on M with respect to B if (4.1.4)–(4.1.6) are fulfilled:

(4.1.4) q(gx) = gq(x)\overline{g} and q(rx) = r^{2}q(x) in A_{q}/Λ = A/(Λ + A_{s}),

(4.1.5) q(x + y) − q(x) − q(y) = B(x, y) in A_{q}/Λ = A/(Λ + A_{s}), and

(4.1.6) q̃(x) + λ\overline{q̃(x)} = B(x, x) in A_{q} = A/A_{s} where q̃(x) is a lifting of q(x),

for all x, y ∈ M, r ∈ R, and g ∈ G. A triple M M M = (M, B, q) consisting of a finitely generated A-module M, a λ-Hermitian form B on M and an A A A-quadratic form q on M with respect to B, is called a doubly parametrized A A A-quadratic module (or simply quadratic module).

Let ((proj)) be the category of all finitely generated projective A-modules, ((s-free)) the category of all finitely generated stably free A-modules, and ((free)) the category of all finitely generated free A-modules. Let C be one of ((proj)), ((s-free)), and ((free)). If M, M′ ∈ C (more precisely Obj(C)) then Mor(M, M′) is the set of all A-linear isomorphisms M → M′. Let Q Q Q(A A A)_{C} be the category of all quadratic modules M M M = (M, B, q) such that M ∈ C, and B is nonsingular. If M M M = (M, B, q), M M M′ = (M′, b′, q′) ∈ Q Q Q(A A A)_{C} then Mor(M M M,M M M′) is the set of all A-linear isomorphisms f : M → M′ such that B(x, y) = B′(f(x), f(y)) and q(x) = q′(f(x)) (∀x, y ∈ M).

Let Θ be a finite G-set. A G-map α : Θ → M (where M is a G-module) will be called a positioning map. Let

Q Q Q(A A A, Θ)_{C}

be the category of all M M M = (M, B, q, α) such that (M, B, q) ∈ Q Q Q(A A A)_{C} and α : Θ → M is a G-map. If M M M = (M, B, q, α), M M M′ = (M′, B′, q′, α′) ∈ Q Q Q(A A A, Θ)_{C} then Mor(M M M,M M M′) is the set of all morphisms f : (M, B, q) → (M′, B′, q′) such that α(x) = α′(f(x)) (∀x ∈ Θ). For M M M = (M, B, q, α), an A-direct summand L of M is called a C-Lagrangian of M M M if L ∈ C, B(L, L) = 0, q(L) = 0, L = L^{⊥}, and α(Θ) ⊂ L, where

L^{⊥} = {x ∈ M | B(x, y) = 0 (∀y ∈ L)}.

If M M M has a C-Lagrangian then M M M is called a C-null module. Define KQ_{0}(A A A, Θ)_{C} to be the Grothendieck group of the category Q Q Q(A A A, Θ)_{C} with respect to orthogonal sum. If C ⫆ D ⫆ ((free)), define

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

In the remainder of this paper we treat only the case that Θ = S and the action of G on S is via conjugation. To M M M = (M, B, q, α) ∈ Q Q Q(A A A, S)_{C}, we associate a function ∇ = ∇_{M M M} : M → Map(S, R/2R) defined by

(4.2)  ∇(x)(s) = [ε(B(α(s) − x, sx))], (x ∈ M, s ∈ S).

where ε : A → R is the ring homomorphism ∑_{g∈G} a_{g}g ↦ a_{1} (a_{g} ∈ R).

Lemma 4.3. Let M M M = (M, B, q, α) ∈ Q Q Q(A A A, S)_{C}. Then for each a, b ∈ R, x, y ∈ M, and s ∈ S, one has the formula

∇_{M M M}(ax + by)(s) = a∇_{M M M}(x)(s) + b∇_{M M M}(y)(s) in R/2R.

Since the proof follows by straightforward calculation from Definition (4.2), we omit it (note that a^{2} ≡ a mod 2R for a ∈ R).

A quadratic module M M M with positioning map is called a special quadratic module if ∇_{M M M} = 0. Let SQ SQ SQ(A A A, S)_{C} be the full subcategory of Q Q Q(A A A, S)_{C} consisting of all special quadratic modules. Define SKQ_{0}(A A A, S)_{C} to be the Grothendieck group of the category SQ SQ SQ(A A A, S)_{C} with respect to orthogonal sum. If C ⫆ D ⫆ ((free)), define

SW Q_{0}(A A A, S)_{C,D} = SKQ_{0}(A A A, S)_{C}/⟨D-null modules in SQ SQ SQ(A A A, S)_{D} ⟩.

Now let n = 2k be an even integer ≧ 6, and λ = (−1)^{k}. Set

(4.4)  W_{n}(A A A, S)_{C} = SW Q_{0}(A A A, S)_{C,((free))}.

Let NSQ NSQ NSQ(A A A, S)_{C} denote the full subcategory of SQ SQ SQ(A A A, S)_{C} consisting of all C-null modules.

Proposition 4.5. NSQ NSQ NSQ(A A A, S)_{((free))} is a cofinal subcategory of SQ SQ SQ(A A A, S)_{C}. That is, each M M M ∈ SQ SQ SQ(A A A, S)_{C} is a direct summand of some N N N ∈ NSQ NSQ NSQ(A A A, S)_{((free))}.

Proof. Let M M M = (M, B, q, α). Since B is nonsingular, M is a selfdual A-module, namely M ≅ M^{#} := Hom_{A}(M, A). Since M is a finitely generated projective A-module, M is a direct summand of A^{m} for large m. Say A^{m} = M ⊕M′. Let H H H(M′) = (M′⊕M′^{#}, B′, q′, 0) be the hyperbolic module: B′((x, f), (x′, f′)) = f(x′) + λ\overline{f′(x)} ∈ A for x, x′ ∈ M′ and f, f′ ∈ M′^{#}; q′((x, f)) = [f(x)] ∈ A/(Λ + A_{s}); and 0 : S → M′ ⊕ M′^{#} is the trivial map. Then M M M ⊕M M M ⊕H H H(M′) has the underlying A-module M ⊕ M^{#} ⊕ M′ ⊕ M′^{#}, namely A^{2m}. It is easy to check that H H H(M) ∈ NSQ NSQ NSQ(A A A, S)_{C}. This allows us to assume that M is a free A-module.

Now let M′ be a copy of M and ψ : M → M′ an A-isomorphism. Define B′ : M′×M′ → A, q′ : M′ → A_{q}/Λ, and α′ : S → M′ as follows: B′(ψ(x), ψ(y)) = −B(x, y), q′(ψ(x)) = −q(x), and α′(s) = ψ(α(s)), for x, y ∈ M and s ∈ S. Set M M M′ = (M′, B′, q′, α′). Obviously M M M′ ∈ SQ SQ SQ(A A A, S)_{((free))}. Now consider N N N = M M M ⊕ M M M′. Then L = {(x, ψ(x)) | x ∈ M} is a ((free))-Lagrangian of N N N. More precisely, one has that

(B ⊕ B′)((x, ψ(x)), (x, ψ(x))) = B(x, x) + B′(ψ(x), ψ(x)) = 0,
(q ⊕ q′)(x, ψ(x)) = q(x) + q′(ψ(x)) = 0, and
(α ⊕ α′)(s) = (α(s), α′(s)) = (α(s), ψ(α(s))) ∈ L.

Moreover L ≅ M is an A-free, direct summand of M ⊕ M′ (M ⊕ 0 is a complementary summand to L). Q.E.D.

It is easy to see that the canonical homomorphism W_{n}(A A A, S)_{((free))} → W_{n}(A A A, S)_{((s−free))} is an isomorphism and that the canonical homomorphism W_{n}(A A A, S)_{((s−free))} → W_{n}(A A A, S)_{((proj))} is injective. We could define W_{n}(A A A, Θ)_{C} for more general Θ, but we omit such generalizations for simplicity. For suitable Q, S, λ, and w, the group W_{n}(A A A, S)_{C} will be called a G-surgery obstruction group.

5. Metabolic planes for special quadratic modules

We construct specific quadratic modules with positioning map whose ∇-invariant is trivial, called special metabolic planes and use them to decompose ((free))-null modules with trivial ∇.

Definition 5.1. Let β = (c, r) be a pair of elements c, r ∈ Map(S, R) such that

(BC) c(s) ≡ r(s) mod 2R for any s ∈ S.

Let x and y be distinct letters. The special metabolic plane

M M M(x, y, β) = (M(x, y), B_{r}, q, α_{c})

associated to β with metabolic basis {x, y} is defined as follows. Let M = M(x, y) be the free R[G]-module with basis {x, y}, i.e. M(x, y) = ⟨x, y⟩_{R[G]}. Let B_{r} : M × M → R[G] be the unique map satisfying Relations (4.1.1)–(4.1.3) and

(5.1.1) B_{r}(x, x) = 0, B_{r}(y, x) = 1, and B_{r}(y, y) = ∑_{g∈S}r(g)g.

Define q : M → R[G \ S]/Λ(G, Q; R) to be the unique map satisfying Relations (4.1.4)– (4.1.5) and

(5.1.2) q(x) = 0 and q(y) = 0.

Clearly, for a, b ∈ R[G],

q(ax + by) = B(by, ax) = a\overline{b} ∈ R[G \ S]/Λ(G, Q; R) = R[G]/(Λ(G, Q; R) + R[S]).

Thus, (4.1.6) is satisfied. Let G act as usual on S by conjugation (hence g · s = gsg^{−1}), and define a map α_{c} : S → M by

(5.1.3) α_{c}(s) = ∑_{g∈G}c(g · s)g^{−1}x,

Clearly α_{c} is a G-map (a positioning map).

Proposition 5.2. Let M M M(x, y, β) be a special metabolic plane as in Definition 5.1. Then M M M(x, y, β) belongs to SQ SQ SQ(A A A, S)_{((free))}.

Proof. By Lemma 4.3, it suffices to prove that ∇_{M M M(x,y,β)}(ax)(s) = 0 and ∇_{M M M(x,y,β)}(ay)(s) = 0 for every a ∈ G and s ∈ S.

The second equality holds because

∇_{M M M(x,y,β)}(ay)(s) = [ε(B_{r}(α_{c}(s) − ay, say))]

= [ε(B_{r}(∑_{g∈G} c(gsg^{−1})g^{−1}x, say))] − [ε(B_{r}(ay, say))]

= [ε(λ ∑_{g∈G} w(g)c(gsg^{−1})sag)] − [ε(∑_{h∈S} r(h)sah\overline{a})]

= [λw(sa)c(a^{−1}sa)] − [w(a)r(a^{−1}sa)] = 0 ∈ R/2R.

The first equality is straightforward to check. Q.E.D.

Lemma 5.3. Let β = (c, r) and β′ = (c′, r′) satisfy Condition (BC). If there exists an a ∈ R^{×} := Unit(R) such that ac(s) = c′(s) and a^{2}r(s) = r′(s) for any s ∈ S then M M M(x, y, β) is isomorphic to M M M(x′, y′, β′).

Proof. Let f : M(x, y) → M(x′, y′) denote the R[G]-linear map determined by the equations f(x) = ax′ and f(y) = a^{−1}y′. Then

f(α_{c}(s)) = f(∑_{g∈G} c(gsg^{−1})g^{−1}x)

= ∑_{g∈G} ac(gsg^{−1})g^{−1}x′ = α_{c′}(s), and

B_{r′}(f(y), f(y)) = B_{r′}(a^{−1}y′, a^{−1}y′)

= a^{−2} ∑_{s∈S} r′(s)s

= ∑_{s∈S} r(s)s = B_{r}(y, y).

Using this, the reader can easily check that f is an isomorphism. Q.E.D.

Proposition 5.4. Suppose R is the ring U^{−1}ℤ of U-fractions of ℤ where U is a multiplicative set in ℤ. Let β = (c, r) be a pair of elements c, r ∈ Map(S, R) satisfying Condition (BC). Then there exists a pair β′ = (c′, r′) of elements c′, r′ ∈ Map(S, ℤ) satisfying (BC) such that M M M(x, y, β) is isomorphic to M M M(x′, y′, β′).

Proof. Since S is finite, there is an integer a ∈ U such that ac, ar ∈ Map(S, ℤ). Set β′ = (ac, a^{2}r). Then by Lemma 5.3, M M M(x, y, β) ≅ M M M(x′, y′, β′). Q.E.D.

Lemma 5.5. Let β = (c, r) and β′ = (c′, r′) satisfy the Condition (BC). If c(s) = c′(s) and r(s) ≡ r′(s) mod 2R for any s ∈ S then M M M(x, y, β) is isomorphic to M M M(x′, y′, β′). Thus the isomorphism class of M M M(x, y, (c, r)) depends only on c.

Proof. By hypothesis, there exists an a ∈ R[S] such that a + λ\overline{a} = ∑_{s∈S}(r′(s) − r(s))s. Let

f : M(x, y) → M(x′, y′) be an R[G]-linear map such that f(x) = x′ and f(y) = y′ − ax′. Then

B_{r′}(f(y), f(y)) = B_{r′}(y′ − ax′, y′ − ax′)

= B_{r′}(y′, y′) − B_{r′}(y′, ax′) − B_{r′}(ax′, y′) + B_{r′}(ax′, ax′)

= (∑_{s∈S} r′(s)s) − a − λ\overline{a} + 0

= ∑_{s∈S} r(s)s = B_{r}(y, y).

Using this, the reader can check easily that f is an isomorphism. Q.E.D.

Theorem 5.6. If M M M = (M, B, q, α) is a ((free))-null, special quadratic module with Lagrangian L then there exist pairs β_{i} = (c_{i}, r_{i}), where c_{i}, r_{i} ∈ Map(S, R) satisfying (BC) (i = 1, ⋯, m = rank_{R[G]}L) such that M M M is isomorphic to M M M(x_{1}, y_{1}, β_{1})⊕⋯⊕M M M(x_{m}, y_{m}, β_{m}).

The result above is proved in the next section.

Corollary 5.7. Suppose R is a ring of fractions of ℤ. If M M M = (M, B, q, α) is a ((free))- null, special quadratic module with Lagrangian L then there exist pairs β_{i} = (c_{i}, r_{i}) where c_{i}, r_{i} ∈ Map(S, ℤ) satisfying (BC) (i = 1, ⋯, m = rank_{R[G]}L) such that M M M is isomorphic to M M M(x_{1}, y_{1}, β_{1}) ⊕ ⋯ ⊕ M M M(x_{m}, y_{m}, β_{m}).

Proof. The result follows immediately from Proposition 5.4 and Theorem 5.6. Q.E.D.

Corollary 5.8. The family of special metabolic planes is cofinal in the category of special quadratic modules.

Proof. The result follows immediately from Proposition 4.5 and Theorem 5.6. Q.E.D.

Corollary 5.9. Suppose R is a ring of fractions of ℤ. Then the family of special metabolic planes M M M(x, y, (c, r)) such that c, r ∈ Map(S, ℤ) is cofinal in the category of special quadratic modules.

Proof. The result follows immediately from Proposition 5.4 and Corollary 5.8. Q.E.D.

6. Decomposition of ((free))-null modules

This section is devoted to the proof of Theorem 5.6.

Let M M M = (M, B, q, α) be a ((free))-null module with Lagrangian L. Let {x_{1}, ⋯, x_{m}} (m = rank_{A}L) be an arbitrary basis of L. Since L is a Lagrangian, the sequence

0 ⟶ L ⟶ M ⟶^{τ} Hom_{A}(L, A) ⟶ 0,

is split-exact over A where τ(y) ∈ Hom_{A}(L, A) (y ∈ M) is given by τ(y)(x) = B(y, x) (x ∈ L). Thus there exist elements y_{i} (i = 1, ⋯, m) in M such that B(y_{i}, x_{i}) = δ_{ij}. By the split-exact sequence above, {x_{i}, y_{i} | 1 ≦ i ≦ m} is a basis of M.

Lemma 6.1 (Orthonormalization of Gram-Schmidt-Wall). Suppose that for some integer k,

B(y_{i}, y_{j}) = 0 (for all i < j ≦ k), and
q(y_{i}) = 0 (for all i ≦ k).

Set

(6.1.1)  y′_{k+1} = y_{k+1} − ({ \atop q(y_{k+1})}x_{k+1} + ∑_{i≦k} B(y_{i}, y_{k+1})x_{i}) ,

where { \atop q(y_{k+1})} ∈ A is a lifting of q(y_{k+1}) ∈ A_{q}/Λ = A/(Λ + A_{s}). Then it follows that B(y′_{k+1}, x_{j}) = δ_{k+1,j} for all j, B(y_{j}, y′_{k+1}) = 0 for all j < k + 1, and q(y′_{k+1}) = 0.

Since the proof is a straightforward calculation, we omit it.

Inductive use of Lemma 6.1 on k produces the next corollary.

Corollary 6.2. For an arbitrary basis {x_{1}, ⋯, x_{m}} of L, there exist elements y_{1}, ⋯, y_{m} ∈ M such that B(y_{i}, x_{j}) = δ_{ij} (for all i, j), B(y_{i}, y_{j}) = 0 (for all i ≠ j), and q(y_{i}) = 0 (for all i).

Theorem 6.3. Let M M M = (M, B, q, α) be a ((free))-null module with Lagrangian L = ⟨x⟩_{A}. Let y ∈ M be an element such that B(y, x) = 1 and q(y) = 0. Then M M M is isomorphic to the special metabolic plane M M M(x, y, β) associated to β = (c_{α}, r), where c_{α}, r ∈ Map(S, R) are determined by the equations

α(s) = ∑_{g∈G} c_{α}(gsg^{−1})g^{−1}x (∀s ∈ S), and B(y, y) = ∑_{s∈S} r(s)s.

For the moment, assume that Theorem 6.3 has been proved.

Proof that Theorem 6.3 ⟹ Theorem 5.6. We shall prove that M M M is isomorphic to an orthogonal sum of special metabolic planes associated to certain β_{i} = (c_{i}, r_{i}) where

(6.4) c_{i}, r_{i} ∈ Map(S, R).

Let {x_{1}, ⋯, x_{m}} be a basis of L and let {y_{1}, ⋯, y_{m}} be as in Corollary 6.2. Set M_{i} = ⟨x_{i}, y_{i}⟩_{A}, B_{i} = B|_{M_{i}} : M_{i} × M_{i} → A, and q_{i} = q|_{M_{i}} : M_{i} → A_{q}/Λ. Let p_{i} : M → M_{i} be the projection with respect to the basis {x_{i}, y_{i} | 1 ≦ i ≦ m}. Set α_{i} = p_{i} ∘ α : S → M_{i}. It is easy to check that M M M_{i} = (M_{i}, B_{i}, q_{i}, α_{i}) ∈ SQ SQ SQ(A A A, S)_{((free))} with ((free))-Lagrangian L_{i} = ⟨x_{i}⟩_{A}. Now use Theorem 6.3 to deduce that each M M M_{i} is isomorphic to a special metabolic plane. Thus M M M = M M M_{1} ⊕ ⋯ ⊕ M M M_{m} is isomorphic to a orthogonal sum of special metabolic planes. Q.E.D.

The rest of this section is devoted to the proof of Theorem 6.3.

Let M M M = (M, B, q, α) and x, y be as in Theorem 6.3. For every s ∈ S, α(s) has the form

(6.5)  α(s) = ∑_{g∈G} a_{α}(s, g)g^{−1}x,

where a_{α}(s, g) ∈ R. As usual G acts on S by conjugation. The isotropy subgroup G_{s} (at s in the G-space S) is {h ∈ G | hsh^{−1} = s}. We define Σ_{G_{s}} ∈ A by

Σ_{G_{s}} = ∑_{h∈G_{s}} h.

Since α is a G-map, α(s) is G_{s}-invariant. Thus α(s) has the form

(6.6)  α(s) = ∑_{gG_{s}∈G/G_{s}} â_{α}(s, gsg^{−1})Σ_{G_{s}}g^{−1}x,

where â_{α}(s, gsg^{−1}) = a_{α}(s, g).

Lemma 6.7. In the above situation, â_{α}(s, gsg^{−1}) = â_{α}(fsf^{−1}, gsg^{−1}) for any f ∈ G.

Proof. Let h ∈ G. Then

hα(s) = ∑_{gG_{s}∈G/G_{s}} â_{α}(s, gsg^{−1})hΣ_{G_{s}}g^{−1}x

= ∑_{gG_{s}∈G/G_{s}} â_{α}(s, gsg^{−1})hΣ_{G_{s}}h^{−1}(gh^{−1})^{−1}x

= ∑_{gG_{s}∈G/G_{s}} â_{α}(s, gsg^{−1})Σ_{G_{hsh^{−1}}} (gh^{−1})^{−1}x

= ∑_{(gh^{−1})G_{hsh^{−1}} ∈G/G_{hsh^{−1}}} â_{α}(s, gsg^{−1})Σ_{G_{hsh^{−1}}} (gh^{−1})^{−1}x

= ∑_{g′G_{hsh^{−1}} ∈G/G_{hsh^{−1}}} â_{α}(s, g′(hsh^{−1})g′^{−1})Σ_{G_{hsh^{−1}}} g′^{−1}x.

On the other hand,

α(hsh^{−1}) = ∑_{g′G_{hsh^{−1}} ∈G/G_{hsh^{−1}}} â_{α}(hsh^{−1}, g′(hsh^{−1})g′^{−1})Σ_{G_{hsh^{−1}}} g′^{−1}x.

Since hα(s) = α(hsh^{−1}), we get

â_{α}(s, g′(hsh^{−1})g′^{−1}) = â_{α}(hsh^{−1}, g′(hsh^{−1})g′^{−1})

for all g′. Substitute now in the equation above f^{−1}, g and fsf^{−1} for h, g′, and s, respectively. Then we obtain that â_{α}(fsf^{−1}, gsg^{−1}) = â_{α}(s, gsg^{−1}). Q.E.D.

Lemma 6.8. If c_{α} : S → R is defined by c_{α}(s) = â_{α}(s, s) then

α(s) = ∑_{g∈G} c_{α}(gsg^{−1})g^{−1}x.

Proof. This is shown by straightforward calculation:

α(s) = ∑_{gG_{s}∈G/G_{s}} â_{α}(s, gsg^{−1})Σ_{G_{s}}g^{−1}x

= ∑_{gG_{s}∈G/G_{s}} â_{α}(gsg^{−1}, gsg^{−1})Σ_{G_{s}}g^{−1}x

= ∑_{gG_{s}∈G/G_{s}} c_{α}(gsg^{−1})Σ_{G_{s}}g^{−1}x

= ∑_{g∈G} c_{α}(gsg^{−1})g^{−1}x. Q.E.D.

Lemma 6.9. If r : G → R is defined by B(y, y) = ∑_{g∈G} r(g)g then r(g) = 0 for all g ∈ G \ S.

Proof. The conclusion follows immediately from the hypothesis q(y) = 0 and the property (4.1.6). Q.E.D.

Putting Lemmas 6.8 and 6.9 together, we get M M M ≅ M M M(x, y, β). This completes the proof of Theorem 6.3.

7. G-Surgery theorem

Throughout this section let n = 2k be an even integer ≧ 6, let X and Y be closed manifolds in Mnf Mnf Mnf_{cp,sg}^{n}(G), let λ = (−1)^{k} and w = w_{X}^{G}, and let R be a ring of fractions of ℤ.

A pair (f, b) is called a G-framed map if f : X → Y is a G-map and b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η) is a G-vector bundle isomorphism (covering the identity map on X) for real G-vector bundles η and ξ over Y. A G-framed map (f, b) is said to be of degree 1 (resp. k-connected) if f is of degree 1 (resp. k-connected).

If V is a real G-module, let ε_{X}(V) denote the product bundle X × V → X with fiber V. Let ℝ be the 1-dimensional, trivial, real G-module.

In the sequel we always assume

(HC) the bundle η is sufficiently large; more precisely, η ⫆ ε_{Y} (ℝ^{n+1}) where n = dim X.

Proposition 10.1 in the appendix demonstrates one advantage of this assumption.

Let I = [0, 1] and let p_{Y} : I × Y → Y be the canonical projection. For a closed subset Z ⫅ X, a cobordism (F, B) : (f, b) ∼ (f′, b′) relative to Z is defined in the usual way:

F : W → (I × Y) (I = [0, 1]) is a G-map such that ∂W = (−X) ∪ X′, F (−X) ⫅ ({0} × Y), F (X′) ⫅ ({1} × Y), F |_{−X} = f, F |_{X′} = f′, (where (I × Z) ⫅ W in a canonical way, and F |_{I×Z} = id_{I} × f|_{Z}); B : T (W) ⊕ (p_{Y} ∘ F)^{∗}η → (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η) is a real G-vector bundle isomorphism such that B|_{−X} = id_{ε_{−X}(ℝ)} ⊕ b and B|_{X′} = id_{ε_{X′} (ℝ)} ⊕ b′, (T (W)|_{I×Z} = ε_{I}(ℝ) × T (X)|_{Z} in a canonical way, and B|_{I×Z} = id_{ε_{I}(ℝ)} × b|_{Z}).

Our first goal is to define a quadruple M M M_{f f f} = (K_{k}(f; R), B_{f}, q_{f f f}, α_{f}) for any k-connected, degree 1, G-framed map fff = (f, b). Let fff = (f, b) be a degree 1, G-framed map and let A A A = (R, G, Q, S, λ, w, S), where Q = Q(G, X) and S = S(G, X). Let

pdual_{X} : H^{∗}(X; R) → H_{∗}(X; R)

denote the Poincaré duality homomorphism, and let

ppair_{X} : H_{k}(X; R) × H_{k}(X, R) → R

denote the Poincaré pairing. For each integer ℓ, define

K_{ℓ}(f; R) = Ker[f_{∗} : H_{ℓ}(X; R) → H_{ℓ}(Y; R)], and
K^{ℓ}(f; R) = Coker[f^{∗} : H^{ℓ}(Y; R) → H^{ℓ}(X; R)].

Suppose that f is k-connected.

This assumption implies by [5, I.2.8] that K_{k}(f; R) = R ⊗_{ℤ} K_{k}(f; ℤ) ≅ K^{k}(f; R) = Hom_{ℤ}(K_{k}(f; ℤ), R) as R-modules and that these modules are finitely generated, free Rmodules. The λ-Hermitian module (H_{k}(X; R), ppair) over R can be decomposed into the orthogonal sum (K_{k}(f; R), ppair|) ⊕ (pdual(Im(f^{∗})), ppair|). It is well-known that (K_{k}(f; R), ppair|) is nonsingular ([5, I.2.9]). Let π_{f} : H_{k}(X; R) → K_{k}(f; R) be the canonical projection, namely

π_{f}(x) = x − pdual_{X} ∘ f^{∗} ∘ pdual_{Y}^{−1} ∘ f_{∗}(x).

We treat first the case to R = ℤ and define B_{f}, q_{f f f}, and α_{f} for K_{k}(f; ℤ). This done, we extend B_{f}, q_{f f f}, and α_{f} to K_{k}(f; R) in the usual way, using the fact that K_{k}(f; R) ≅ R ⊗_{ℤ} K_{k}(f,; ℤ).

For the moment we forget the G-action on X and apply the ordinary surgery theory of C. T. C. Wall. Since f : X → Y is k-connected, the canonical map π_{k+1}(f) → K_{k}(X; ℤ) is surjective. Thus each element x ∈ K_{k}(X; ℤ) can be represented by a continuous map h′_{x} : S^{k} → X such that f ∘ h′_{x} is null homotopic in Y. This h′_{x} can be approximated by an immersion. Since f ∘ h′_{x} is null homotopic, h′_{x}^{∗}(T (X) ⊕ f^{∗}η) ≅ (f ∘ h′_{x})^{∗}(ξ ⊕ η) is a trivial bundle. Thus, it follows from Hirsch’s immersion classification theorem that the map h′_{x} is homotopic to an immersion h_{x} with trivial normal bundle in X. Moreover the regular homotopy class of h_{x} in X is uniquely determined by x (providing, of course, f and b are fixed).

It is well known that ppair(x, y) = ±intsec(h_{x}, h_{y}) (∀x, y ∈ K_{k}(f; ℤ)). The sign ± is determined by the definitions of ppair and intsec. We shall adopt definitions such that ppair(x, y) = (−1)^{k}intsec(h_{x}, h_{y}). (The sign will not be essential in our arguments. A reader preferring definitions of ppair and intsec such that ppair(x, y) = intsec(h_{x}, h_{y}) can easily modify the arguments.)

Reimpose now the G-action on X. Define B_{f} : K_{k}(f; ℤ) × K_{k}(f; ℤ) → ℤ[G] by

B_{f}(x, y) = ∑_{g∈G} intsec(h_{x}, g^{−1}h_{y})g (= (−1)^{k} ∑_{g∈G} ppair(x, g^{−1}y)g) .

Then (K_{k}(f; ℤ), B_{f}) is a nonsingular λ-Hermitian module over ℤ[G] by [5, I.2.9] and [1, (1.2.4)].

Define q_{f f f} : K_{k}(f; ℤ) → ℤ[G \ S]/Λ(G, X; ℤ) by

q_{f f f}(x) = μ_{X}(h_{x}).

By Hypothesis (2.1.5), each X_{γ} (γ ∈ Θ(G, X, k)) has the orientation class ori(X_{γ}) ∈ H_{k}(X_{γ}; ℤ). Let j_{γ} : X_{γ} → X be the canonical inclusion. Adopting the identification in Lemma 2.6, define α_{f} : S = Θ(G, X, k) → K_{k}(f; ℤ) by

α_{f}(s) = α_{f}(γ(s)) = π_{f} ∘ j_{γ(s)}_{∗}(ori(X_{γ(s)})).

By (2.1.5), α_{f} is a G-map.

This completes the definition of the quadruple

M M M_{f f f} = (K_{k}(f; R), B_{f}, q_{f f f}, α_{f})

for R = ℤ.

We consider next the case of a general R. There is a canonical homomorphism φ : K_{k}(f; ℤ) → K_{k}(f; R) and the induced R-homomorphism R ⊗ φ : R ⊗ K_{k}(f; ℤ) → K_{k}(f; R) is an isomorphism by the universal coefficient theorem [5, I.2.8]. Thus we can extend B_{f} above to a pairing B_{f} : K_{k}(f; R) × K_{k}(f; R) → R[G] by using the rule B_{f}(rx, r′y) = rr′B_{f}(x, y) (r, r′ ∈ R, x, y ∈ K_{k}(f; R)), q_{f f f} above to a map q_{f f f} : K_{k}(f; R) → R[G]_{q}/Λ(G, X; R) by using the rule q_{f}(rx) = r^{2}q_{f}(x) (r ∈ R, x ∈ K_{k}(f; R)), and α_{f} above to a function α_{f} : S → K_{k}(f; R) by composing it with φ. It is straightforward to check that B_{f} is a nonsingular form over R[G], that α_{f} is a G-map, and that B_{f} and q_{f f f} satisfy (4.1.1)–(4.1.6).

Lemma 7.1. Let X and Y be closed G-manifolds in Mnf Mnf Mnf_{cp,sg}^{n}(G) (n = 2k ≧ 6) and let fff = (f, b) be a k-connected, degree 1, G-framed map. Let A A A = (R, G, Q, S, λ, w) where Q = Q(G, X), S = S(G, X), λ = (−1)^{k}, and w = w_{X}^{G}. Let C = ((proj)), ((s − free)), or ((free)). Suppose K_{k}(f; R) ∈ C. Then the quadruple M M M_{f f f} = (K_{k}(f; R), B_{f}, q_{f f f}, α_{f}) belongs to SQ SQ SQ(A A A, S)_{C}.

Proof. Set ∇ = ∇_{M M M_{_{f}_{f}_{f}}}. It suffices to show that ∇(x)(s) = 0 for each x ∈ K_{k}(f; R) and s ∈ S = S(G, X).

Let j_{γ(s)} : X_{γ(s)} → X be the canonical inclusion and let h_{x} : S^{k} → X be an immersion with trivial normal bundle, representing x. Without loss of generality, we may suppose that h_{x}, sh_{x}, and j_{γ(s)} transversally intersect one another (cf. Lemma 9.1). If z ∈ X is an intersection point of h_{x} and sh_{x} then so is sz ∈ X. Thus intsec(h_{x}, sh_{x}) ≡ intsec(h_{x}, j_{γ(s)}) mod 2. It is obvious that intsec(h_{x}, j_{γ(s)}) ≡ intsec(sh_{x}, j_{γ(s)}) mod 2. Thus for R = ℤ we obtain using (4.2) that

∇(x)(s) = [ε(B_{f}(α_{f}(s) − x, sx))]
= [ε(B_{f}(α_{f}(s), sx))] − [ε(B_{f}(x, sx))]
= [intsec(j_{γ(s)}, sh_{x})] − [intsec(h_{x}, sh_{x})]
= 0 in ℤ/2ℤ.

Consider now the general case. Clearly for each x ∈ K_{k}(f; R), there are elements a ∈ R and y ∈ K(f; ℤ) such that x = ay. By Lemma 4.3 , ∇(ay) = a∇(y) and by the case R = ℤ above, ∇(y) = 0. Q.E.D.

Definition 7.2. In the situation of Lemma 7.1, define σ(fff) to be the element in W_{n}(A A A, S)_{C} determined by the quadruple M M M_{f f f}.

Theorem 7.3. Let X and Y be closed G-manifolds in Mnf Mnf Mnf_{cp,sg}^{n}(G) (n = 2k ≧ 6) and let fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η)) be a k-connected, degree 1 G-framed map. Suppose that K_{k}(f; R) belongs to C. If σ(fff) = 0 in W_{n}(A A A, S)_{C} then fff can be converted by G-surgery of isotropy type {1} to a k-connected, degree 1, G-framed map fff′ = (f′ : X′ → Y, b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) (thus fff ∼ fff′ rel. Sing(G, X)) such that f′ : X′ → Y is an R-homology equivalence.

This will be proved in the next section.

Concerning the assumption in Theorem 7.3 that K_{k}(f; R) belongs to C, the following is known.

Remark 7.4. Let f : X → Y be a k-connected, degree 1 G-map. Then the following are true.

(7.4.1) If f^{P} : X^{P} → Y^{P} is an R_{(p)}-homology equivalence for any p-subgroup P ≠ {1} of G (where p ranges over the set of all primes dividing |G|) then K_{k}(f; R) is a projective R[G]-module.

(7.4.2) If f^{H} : X^{H} → Y^{H} is an R-homology equivalence for any hyperelementary subgroup H ≠ {1} of G then K_{k}(f; R) is a stably free R[G]-module.

Proof of Theorem 1.1. Since Y is 1-connected and f : X → Y is k-connected, X is 1-connected. Condition (2.1.1) follows from (1.1.1); Conditions (2.1.2), (2.1.3) and (2.1.5) follow from (1.1.2);Condition (2.1.4) follows from (1.1.3). Thus X and Y belong to Mnf Mnf Mnf_{cp,sg}^{n}(G) (see §4). Since f has degree 1, it follows that w_{X}^{G} = w_{Y}^{G}. Set W (G, Y; R) = W_{n}(A A A, S)_{((s−free))} (cf. (4.4)) for A A A = (R, G, Q, S, λ, w_{Y}^{G}). Theorem 1.1 follows now from Theorem 7.3. Q.E.D.

8. Algebraic triviality and geometric deformation

This section is devoted to the proof of Theorem 7.3.

Throughout the section, X and Y are closed G-manifolds in Mnf Mnf Mnf_{cp,sg}^{n}(G) (n = 2k ≧ 6), and fff = (f, b) is a k-connected, degree 1, G-framed map consisting of f : X → Y and b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η). We set λ = (−1)^{k}, w = w_{X}^{G}, Q = Q(G, X), S = S(G, X), and A A A = (R, G, Q, S, λ, w).

Theorem 8.1. Let K_{k}(f; R) ∈ C and let β = (c, r), (r, c ∈ Map(S, ℤ)) be a pair such that c(s) ≡ r(s) mod 2ℤ for all s ∈ S = S(G, X). Then fff = (f, b) can be converted by G-surgery of isotropy type ({1}) to a k-connected, degree 1, G-framed map fff′ = (f′, b′) (f′ : X′ → Y and b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) such that K_{k}(f′; R) ∈ C and M M M_{f f f}′ ≅ M M M_{f f f} ⊕ M M M(x, y, β).

This will be proved in §9.

Proof of Theorem 7.3. In outline, the proof proceeds as follows. First, we show using Theorems 5.6 and 8.1 that σ(fff) = 0 (fff = (f, b)) implies M M M_{f f f} has a free Lagrangian L after suitable G-surgery on fff. Second, we show using Theorem 2.8 that the elements x_{1}, ⋯, x_{m} of an R[G]-basis of L can be represented by smooth embeddings h_{1}, ⋯, h_{m} : S^{k} → X with trivial normal bundles such that gIm(h_{i}) ∩ g′Im(h_{j}) = ∅ unless i = j and g = g′ ∈ G. Third, we perform G-surgery along the h_{i}’s and fourth, check that the resulting fff′ = (f′, b′) has the desired properties.

We shall prove first the case R = ℤ and then show how this proof can be modified in the case of a general R. The case R = ℤ is divided into 4 steps corresponding to the 4 steps in the outline above.

Step 1. We reduce the proof to the case that M M M_{f f f} is a ((free))-null module. Suppose σ(fff) = 0. By definition, there exist M M M ∈ SQ SQ SQ(A A A, S)_{C} and N N N ∈ SQ SQ SQ(A A A, S)_{((free))} such that N N N is a ((free))-null module and

M M M_{f f f} ⊕ M M M ≅ N N N ⊕ M M M.

Since NSQ NSQ NSQ(A A A, S)_{((free))} is cofinal in SQ SQ SQ(A A A, S)_{C} (Proposition 4.5), we may assume that M M M is a ((free))-null module. Thus N N N ⊕ M M M is a ((free))-null module. By Theorem 5.6, M M M is isomorphic to a orthogonal sum of special metabolic planes. Thus by applying Theorem 8.1, we may continue the proof under the hypothesis that M M M_{f f f} is a ((free))-null module.

Let L ⊂ K_{k}(f; R) be a ((free))-Lagrangian of M M M_{f f f} and let {x_{1}, ⋯, x_{m}} be an R[G]-basis of L.

Step 2. We find nice embeddings S^{k} → Free(G, X) representing the x_{i}’s. For each i (1 ≦ i ≦ m), there is a smooth immersion h_{i} : S^{k} → X with trivial normal bundle, representing x_{i}. Since by Lemmas 2.2 and 2.6 dim Fix_{G}(> ρ_{Π}(γ(s)), X_{γ(s)}) ≦ k − 2 for all s ∈ S(G, X), we may assume that Im(h_{i}) ∩ Fix_{G}(> ρ_{Π}(γ(s)), X_{γ(s)}) = ∅ for all i (1 ≦ i ≦ m) and s ∈ S(G, X). Let j_{γ(s)} : X_{γ(s)} → X be the canonical inclusion. Since L is a Lagrangian, L ⊃ Im(α_{f}). Thus B_{f}(α_{f}(s), x_{i}) = 0. By definition,

(8.2)  α_{f}(s) = π_{f}(j_{γ(s)}_{∗}(ori(X_{γ(s)}))).

Since K_{k}(f; R) is orthogonal to pdual(Im(f^{∗})) under ppair, it follows that

ppair(y, x) = ppair(π_{f}(y), x) (∀y ∈ H_{k}(X; R) and ∀x ∈ K_{k}(f; R)).

Since ppair is equal to intsec up to sign, the equality (8.2) implies intsec_{G}(j_{γ(s)}, h_{i}) = 0. By Lemma 2.2, Fix_{G}(= ρ_{Π}(γ(s)), X_{γ(s)}) is connected whenever s ∈ S(G, X). Thus if a, b ∈ Im(h_{i}) ∩ X_{γ(s)} have opposite intersection numbers, we can take a path from a to b in Im(h_{i}) and another in Fix_{G}(= ρ_{Π}(γ(s)), X_{γ(s)}). Apply now Theorem 6.6 of [12] (a procedure for cancelling intersection points with opposite intersection numbers) to deduce that h_{i} is regularly homotopic to an immersion h′_{i} such that Im(h′_{i}) ∩ X_{γ(s)} = ∅ for all s ∈ S(G, X). Replace h_{i} by h′_{i}, 1 ≦ i ≦ m. Then for all i and j,

(8.3)  intsec_{G}(h_{i}, h_{j}) = 0, and

(8.4)  μX̂(h_{i}) = 0,

where X̂ = X \ (⋃_{γ} X_{γ}^{H}) (γ runs over Θ(G, X, k)). By [12, Theorem 6.6], the vanishing property (8.3) for i ≠ j allows us to assume that gIm(h_{i}) ∩ g′Im(h_{j}) = ∅ (i ≠ j) for all g, g′ ∈ G. Next apply Theorem 2.8 for f = h_{i}. This allows to assume that each h_{i} is a smooth embedding such that Im(h_{i}) ∩ gIm(h_{i}) = ∅ for all g ∈ G \ {1}. Thus each x_{i} is represented by a embedding h_{i} with trivial normal bundle such that gIm(h_{i})∩g′Im(h_{j}) = ∅ unless i = j and g = g′. In particular, Im(h_{i}) ⊂ Free(G, X).

Step 3. We construct fff′ = (f′, b′). There will be no essential differences here from the corresponding step in Wall’s ordinary surgery theory. Perform G-surgery on fff along the embeddings h_{1}, ⋯, h_{m}. Let F = id_{I} × f : I × X → I × Y and let B = id_{ε_{I×X}(ℝ)} ⊕ (p_{X}^{∗}b) : T (I ×X)⊕(f ∘p_{X})^{∗}η → (f ∘p_{X})^{∗}(ε_{Y} (ℝ)⊕ξ ⊕η) where we identify T (I ×X) = ε_{I×X}(ℝ)⊕ p_{X}^{∗}T (X) and p_{X} : I × X → X is the canonical projection. The embeddings h_{i} : S^{k} → X = {1} × X can be extended to framed embeddings H_{i} : S^{k} × D^{k} → {1} × X such that gIm(H_{i}) ∩ g′Im(H_{j}) = ∅ unless i = j and g = g′. Define ind^{G}H_{i} : G × S^{k} × D^{k} → {1} × X by ind^{G}H_{i}(g, s, d) = gH_{i}(s, d) (g ∈ G, s ∈ S^{k}, and d ∈ D^{k}). Construct the attaching space

\begin{aligned}W & = W (H_{1}, ⋯, H_{m}) \\ & = (I × X) ∪_{ind^{G}H_{1}∪···∪ind^{G}H_{m}} \{(G × D^{k+1} × D^{k})_{1} ∪ ⋯ ∪ (G × D^{k+1} × D^{k})_{m}\} . \end{aligned}

Define X′ by ∂W = ({0} × X) ∐ X′. Since each f ∘ h_{i} is null homotopic, there is a map d_{i} : D^{k+1} → Y such that d_{i}(x) = f ∘ h_{i}(x) for all x ∈ S^{k}. The G-map F : I × X → I × Y is extensible to a G-map F′ : W → I × Y such that F′((G × D^{k+1} × D^{k})_{i}) ⊂ {1} × Y and F′(1, x, 0) = (1, d_{i}(x)) for (1, x, 0) ∈ {1} × D^{k+1} × {0} ⊂ (G × D^{k+1} × D^{k})_{i}. If we choose appropriate H_{i}’s then the bundle isomorphism B is extensible to B′ : T (W) ⊕ (p_{Y} ∘ F′)^{∗}η → (p_{Y} ∘ F′)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η). Let f′ = F′|_{X′} : X′ → {1} × Y = Y and let b′′ = B′|_{X′} : ε_{X′}(ℝ) ⊕ T (X′) ⊕ f′^{∗}η → f′^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η). Since η satisfies (HC) in §7, b′′ is G-regularly homotopic to id_{ε_{X′} (ℝ)} ⊕ b′, where id_{ε_{X′} (ℝ)} : ε_{X′}(ℝ) → f′^{∗}ε_{Y} (ℝ) is the canonical isomorphism and b′ : T (X′)⊕f′^{∗}η → f′^{∗}(ξ ⊕η) (cf. §10). We have just obtained a G-framed map (f′, b′).

Step 4. We prove that the f′ obtained in Step 3 is a k-connected, degree 1, R-homology equivalence. These properties are independent of the G-action on X and Y in the following sense. We obtained the G-manifold X′ and the G-map f′ by G-surgery. But forgetting G-actions, these are obtained by ordinary surgery on res_{{1}}^{G}X along the basis {e_{i,g} | g ∈ G, 1 ≦ i ≦ m} (where e_{i,g} = gx_{i}) of the Lagrangian res_{{1}}^{G}L for

res_{{1}}^{G}(K_{k}(f; R), B_{f}, q_{f f f}).

It is obvious that f′ has degree 1. Furthermore the basis {e_{i,g} | 1 ≦ i ≦ m, g ∈ G} possesses complementary basis elements f_{i,g} such that

(8.5)

(res_{{1}}^{G}B_{f})(f_{i,g}, e_{i′,g′}) = δ_{(i,g),(i′,g′)} ∈ R,
(res_{{1}}^{G}B_{f})(f_{i,g}, f_{i′,g′}) = 0, and
(res_{{1}}^{G}q_{f f f})(f_{i,g}) = 0 ∈ R/(1 − λ)R,

where (res_{{1}}^{G}B_{f})(x, y) = ε ∘ B_{f}(x, y) (x, y ∈ K_{k}(f, R)) (cf. (4.2)). Thus, the arguments in [19, pp.51–52] imply that f′ is a k-connected, R-homology equivalence. Thus we have proved Theorem 7.3 in the case R = ℤ.

The case of a general R is proved as above, except one has to take a little extra care at the three places.

The first is in Step 1. Here we should replace the application of Theorem 5.6 by one of Corollary 5.7.

The second is just after Step 1. There we used an arbitrary basis {x_{1}, ⋯, x_{m}}. However in the general case, we should choose the basis such that each x_{i} ∈ K_{k}(f; R) = R⊗K_{k}(f; ℤ) lies in the image of K_{k}(f; ℤ) under the canonical homomorphism. Furthermore if 2 ∈ R^{×} then we can and should assume that each x_{i} = 2v_{i} for some v_{i} in the image of K_{k}(f; ℤ). This will guarantee that if X̂ and h′_{i} are as in Step 2 then μX̂(h′_{i}) = 0.

The third is in Step 4. In the general case, the complementary basis elements f_{i,g} should be taken so that they also lie in the image of K_{k}(f; ℤ) and (8.5) should be replaced by

(8.6)

(res_{{1}}^{G}B_{f})(f_{i,g}, e_{i,g}) are integers invertible in R,
(res_{{1}}^{G}B_{f})(f_{i,g}, e_{i′,g′}) = 0 if (i, g) ≠ (i′, g′),
(res_{{1}}^{G}B_{f})(f_{i,g}, f_{i′,g′}) = 0 if (i, g) ≠ (i′, g′), and
(res_{{1}}^{G}q_{f f f})(f_{i,g}) = 0 ∈ R/(1 − λ)R.

With these modifications, Step 1 – Step 4 will prove Theorem 7.3 in the general case. Q.E.D.

9. Special metabolic stabilization in G-surgery This section is devoted to the proof of Theorem 8.1. If M is an oriented smooth manifold of dimension m_{1} + m_{2}, if M_{1} and M_{2} are oriented

submanifolds of M of dimension m_{1} and m_{2}, respectively, and if M_{1} and M_{2} transversally intersect at only finitely many points of Int(M), let M_{1}·M_{2} denote the intersection number of M_{1} and M_{2}.

Lemma 9.1. Let ⟨s⟩ be a group of order 2 and let ℝ (resp. ℝ_{±}) be the 1-dimensional, real ⟨s⟩-module with trivial (resp. nontrivial) ⟨s⟩-action. Let M = ℝ^{k} ⊕ ℝ_{±}^{k} with standard orientation. Let

M_{1} = {(x_{1}, ⋯, x_{k}, y_{1}, ⋯, y_{k}) ∈ ℝ^{k} ⊕ ℝ_{±}^{k} | x_{1}, ⋯, x_{k} ∈ ℝ; y_{1} = x_{1}, ⋯, y_{k} = x_{k}}.

Then M_{1} · sM_{1} = λ (where λ = (−1)^{k}).

Proof. It is clear that

sM_{1} = {(x_{1}, ⋯, x_{k}, −y_{1}, ⋯, −y_{k}) | x_{1}, ⋯, x_{k} ∈ ℝ; y_{1} = x_{1}, ⋯, y_{k} = x_{k}}.

Thus the matrix corresponding to the standard ordered basis of M_{1} ⊕ sM_{1} is

P = ( \begin{matrix}1 & 0 & ⋯ & 0 & 1 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 & 0 & 1 & ⋯ & 0 \\ 0 & 0 & ⋯ & 1 & 0 & 0 & ⋯ & 1 \\ 1 & 0 & ⋯ & 0 & −1 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 & 0 & −1 & ⋯ & 0 \\ 0 & 0 & ⋯ & 1 & 0 & 0 & ⋯ & −1 \end{matrix} ) .

Since det(P) = 2^{k}(−1)^{k}, we obtain M_{1} · sM_{1} = λ.

Let fff = (f, b), where f : X → Y, b : T (X) ⊕ f^{∗}(η) → f^{∗}(ξ ⊕ η), and let β = (c, r) be as in Theorem 8.1. In particular, c : S → ℤ and r : S → ℤ satisfy the property c(g) ≡ r(g) mod 2ℤ for all g ∈ S = S(G, X). It is helpful to prove first a special case of Theorem 8.1 in order to grasp an outline of the proof.

Special Case. Here we assume that ∃s ∈ S such that ∑_{g∈S}c(g)g = s. Let γ ∈ Θ(G, X, k)

such that ρ_{Π}(γ) ∋ s (cf. Lemma 2.6). Take a point a ∈ X_{γ} such that G_{a} = ρ_{Π}(γ). The canonical inclusion X_{γ} → X is denoted by j_{γ}. Let ν = ν(X_{γ}, X) be the normal bundle of X_{γ} and regard it as an N_{G}(ρ_{Π}(γ))-tubular neighborhood of X_{γ}. Note that ρ_{Π}(γ) = G_{s} (the centralizer of s). Take a neighborhood E (≅ ℝ^{k}) of a in Fix_{G}(= ρ_{Π}(γ), X_{γ}) such that E ∩ gE ≠ ∅ ⟹ g ∈ ρ_{Π}(γ). Then ν|_{E} is a ρ_{Π}(γ)-neighborhood of a in X, which is ρ_{Π}(γ)-diffeomorphic to E × V, where V = ν|_{a} is a k-dimensional real ρ_{Π}(γ)-representation space. Note that res_{⟨s⟩}^{ρ(γ)}V ≅ ℝ_{±}^{k}. Regard the point a as the origin 0 in E × V. Let Δ : E → V be an ℝ-linear map such that Ker(Δ) = {0}. Then the graph Graph(Δ) of Δ is a k-dimensional linear subspace of E × V. We choose Δ so that Graph(Δ) is M_{1} in Lemma 9.1 when the group action is restricted to ⟨s⟩. We orient Graph(Δ) so that the ordered direct sum T_{a}(X_{γ}) ⊕ T_{a}(Graph(Δ)) has the same orientation as T_{a}(X). Let δ > 0 be a small real number and let D_{δ}(Graph(Δ)) be the closed disk of Graph(Δ) with radius δ centered at the origin (i.e. a). Take an orientation preserving (linear) diffeomorphism h′_{D} : D^{k} → D_{δ}(Graph(Δ)) such that h′_{D}(0) = a. Fix a small real number δ′ such that 0 < δ′ ≪ δ. There is a δ′-approximation h_{D} : D^{k} → ν|_{E} of h′_{D} such that h_{D} is also a smooth embedding, h_{D}(x) = h′_{D}(x) if ∥x∥ ≦ 1/2, and h := h_{D}|_{S^{k−1}} : S^{k−1} = ∂D^{k} → X satisfies the condition that if gh(x) = g′h(x′) (g, g′ ∈ G and x, x′ ∈ S^{k−1}) then g = g′ and x = x′. Set D = Im(h_{D}). Then it follows that

(9.2) the intersection number X_{γ} · D = 1.

Since h extends to h_{D}, the normal bundle of h is trivial. There is an orientation-preserving, smooth embedding H : S^{k−1} × D^{k+1} → Free(G, X) such that h = H|_{S^{k−1}×{0}} and such that if gH(x) = g′H(x′) (g, g′ ∈ G and x, x′ ∈ S^{k−1} × D^{k+1}) then g = g′ and x = x′. Thus,

ind^{G}H : G × S^{k−1} × D^{k+1} → Free(G, X), (g, x) ↦ gH(x) (g ∈ G, x ∈ S^{k−1} × D^{k+1})

is a smooth embedding.

Perform G-surgery on X along h as follows. Let I = [0, 1] and W = I × X. Regard ind^{G}H as a map to {1} × X. Construct the attaching space

W′ = W ∪_{ind^{G}H} (G × D^{k} × D^{k+1}).

Define X′ by ∂W′ = ({0} × X) ∪ X′ (disjoint union). Then the map F = id_{I} × f : W = I × X → I × Y is extensible to a G-map F′ : W′ → I × Y such that F′(X′) ⫅ {1} × Y and F′(g, p, 0) = gh_{D}(φ(p)) for g ∈ G, p ∈ D^{k} and 0 the origin of D^{k+1}, where φ : D^{k} → D^{k} is the usual orientation reversing diffeomorphism from the upper hemisphere to the lower hemisphere. Define f′ : X′ → Y by f′ = F′|_{X′} → ({1} × Y) = Y. In addition, B = id_{ε_{I}(ℝ)} × b is extensible to a G-vector bundle isomorphism B′ : T (W′) ⊕ (p_{Y} ∘ F′)^{∗}η → (p_{Y} ∘ F′)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η). Define b′′ : ε_{X′}(ℝ) ⊕ T (X′) ⊕ f′^{∗}η → f′^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η) by b′′ = B′|_{X′}. Since η is large (i.e. satisfies (HC) in §7), b′′ is G-regularly homotopic to id_{ε_{X′} (ℝ)} ⊕ b′ where b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η) (cf. Proposition 10.1). Let fff′ = (f′, b′).

We shall show that fff′ satisfies the conclusion of Theorem 8.1.

If we forget the G-actions on X and X′ then

(9.3) X′ = [{X#({g_{1}} × S^{k} × S^{k})}# ⋯]#({g_{|G|}} × S^{k} × S^{k}) (iterated connected sum),

where {g_{1}, ⋯, g_{|G|}} = G and g_{1} = 1. Clearly, X′ is 1-connected and f′ is of degree 1.

Our next goal is to obtain elements x and y of K_{k}(f′; ℤ) such that B_{f′}(x, x) = 0, q_{f f f}′(x) = 0, B_{f′}(y, x) = 1, and q_{f f f}′(y) = 0. This will be done by the procedure. First we define an element −x ∈ K_{k}(f′; ℤ) such that B_{f′}(−x, −x) = 0 and q_{f f f}′(−x) = 0. The element x we are seeking is then defined to be −(−x). Next we define an element −λz ∈ K_{k}(f′; ℤ) such that B_{f′}(−λz, x) = −λ and q_{f f f}′(−λz)(g) = 0 for all g ∈ {1}∪G(2)\(Q∪S). Set z = −λ(−λz). Then B_{f′}(z, x) = 1 and q_{f f f}′(z)(g) = 0 for all g ∈ {1}∪G(2)\(Q∪S). By the orthonormalization procedure in Lemma 6.1, there is an element v ∈ ℤ[G\({1}∪G(2))] such that q_{f f f}′(z + vx) = 0. Now we define y = z + vx. It follows that B_{f′}(y, x) = 1 and that the elements x and y have the properties sought above. To define the elements −x and −λz in K_{k}(f′; ℤ), we construct first embeddings j_{−x}, j_{−λz} : S^{k} → X′ and then set −x (resp. −λz) to be the image under the homomorphism (j_{−x})_{∗} : H_{k}(S^{k}; ℤ) → H_{k}(X′; ℤ) (resp. (j_{−λz})_{∗} : H_{k}(S^{k}; ℤ) → H_{k}(X′; ℤ)) of the orientation class of S^{k}. Fix a point pt ∈ S^{k−1} and define j_{−x} : S^{k} = ∂D^{k+1} → X′ by j_{−x}(z) = H(pt, z) (z ∈ S^{k}). The map j_{−x} will be regarded as the meridian {1} × {pt} × S^{k} in (9.3). Clearly intsec_{G}(j_{−x}, j_{δ}) = 0 (∀δ ∈ Θ(G, X′, k) = Θ(G, X, k)) (equivalently B_{f′}(x, α_{f′}(g)) = 0 (∀g ∈ S)). As j_{−x} is an embedding with trivial normal bundle such that Im(j_{−x}) ∩ gIm(j_{−x}) = ∅ whenever g ∈ G \ {1}, it follows that intsec_{G}(j_{−x}, j_{−x}) = 0 (equivalently B_{f′}(x, x) = 0), and q_{f f f}′(x) = 0. By choosing Im(H) sufficiently thin, we may suppose that D′ = D\H(S^{k−1}×Int(D^{k+1})) is diffeomorphic to the closed disk of dimension k. Define k_{−} : D_{−}^{k} → W and k_{+} : D_{+}^{k} → W′ by

\begin{matrix}k_{−} : D_{−}^{k} = D^{k} ⟶^{h_{D}} X = \{1\} × X ↪ W, and \\ k_{+} : D_{+}^{k} = D^{k} × \{0\} ↪ \{1\} × D^{k} × D^{k+1} ↪ W′. \end{matrix}

Define j′ : S^{k} = D_{−}^{k}∪D_{+}^{k} → W′ by gluing k_{−} and k_{+}. Pushing j′ into X′ within the handle {1} × D^{k} × D^{k+1}, we obtain an isotopy from j′ to a smooth embedding j_{−λz} : S^{k} → X′. We may assume that Im(j_{−λz}) = D′∪({1}×D^{k} ×{pt′}) for some pt′ ∈ S^{k}. The embedding j_{−λz} will be regarded as the longitude {1} × S^{k} × {pt′} in (9.3). Clearly B_{f′}(z, α_{f′}(g)) = 0 (∀g ∈ S \ {s}), and (q_{f f f}′(z))(g) = 0 (∀g ∈ {1} ∪ G(2) \ (Q ∪ S)). Moreover

intsec(j_{−λz}, j_{−x}) = \begin{matrix}ori & (T_{(1,pt,pt′)}(\{1\} × S^{k} × \{pt′\}) ⊕ T_{(1,pt,pt′)}(\{1\} × \{pt\} × S^{k})) \\ ori & (T_{(1,pt,pt′)}(∂(\{1\} × S^{k} × D^{k+1}))) \end{matrix}

= \frac{ori (T_{(1,pt,pt′)}(\{1\} × S^{k} × S^{k}))}{(−1)^{k}ori (T_{(1,pt,pt′)}(\{1\} × S^{k} × ∂D^{k+1}))}

= (−1)^{k} = λ

and by construction intsec(j_{−λz}, g^{−1}j_{−x}) = 0 for all g ≠ 1. Thus B_{f′}(z, x) = 1.

Let the elements x, y, z ∈ K_{k}(f′; ℤ) be determined by the procedures above. Obviously ⟨x, z⟩_{ℤ[G]} = ⟨x, y⟩_{ℤ[G]}. We claim that K_{k}(f′; R) ≅ K_{k}(f; R) ⊕ ⟨x, z⟩_{R[G]}. This can be shown by the following standard argument.

We identify X with {0} × X. Then K_{i}(F′, f; ℤ) = K_{i}(F′, F; ℤ) = H_{i}(W′, W; ℤ) ≅ ℤ[G] (resp. 0) if i = k (resp. i ≠ k). Since we can regard the element z to be in H_{k}(W′, W; ℤ), we may identify K_{i}(F′, f; ℤ) with ⟨z⟩_{ℤ[G]}. Consider the Mayer-Vietoris exact sequence

⋯ ⟶ K_{i+1}(F′, f; ℤ) ⟶ K_{i}(f; ℤ) ⟶^{(κ_{1})_{i}} K_{i}(F′; ℤ) ⟶^{(τ_{1})_{i}} K_{i}(F′, f; ℤ) ⟶ · · · .

If i ≠ k then K_{i}(f; ℤ) = 0 and K_{i}(F′, f; ℤ) = 0 and so K_{i}(F′; ℤ) = 0. Since we can regard z ∈ K_{k}(F′; ℤ), there is a ℤ[G]-splitting σ_{1} : K_{k}(F′, f; ℤ) → K_{k}(F′; ℤ) for τ_{1} = (τ_{1})_{k} such

that σ_{1}(z) = z. Thus K_{k}(f; ℤ) ⊕ ⟨z⟩_{ℤ[G]} ≅^{κ_{1}⊕ι} K_{k}(F′; ℤ) where ι : ⟨z⟩_{ℤ[G]} → K_{k}(F′; ℤ) is

the canonical inclusion. Since W ≅_{G} W′ ∪ (G × D^{k+1} × D^{k}) (the dual-handle attachment) ≃_{G} W′∪(G×D^{k+1}×{0}), it follows that K_{i}(F′, f′; ℤ) = H_{i}(G×D^{k+1}×D^{k}, G×S^{k}×D^{k}; ℤ) ≅ ℤ[G] (resp. 0) if i = k + 1 (resp. i ≠ k + 1). (This can be shown also using the universal coefficient theorem and the Poincaré-Lefschetz duality, cf. [5, I.2.8].) Now consider the Mayer-Vietoris exact sequence

⋯ ⟶ K_{i+1}(F′, f′; ℤ) ⟶^{(∂_{2})_{i+1}} K_{i}(f′; ℤ) ⟶^{(κ_{2})_{i}} K_{i}(F′; ℤ) ⟶ K_{i}(F′, f′; ℤ) ⟶ · · · .

If i ≠ k then K_{i+1}(F′, f′; ℤ) = 0 and K_{i}(F′; ℤ) = 0 and so K_{i}(f′; ℤ) = 0. Hence f′ is k-connected. Consider the short exact sequence

0 ⟶ K_{k+1}(F′, f′; ℤ) ⟶^{∂_{2}} K_{k}(f′; ℤ) ⟶^{κ_{2}} K_{k}(F′; ℤ) ⟶ 0

obtained from the long exact sequence above, where ∂_{2} = (∂_{2})_{k+1} and κ_{2} = (κ_{2})_{k}. Note that ∂_{2}(K_{k+1}(F′, f′; ℤ)) = ⟨x⟩_{ℤ[G]}. Since K_{k}(F′; R) = R ⊗ K_{k}(F′; ℤ) is a projective R[G]- module, there is an R[G]-splitting σ_{2} : K_{k}(F′; R) → K_{k}(f′; R) for κ_{2} (more precisely for R ⊗ κ_{2}) such that σ_{2}(z) = z. Putting all this together, we get an R[G]-isomorphism ω : K_{k}(f; R) ⊕ ⟨x, z⟩_{R[G]} → K_{k}(f′; R) such that ω(u, v_{1}x, v_{2}z) = σ_{2}(κ_{1}(u)) + v_{1}x + v_{2}z for u ∈ K_{k}(f; R), v_{1}, v_{2} ∈ R[G].

Let

M′′ = ⟨x, y⟩_{R[G]} = ⟨x, z⟩_{R[G]} ⊂ K_{k}(f′; R),
M_{0} = M′′^{⊥} = {u_{1} ∈ K_{k}(f′; R) | B_{f′}(u_{2}, u_{1}) = 0 for all u_{2} ∈ M′′}, and
M_{1} = Im(σ_{2} ∘ κ_{1} : K_{k}(f; R) → K_{k}(f′; R)).

Let p′′ : K_{k}(f′; R) → M′′ be the projection associated to the decomposition K_{k}(f′; R) = M_{0} ⊕ M′′ and let p_{i} : K_{k}(f′; R) → M_{i} (i = 0, 1) be the projections associated to the decompositions K_{k}(f′; R) = M_{i} ⊕ M′′ (i = 0, 1) respectively. By construction, M_{1} ⊂ ⟨x⟩_{R[G]}^{⊥}. Thus

(9.4)  M_{1} + ⟨x⟩_{R[G]} = ⟨x⟩_{R[G]}^{⊥} and q_{f f f}′(x) = 0.

Thus the isomorphism class of (M_{1}, B_{f′}|_{M_{1}}, q_{f f f}′|_{M_{1}}) is independent of the choice of σ_{2}. For each element a of K_{k}(f; R), take an smooth immersion h_{a} : S^{k} → X with trivial normal bundle, representing a. Take h_{a} so that Im(h_{a}) ∩ Im(ind^{G}H) = ∅. Then h_{a} can be regarded as an immersion to X′. Let a′ ∈ K_{k}(f′; R) be the element represented by h_{a}. Clearly it follows that σ_{2} ∘ κ_{1}(a) ≡ a′ mod ⟨x⟩_{R[G]}. For a, b ∈ K_{k}(f; R), one can compute B_{f′}(σ_{2} ∘κ_{1}(a), σ_{2} ∘κ_{1}(b)) (resp. q_{f f f}′(σ_{2} ∘κ_{1}(a))) by using (9.4) and counting the equivariant intersection number of h_{a} and h_{b} (resp. the equivariant selfintersection number of h_{a}). This makes it clear that σ_{2} ∘ κ_{1} is an isomorphism (K_{k}(f; R), B_{f}, q_{f f f}) ≅ (M_{1}, B_{f′}|_{M_{1}}, q_{f f f}′|_{M_{1}}). The map p_{0}|_{M_{1}} : M_{1} → M_{0} is determined by the formula

p_{0}(u) = u − B_{f′}(y, u)x (for u ∈ M_{1}).

Again by (9.4), p_{0}|_{M_{1}} is an isomorphism (M_{1}, B_{f′}|_{M_{1}}, q_{f f f}′|_{M_{1}}) ≅ (M_{0}, B_{f′}|_{M_{0}}, q_{f f f}′|_{M_{0}}). Thus p_{0} ∘ σ_{2} ∘ κ_{1} : K_{k}(f; R) → M_{0} is an isomorphism (K_{k}(f; R), B_{f}, q_{f f f}) ≅ (M_{0}, B_{f′}|_{M_{0}}, q_{f f f}′|_{M_{0}}).

Set α′′ = p′′ ∘ α_{f′}, and let g ∈ S. Obviously, α_{f′}(g) = p_{0}(α_{f′}(g)) + α′′(g). ¿From the equation B_{f}(a, α_{f}(g)) = B_{f′}(a′, α_{f′}(g)), it follows that σ_{2} ∘ κ_{1}(α_{f}(g)) = p_{1}(α_{f′}(g)). Note that p_{0}(p_{1}(u)) = p_{0}(u) for all u ∈ K_{k}(f′; R). Thus p_{0}∘p_{1}∘α_{f′} = p_{0}∘α_{f′}. Since p_{0}|_{M_{1}} : (M_{1}, B_{f′}|_{M_{1}}, q_{f f f}′|_{M_{1}}) ≅ (M_{0}, B_{f′}|_{M_{0}}, q_{f f f}′|_{M_{0}}), we obtain p_{0}∘σ_{2}∘κ_{1} : (K_{k}(f; R), B_{f}, q_{f f f}, α_{f}) ≅ (M_{0}, B_{f′}|_{M_{0}}, q_{f f f}′|_{M_{0}}, p_{0} ∘ α_{f′}).

Let B′′ = B_{f′}|_{M′′}, q′′ = q_{f′}_{f′}_{f′}|_{M′′}. By Theorem 6.3, (M′′, B′′, q′′, α′′) ≅ M M M(x, y, β′′) where β′′ = (c′′, r′′) (c′′, r′′ ∈ Map(S, ℤ)) is determined by the equations

α′′(t) = ∑_{g∈G} c′′(gtg^{−1})g^{−1}x (∀t ∈ S) and B_{f′}(y, y) = ∑_{t∈S} r′′(t)t.

Next we compute that c′′ = c and calculate r′′.

Since B_{f′}(x, α_{f′}(s)) = 0, α′′(s) = ux for some u ∈ ℤ[G]. Since B_{f′}(z, x) = 1, we obtain α′′(s) = B_{f′}(z, α′′(s))x = B_{f′}(z, α_{f′}(s))x. Furthermore intsec(j_{γ}, j_{−λz}) = −1 (ρ_{Π}(γ) ∋ s), because D = Im(h_{D}) is identified with the lower hemisphere of Domain(j_{−λz}) by an orientation reversing diffeomorphism. If g ∈ G then intsec(j_{γ}, g^{−1}j_{−λz}) = w(g)intsec(gj_{γ}, j_{−λz}) and this is trivial if g ∉ G_{s}. Thus intsec_{G}(j_{γ}, j_{−λz}) = − ∑_{g∈G_{s}} w(g)g. This implies B_{f′}(α_{f′}(s), −λz) = − ∑_{g∈G_{s}} w(g)g. Hence

(9.5)  B_{f′}(z, α_{f′}(s)) = ∑_{g∈G_{s}} g = ∑_{g∈G_{s}} g^{−1} (= Σ_{G_{s}}).

Thus

(9.6)  α′′(s) = ∑_{g∈G_{s}} g^{−1}x.

Furthermore if t ∈ S and t ≠ s then α′′(t) = 0, since for γ′ ≠ γ and g ∈ G, intsec(j_{γ′}, gj_{−λz}) = 0. Clearly c′′(s) = 1 and c′′(t) = 0 if t ≠ s. Thus c′′ = c holds.

Since Im(j_{−λz})∩gIm(j_{−λz}) = ∅ whenever g^{2} = 1 and g ≠ s, 1, it is clear that B_{f′}(z, z) = 0 in ℤ[{1}∪G(2)\{s}] = ℤ[G]/ℤ[G\({1}∪G(2)\{s})]. By Lemma 9.1, we may suppose that intsec(j_{−λz}, sj_{−λz}) = λ. Then B_{f′}(z, z) = λs in ℤ[{1} ∪ G(2)] = ℤ[G]/ℤ[G \ ({1} ∪ G(2))]. ¿From the equations q_{f f f}′(y) = 0 and y = z + vx (v ∈ ℤ[G \ ({1} ∪ G(2))]), it follows that

(9.7)  B_{f′}(y, y) = λs

Thus r′′(s) = λ and r′′(t) = 0 for t ≠ s. This completes the calculation of r′′.

Since c′′(g) = c(g) (and r′′(g) ≡ r(g) mod 2ℤ for all g ∈ S), it follows from Lemma 5.5 that M M M(x, y, β′′) ≅ M M M(x, y, β) and hence (M′′, B′′, q′′, α′′) ≅ M M M(x, y, β).

Consequently M M M_{f f f}′ ≅ M M M_{f f f} ⊕ M M M(x, y, β), and we have proved Theorem 8.1 in the special case cited above.

General Case. Let S_{+} = {s ∈ S | c(s) > 0}, S_{−} = {s ∈ S | c(s) < 0} and set S′ = S_{+} ∪ S_{−}. Let Γ = {(s, i) | s ∈ S′, 1 ≦ i ≦ |c(s)| }. For each s ∈ S′, take |c(s)| distinct points x(s, 1), ⋯, x(s, |c(s)|) of X_{γ(s)} (γ(s) ∈ Θ(G, X, k) and ρ_{Π}(γ(s)) ∋ s) such that G_{x(s,i)} = ρ_{Π}(γ(s)). Furthermore we can choose these points so that if (s, i) ≠ (s′, i′) then Gx(s, i) ∩ Gx(s′, i′) = ∅. Take neighborhoods E_{(s,i)} (≅ ℝ^{k}) of x(s, i) in X_{γ(s)}, respectively. Then each ν(X_{γ(s)}, X)|_{E_{(s,i)}} is a neighborhood of x(s, i) which is diffeomorphic to E_{(s,i)} × V_{(s,i)}, where V_{(s,i)} = ν(X_{γ(s)}, X)|_{x(s,i)}. We may assume that Gν(X_{γ(s)}, X)|_{E_{(s,i)}} ∩ Gν(X_{γ(s′)}, X)|_{E_{(s′,i′)}} = ∅ whenever (s, i) ≠ (s′, i′) and that if ν(X_{γ(s)}, X)|_{E_{(s,i)}} ∩ gν(X_{γ(s)}, X)|_{E_{(s,i)}} ≠ ∅ then g ∈ ρ_{Π}(γ(s)). Let Δ_{(s,i)} : E_{(s,i)} → V_{(s,i)} be ℝ-linear maps such that Ker(Δ_{(s,i)}) = {0}. The graphs Graph(Δ_{(s,i)}) are k-dimensional linear subspaces of E_{(s,i)} × V_{(s,i)}, respectively. For each (s, i) ∈ S_{+} (resp. S_{−}), we orient Graph(Δ_{(s,i)}) so that the ordered direct sum T_{x(s,i)}(X_{γ(s)})⊕T_{x(s,i)}(Graph(Δ_{(s,i)})) has the same orientation (resp. opposite orientation) as T_{x(s,i)}(X). Take orientation preserving (linear) diffeomorphisms h′_{D_{(s,i)}} : D^{k} → D_{δ}(Graph(Δ_{(s,i)})) such that h′_{D_{(s,i)}}(0) = x(s, i). For each (s, i), there is a δ′-approximation h_{D_{(s,i)}} : D^{k} → ν(X_{γ(s)}, X)|_{E_{(s,i)}} of h′_{D_{(s,i)}} such that h_{D_{(s,i)}} is also a smooth embedding, that h_{D_{(s,i)}}(x) = h′_{D_{(s,i)}}(x) if ∥x∥ ≦ 1/2, and that h_{(s,i)} := h_{D_{(s,i)}}|_{S^{k−1}} : S^{k−1} → X satisfies the property that if gh_{(s,i)}(x) = g′h_{(s,i)}(x′) (g, g′ ∈ G and x, x′ ∈ S^{k−1}) then g = g′ and x = x′. Set D_{(s,i)} = Im(h_{D_{(s,i)}}). Instead of (9.2), we have now

(9.2′) the intersection number X_{γ(s)} · D_{(s,i)} = sign(c(s)).

Let ord : Γ → {1, ⋯, |Γ|} be a bijection. For each i = 1, ⋯, |Γ| − 1, take a kdimensional band B_{i} ≅ I × D^{k−1} (in general position in Free(G, X) \ (⋃_{t} Int(D_{ord^{−1}(t)}))) connecting ∂D_{ord^{−1}(i)} with ∂D_{ord^{−1}(i+1)}. This done, we obtain an embedded k-dimensional closed disk

D = D_{ord^{−1}(1)} ∪ B_{1} ∪ D_{ord^{−1}(2)} ∪ ⋯ ∪ B_{|Γ|−1} ∪ D_{ord^{−1}(|Γ|)}

in X. The bands B_{i} should be taken so that ∂D is the oriented connected sum of the oriented ∂D_{(s,i)}’s. Let h_{D} : (D^{k} ≅) D → X be the canonical inclusion and set h = h_{D}|_{∂D} : (S^{k−1} ≅) ∂D → Free(G, X). Without loss of generality, we may assume that if gh(x) = g′h(x′) (g, g′ ∈ G and x, x′ ∈ S^{k−1}) then g = g′ and x = x′. There is a smooth embedding H : S^{k−1} × D^{k+1} → Free(G, X) such that h(x) = H(x, 0) for all x ∈ S^{k−1} and such that the induced G-map ind^{G}H : G × S^{k−1} × D^{k+1} → Free(G, X) is an embedding. Construct the following spaces and maps as in Special Case: W′ = W ∪_{ind^{G}H} (G × D^{k} × D^{k+1}), X′, F, F′, f′ : X′ → Y, and b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η). Then set fff′ = (f′, b′). As in Special Case, K_{k}(f′; R) ≅ K_{k}(f; R) ⊕ ⟨x, z⟩_{R[G]}. Moreover, −x and −λz have geometric realizations by embeddings j_{−x}, j_{−λz} : S^{k} → X′, respectively, and B_{f′}(x, x) = 0, B_{f′}(z, x) = 1, q_{f f f}′(x) = 0, (q_{f f f}′(z))(g) = 0 (∀g ∈ {1}∪G(2)\(Q∪S)), and there is an element v ∈ ℤ[G\({1}∪G(2))] such that y = z+vx satisfies q_{f f f}′(y) = 0. For each s ∈ S, let j_{γ(s)} : X_{γ(s)} → X be the canonical inclusion. Then intsec(j_{γ(s)}, j_{−λz}) = −c(s) and hence

intsec_{G}(j_{γ(s)}, j_{−λz}) = ∑_{g∈G} intsec(j_{γ(s)}, g^{−1}j_{−λz})g

= ∑_{g∈G} w(g)intsec(gj_{γ(s)}, j_{−λz})g

= ∑_{g∈G} w(g)intsec(j_{gγ(s)}, j_{−λz})g

= ∑_{g∈G} w(g)intsec(j_{γ(gsg^{−1})}, j_{−λz})g

= ∑_{g∈G} w(g) ( −c(gsg^{−1})g) .

Thus

B_{f′}(z, α_{f′}(s)) = λ\overline{B_{f′}(α_{f′}(s), z)}

= −\overline{B_{f′}(α_{f′}(s), −λz)}

= −\overline{∑_{g∈G} w(g) (−c(gsg^{−1})g)}

= ∑_{g∈G} c(gsg^{−1})g^{−1}.

The equality (9.5) is replaced by the equality

(9.5′)  \begin{aligned}B_{f′}(z, α_{f′}(s)) & = ∑_{g∈G} c(gsg^{−1})g^{−1} \\ ( & = ∑_{gG_{s}∈G/G_{s}} c(gsg^{−1})Σ_{G_{gsg^{−1}}} g^{−1}). \end{aligned}

Let M′′ = ⟨x, y⟩_{R[G]} (= ⟨x, z⟩_{R[G]}) and M_{0} = M′′^{⊥}. Let p_{0} : K_{k}(f′; R) → M_{0} and p′′ : K_{k}(f′; R) → M′′ denote the projections associated to the decomposition K_{k}(f′; R) = M_{0} ⊕ M′′. Let α′′ = p′′ ∘ α_{f′}, B′′ = B_{f′}|_{M′′}, and q′′ = q_{f′}_{f′}_{f′}|_{M′′}. By Theorem 6.3, (M′′, B′′, q′′, α′′) ≅ M M M(x, y, β′′) where β′′ = (c′′, r′′) ( c′′, r′′ ∈ Map(S, ℤ)) is determined by the equation

α′′(s) = ∑_{g∈G} c′′(gsg^{−1})g^{−1}x (∀s ∈ S), and B_{f′}(y, y) = ∑_{s∈S} r′′(s)s.

As in Special Case, we compute that c′′ = c. Since α′′(s) = B_{f′}(z, α_{f′}(s))x, it follows that

(9.6′)  α′′(s) = ∑_{g∈G} c(gsg^{−1})g^{−1}x

and hence that c′′ = c.

Next we calculate r′′. By Lemma 9.1, we may assume that intsec(j_{−λz}, sj_{−λz})) = λ|c(s)| for all s ∈ S. Since (q_{f f f}′(z))(g) = 0 for any g ∈ {1} ∪ G(2) \ (Q ∪ S), it follows that

B_{f′}(z, z) = ∑_{s∈S}λ|c(s)|s in ℤ[{1} ∪ G(2)] = ℤ[G]/ℤ[G \ ({1} ∪ G(2))]. As q_{f f f}′(y) = 0 and

y = z + vx for some v ∈ ℤ[G \ ({1} ∪ G(2))], we have

(9.7′)  B_{f′}(y, y) = ∑_{s∈S}λ|c(s)|s.

Thus for all s ∈ S, r′′(s) = λ|c(s)| and r′′(s) ≡ r(s) mod 2ℤ.

Since c′′ = c (and r′′ ≡ r mod 2), Lemma 5.5 implies M M M(x, y, β′′) ≅ M M M(x, y, β).

By the same arguments as in Special Case, we can check

(K_{k}(f; R), B_{f}, q_{f f f}, α_{f}) ≅ (M_{0}, B_{f′}|_{M_{0}}, q_{f f f}′|_{M_{0}}, p_{0} ∘ α_{f′})

and conclude M M M_{f f f}′ ≅ M M M_{f f f} ⊕ M M M(x, y, β). Q.E.D.

10. Appendix

We have invoked Assumption (HC) (see §7) in order to apply the next proposition.

Proposition 10.1. Let M be an n-dimensional, G-CW-complex and let η and η′ be real G-vector bundles with G-invariant Riemannian metrics over M. If η ⫆ ε_{M}(ℝ^{n+1}) then any G-vector bundle isomorphism b : ε_{M}(ℝ)⊕η → ε_{M}(ℝ)⊕η′ (⊕ denotes orthogonal sum) is G-regularly homotopic to a G-vector bundle isomorphism id_{ε_{M} (ℝ)} ⊕ b′ where b′ : η → η′.

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

We shall prove Proposition 10.1 by double induction on n and the number of isotropy types of n-dimensional cells. Suppose M = M′ ∪ ⋃_{γ}(G/H × D_{γ}^{n}) where D_{γ}^{n} = D^{n}, and invoke the induction hypothesis that b|_{M′} has the form id_{ε_{M′} (ℝ)} ⊕ b′′, where b′′ : η|_{M′} → η′|_{M′}. Under this hypothesis, we shall find b′ as in the conclusion of the proposition. For fixed γ, set E = H/H ×Int(D_{γ}^{n}). Then b(ε_{M}(ℝ)|_{\overline{E}\E}) = ε_{M}(ℝ)|_{\overline{E}\E}, but it is not necessary that

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

Let b^{H} : ε_{M^{H}} (ℝ)⊕η^{H} → ε_{M^{H}} (ℝ)⊕η′^{H} be the restriction of b to the H-fixed point set. Then b|_{M^{H}} is decomposed into 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 deforming b to satisfy (10.2) lies in π_{n−1}(S^{m−1}), where m = fiber-dim(η^{H}) + 1. Since fiber-dim(η^{H}) ≧ fiber-dim(η^{G}) ≧ n, the obstruction group π_{n−1}(S^{m−1}) is trivial. Hence the obstruction σ vanishes. If (10.2) is satisfied for all γ then b(ε_{M}(ℝ)) = ε_{M}(ℝ). Since b is metric preserving, we have b(η) ⫅ η′. Moreover, we can arrange b so that b|_{ε_{M} (ℝ)} = id_{ε_{M} (ℝ)}, since fiber-dim(ε_{M}(ℝ) ⊕ ε_{M}(ℝ^{n+1})) ≧ 2. Q.E.D.

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