FORUM MATH. 1 (2003),

THE DIMENSION OF SPHERES WITH

SMOOTH ONE FIXED POINT ACTIONS

ANTHONY BAK AND MASAHARU MORIMOTO

(Communicated by Andrew Ranicki )

Abstract. The article proves that there are smooth one fixed point actions of the alternating group of degree 5 on the 8-dimensional sphere. It follows that a sphere has a smooth one fixed point action of some finite group if and only if the dimension of the sphere is greater than or equal to 6.

Acknowledgement. The second author was partially supported by Grant-in-Aid for Scientific Research (KAKENHI). The first author was partially supported by INTAS 00-566.

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

1. Introduction

This paper completes research stretching over several decades to determine which spheres have smooth one fixed point actions of finite groups. Except for the 8-dimensional sphere, the answer has been known for about a decade; namely if n ≠ 8 then the standard n-dimensional sphere S^{n} has a smooth one fixed point action of some finite group if and only if n ≥ 6. In the current paper, it will be shown that S^{8} has a smooth one fixed point action of the alternating group A_{5} of degree 5. Recall that an action is called a one fixed point action, if there is exactly one point which is left fixed by every element of the group.

We review some of the major steps of the past.

A fundamental problem in the theory of transformation groups is determining fixed point sets of smooth actions of finite groups on disks and spheres. R. Oliver showed in [28, Theorem 7] that a finite group can act smoothly, without fixed points on some (closed) disk of undetermined dimension if and only if it is not mod p hyperelementary for all primes p. A finite group G is called mod p hyperelementary, if it has a normal p-subgroup P such that the quotient G/P is hyperelementary. The result provides a necessary condition for a finite group to act smoothly, with exactly one fixed point on some sphere of undetermined dimension.

E. Stein was the first to find a smooth one fixed point action on a sphere, namely he showed that S^{7} admits a smooth one fixed point action of the special linear group SL(2, ℤ_{5}) [37, Proposition 4.3]. Next T. Petrie proved in [34, Theorem A] that a finite, odd order abelian group having at least three noncyclic Sylow subgroups, e.g. ℤ_{pqr} × ℤ_{pqr} where p, q, and r are distinct odd primes, has a smooth one fixed point action on some higher dimensional spheres. The same conclusion holds for any finite group which is not mod p hyperelementary for all primes p, thanks to [13, Theorem A] and [12, Theorem A]. The result is the strongest possible one, in view of Oliver’s work on disks.

Aside from Stein’s result, all of the above smooth one fixed point theorems tell us only that there exist higher dimensional spheres which have smooth one fixed point actions of finite groups which are not mod p hyperelementary for all primes p. E. Laitinen and P. Traczyk posed in [14] the problem whether or not S^{6} has a smooth one fixed point action of A_{5}. This was answered affirmatively by the second author. In fact he proved in [15, Theorem A], [17, Theorem 0.1], and [19, Theorem A] that if n ≥ 6 and n ≠ 7, 8 then S^{n} has smooth one fixed point actions of A_{5}. Thereafter it was shown in a joint paper [2, Theorem 7] that S^{7} has a smooth one fixed point action of A_{5}. This left S^{8} as the only unsolved case for n ≥ 6. On the other hand, it has been shown in M. Furuta [11, Theorem 0.1] (cf. [16, Theorem A]), Buchdahl-Kwasik-Schultz [5, Theorem I.1, II.1, and II.4], and DeMichels [6, Theorem 5.1] that if 0 ≦ n ≦ 5 then S^{n} does not have a smooth one fixed point action of any finite group.

It is worthwhile remarking that the existence of a smooth one fixed point action of A_{5} on S^{6} implies that for any n ≥ 7, there is a continuous one fixed point action of A_{5} on S^{n}, which is everywhere smooth, except possibly at the fixed point. This fact motivated Buchdahl-Kwasik-Schultz to show in [5, Theorem II.5] that for each n ≥ 6, there is a locally linear, one fixed point action of A_{5} on S^{n}. Their arguments, however, do not provide ways of constructing smooth one fixed point actions on spheres.

The advances above of T. Petrie and the current authors depend on parallel developments in equivariant surgery theory. In Petrie’s case, it was his invention [33] (cf. [9], [35]) of equivariant surgery theory under the so-called strong gap hypothesis for singular sets. Here the surgery groups are those of Wall. In the authors’ case, it was the second author’s replacement of the strong gap hypotheses in Petrie’s theory, by the gap hypothesis. Here the surgery groups are those of the first author. They are generalizations of Wall’s groups, which are obtained by using form parameters to adjust the quadratic form. The procedure for applying equivariant surgery to one fixed point problems starts with a smooth two fixed point action satisfying one of the gap hypotheses and then uses equivariant surgery to remove one of the fixed points. However, for the current problem of establishing a smooth one fixed point action of A_{5} on S^{8}, we were not able to find a smooth two fixed point action of A_{5} on S^{8}, satisfying the gap hypothesis, but only smooth two fixed point actions of A_{5} on S^{8}, whose singular sets have dimension up to and including 4, which is half of the dimension of S^{8}. This motivated us to develop in [4] equivariant surgery for smooth actions having dimensions of singular sets up to and including half the dimension of the manifold. Here the surgery groups generalize those above by using two form parameters to adjust simultaneously the quadratic and Hermitian forms, rather than just one parameter for adjusting the quadratic form. The resulting surgery theory is applied in the current paper to prove the theorem below. It played also a central role in establishing the smooth analog of Oliver’s theorem in papers [12] and [13] above.

Theorem 1.1. There are smooth one fixed point actions of A_{5} on S^{8}.

The following corollary follows immediately from the above.

Corollary 1.2. For a natural number n, the following are equivalent:

(1) There is a smooth one fixed point action of some finite group on S^{n}.

(2) There is a smooth one fixed point action of A_{5} on S^{n}.

(3) n ≥ 6.

The proof of Theorem 1.1 is organized as follows. In Section 2, we recall the Burnside ring and certain facts concerning it. This includes showing the existence of a certain idempotent. In Section 3, we recall equivariant framed maps and use the idempotent above to construct a G-equivariant framed map, whenever G is nontrivial and perfect. In Section 4, we prove Theorem 1.1. We start by showing that a certain two fixed point, linear action of A_{5} on S^{8} has only singular sets of dimension ≦ 4. Then the results of Section 3 are used to construct a good A_{5}-framed map to S^{8}. In order to show that the surgery obstruction of the map vanishes, we need to know that the map has certain properties. These are isolated in Proposition 4.1 whose proof is postponed to Section 5. Applying the equivariant surgery theory in [4], we convert the map to a homotopy equivalence whose source is a homotopy sphere of dimension 8 with exactly one fixed point. Using Stein’s trick of taking the equivariant connected sum of copies of this manifold, we obtain a smooth one fixed point action of A_{5} on the standard 8-dimensional sphere S^{8}. This completes the proof of Theorem 1.1. In section 5, we develop a certain way of performing equivariant surgery which we call the reflection method and apply this to prove Proposition 4.1.

2. The Burnside ring and an idempotent

Let G be a finite group. Let Ω(G) denote the Burnside ring of G, namely the Grothendieck group of the category of finite G-sets with addition and multiplication induced by disjoint union and Cartesian product, respectively. For any subgroup H ⊂ G, there is a canonical homomorphism χ_{H} : Ω(G) → ℤ defined by

χ_{H}(α) = |A^{H}| − |B^{H}|

where A and B are finite G-sets, α = [A]−[B], A^{H} and B^{H} denote the sets of elements in A and B respectively which are left pointwise fixed by the action of H, and |X| denotes the number of elements of X. By [8, IV, Theorem 5.7], we obtain

Lemma 2.1. Suppose G is a finite nontrivial perfect group. Then there exists an idempotent α in Ω(G) such that χ_{G}(α) = 1 and χ_{H}(α) = 0 for any H ≠ G.

For a finite G-CW complex X, the n-th equivariant cohomology ω_{G}^{n}(X) of X is defined to be

\underrightarrow{lim}_{m}[(ℝ^{n_{1}} ⊕ ℂ[G]^{m})^{+} ∧ (X ⨿ \{pt\}), (ℝ^{n_{2}} ⊕ ℂ[G]^{m})^{+}]_{0}^{G},

where n = n_{2} − n_{1} ∈ ℤ and W^{+} denotes the one point compactification of W, ℂ[G] the complex regular representation of G, and [A, B]_{0}^{G} the set of G-homotopy classes of basepoint preserving G-maps from A to B. We refer the reader to [8, pp.140–141] for details concerning ω_{G}^{n}(X), e.g. its group structure. For the special case where X is a point, we obtain

ω_{G}^{0}(pt) = \underrightarrow{lim}_{m}[(ℂ[G]^{m})^{+}, (ℂ[G]^{m})^{+}]_{0}^{G} (≅ [S(ℝ ⊕ ℂ[G]), S(ℝ ⊕ ℂ[G])]_{0}^{G})

and this becomes a ring whose multiplication is induced by taking composition of maps (cf. [25]). Furthermore ω_{G}^{n}(X) is a module over ω_{G}^{0}(pt), with multiplication defined as follows. If [f : (ℂ[G]^{m})^{+} → (ℂ[G]^{m})^{+}] ∈ ω_{G}^{0}(pt) and [h : (ℝ^{n_{1}} ⊕ ℂ[G]^{m})^{+} ∧ (X ⨿ {pt}) → (ℝ^{n_{2}} ⊕ ℂ[G]^{m})^{+}] ∈ ω_{G}^{n}(X), then the product [f][h] ∈ ω_{G}^{n}(X) is defined to be the Ghomotopy class including the composition (id_{ℝ^{n_{2}}}^{+} ∧ f) ∘ h. For each subgroup H of G, there is a canonical homomorphism deg_{H} : ω_{G}^{0}(pt) → ℤ defined by

deg_{H}([f]) = deg(f^{H} : S(ℝ ⊕ ℂ[G])^{H} → S(ℝ ⊕ ℂ[G])^{H}).

By Segal’s theorem [36, Corollary], there is a canonical ring isomorphism Ω(G) → ω_{G}^{0}(pt) such that the diagram

\xymatrix{Ω(G) \ar[rr]^{≅} \ar[dr]_{χ_{H}} &  & ω_{G}^{0}(pt) \ar[dl]^{_{deg_{H}}} \\  & ℤ}

commutes for every H ⊂ G and we identify Ω(G) with ω_{G}^{0}(pt) via this isomorphism.

Lemma 2.2 ([33, Lemma 1.8], [34, Lemma 1.6]). Let G and α be as in the previous lemma. For any finite G-CW complex X and any integer n, the restriction homomorphism α^{−1}ω_{G}^{n}(X) → α^{−1}ω_{G}^{n}(X^{G}) is an isomorphism, where α^{−1}ω_{G}^{n}(X) denotes the localization of ω_{G}^{n}(X) with respect to the mutiplicatively closed set {α}.

Lemma 2.3 ([33, Corollary 1.9]). Let G = A_{5} and let Y be a finite G-CW complex such that Y^{G} = {y_{+}, y_{−}}. The ω_{G}^{0}(pt)-module ω_{G}^{0}(Y^{G}) is free of rank 2 and we identify ω_{G}^{0}(y^{G}) = Ω(G) ⊕ Ω(G). The assertion is that there exists an element ω ∈ ω_{G}^{0}(Y) such that

ω|_{Y^{G}} = ([G/G] − α, [G/G]) ∈ Ω(G) ⊕ Ω(G),

where α is the element appearing in Lemma 2.1.

Define the subset G(2) of G by

G(2) = {g ∈ G | g^{2} = e, g ≠ e}.

It is regarded as a G-set via conjugation. Let SGW_{0}(ℤ, G, G(2)) denote the special Grothendieck–Witt ring defined in [12, p.509]. By definition, the multiplicative identity element 1_{G} of SGW_{0}(ℤ, G, G(2)) is the equivalence class of (ℤ, B, α), where B : ℤ×ℤ → ℤ is the canonical multiplication and α : G(2) → ℤ is the map g ↦ 1.

Let subgp(G) denote the category of subgroups of G with morphisms defined by inclusion and conjugation. For a precise definition, see [1]. Let M : subgp(G) → ((abelian groups)) be a Mackey functor. By definition M is a pair of functors consisting of a covariant functor and a contravariant functor, both taking the same value on objects. It is assumed that the pair forms a bifunctor and satisfies the Mackey subgroup property. See [1]. If K → H is a morphism in subgp(G) then Ind_{K}^{H} denotes as usual the covariant map Ind_{K}^{H} : M(K) → M(H) and is called induction. If K → H is as above then Res_{K}^{H} denotes as usual the contravariant map Res_{K}^{H} : M(H) → M(K) and is called restriction.

Let R : subgp(G) → ((rings)) be a Green ring functor [1] (cf. also [7, p.165]). By definition, R is a Mackey functor whose contravariant part delivers ring homomorphisms, but whose covariant part only group homomorphisms, and satisfies the Frobenius reciprocity law. A Mackey functor M is called an R-module or a Green module over R, if for each H ⊂ G, M(H) is an R(H)-module and the following two properties are satisfied:

(2.1)  (Ind_{K}^{H}r) · x = Ind_{K}^{H}(r · Res_{K}^{H}x) for all r ∈ R(K) and x ∈ M(H).
(2.2)  r · (Ind_{K}^{H}x) = Ind_{K}^{H}((Res_{K}^{H}r) · x) for all r ∈ R(H) and x ∈ M(K).

Lemma 2.4. Let G be a nontrivial finite perfect group, α the idempotent in Lemma 2.1, and 1_{G} the multiplicative identity of the ring SGW_{0}(ℤ, G, G(2)). Then, α1_{G} = 0 and hence

SGW_{0}(ℤ, G, G(2)) = ∑_{H⊊G}Ind_{H}^{G}(SGW_{0}(ℤ, H, H(2))).

Consequently, for a module M(−) over SGW_{0}(ℤ,−,−(2)) the map

Res : M(G) → ⨁_{H⊊G}M(H)

is injective.

Remark. For the lemma above, the property (2.2) is unnecessary.

Proof. By definition, Res_{H}^{G}([G/G] − α) = 0 for all H ⊊ G. By [12, Proposition 6.4], we get

([G/G] − ([G/G] − α))^{2}1_{G} = 0,

which implies α1_{G} = 0. Since [G/G] − α = ∑_{H⊊G} a_{H}[G/H] for some integers a_{H}, we obtain

1_{G} = ([G/G] − α)1_{G} = ∑_{H⊊G}a_{H}Ind_{H}^{G}1_{H}.

The remainder of the lemma follows from the equality

x = ∑_{H⊊G}a_{H}(Ind_{H}^{G}1_{H}) · x = ∑_{H⊊G}a_{H}Ind_{H}^{G}(1_{H} · Res_{H}^{G}x)

for x ∈ M(G). □

3. Equivariant framed maps

For a finite group G, a pair (f, b) consisting of a G-map f : X → Y between G-manifolds and an isomorphism b : T (X) ⊕ ε_{X}(ℝ^{m}) → f^{∗}T (Y) ⊕ ε_{X}(ℝ^{m}) of real G-vector bundles, for some m, is called a G-framed map from X to Y. Here ε_{X}(ℝ^{m}) denotes the product bundle over X with fiber ℝ^{m}. If the degree of f is one then (f, b) is also said to be of degree one. In the case where X and Y are closed, a G-framed cobordism (F, B) between degree one, G-framed maps (f, b) and (f′, b′) is by definition a pair consisting of a degree one G-map

F : (W, ∂W) → (I × Y, Y × ∂I),

where I = [0, 1], and a G-isomorphism

B : T (W) ⊕ ε_{W} (ℝ^{m}) → F^{∗}(T (I × Y)) ⊕ ε_{W} (ℝ^{m})

satisfying the following conditions:

(1) ∂W = (−X) ⨿ X′ as oriented G-manifolds,

(2) F |_{(−X)} = f as G-maps (−X) → Y × {0},

(3) F |_{X′} = f′ as G-maps X′ → Y × {1},

(4) B|_{(−X)} = −id_{ε_{(−X)}(ℝ)} ⊕ b as G-isomorphisms

ε_{(−X)}(ℝ) ⊕ T (−X) ⊕ ε_{(−X)}(ℝ^{m}) → ε_{(−X)}(ℝ) ⊕ f^{∗}T (Y) ⊕ ε_{(−X)}(ℝ^{m}), and

(5) B|_{X′} = id_{ε_{X′}(ℝ)} ⊕ b′ as G-isomorphisms

ε_{X′}(ℝ) ⊕ T (X′) ⊕ ε_{X′}(ℝ^{m}) → ε_{X′}(ℝ) ⊕ f′^{∗}T (Y) ⊕ ε_{X′}(ℝ^{m})

(after stabilizing b and b′ if necessary), where (−X) is a copy of X but with opposite orientation.

Let V be a real G-module with G-invariant inner product and set Y = S(ℝ ⊕ V). The north and south poles of Y, namely the points (1, 0), (−1, 0) ∈ ℝ ⊕ V, will be denoted by y_{+} and y_{−}, respectively. We have the canonical isomorphisms

T (Y) ⊕ ε_{Y} (ℝ) ≅ ε_{Y} (ℝ) ⊕ T (Y) ≅ ν(Y, ℝ ⊕ V) ⊕ T (Y) ≅ T (ℝ ⊕ V)|_{Y} ≅ ε_{Y} (ℝ ⊕ V).

Define b_{+} : T (Y) ⊕ ε_{Y} (ℝ) → T (Y) ⊕ ε_{Y} (ℝ) to be the identity map and Define b_{−} : T (Y) ⊕ ε_{Y} (ℝ) → T (Y) ⊕ ε_{Y} (ℝ) by

b_{−}(y, (r, v)) = (y, (−r, v))

for y ∈ Y, r ∈ ℝ and v ∈ V. From the above we obtain G-framed maps 111_{Y,G} := (id_{Y} , b_{+}) and −111_{Y,G} := (id_{Y} , b_{−}).

Fix small G-invariant disk neighborhoods U_{Y,+} and U_{Y,−} of y_{+} and y_{−} in Y, respectively. Let ℤ_{≥0} denote the set of nonnegative integers. For a set A of subgroups of G and maps φ_{+}, φ_{−} : A → ℤ_{≥0}, define the G-manifold Z_{φ_{+},φ_{−}} by

Z_{φ_{+},φ_{−}} = ∐_{H∈A} ( ∐_{i=1}^{φ_{+}(H)}(G ×_{H} U_{Y,+,H,i}) ⨿ ∐_{i=1}^{φ_{−}(H)}(G ×_{H} U_{Y,−,H,i})) ,

where U_{Y,+,H,i} and U_{Y,−,H,i} are copies of U_{Y,+} and −U_{Y,+} respectively. Define the G-framed map fff_{φ_{+},φ_{−}} from Z_{φ_{+},φ_{−}} to U_{U,+} by

fff_{φ_{+},φ_{−}} = ⋃_{H∈A} ( ⋃_{i=1}^{φ_{+}(H)}(G ×_{H} (111_{Y,+,H,i}|_{U_{Y,+,H,i}})) ∪ ⋃_{i=1}^{φ_{−}(H)}(G ×_{H} (111_{Y,−,H,i}|_{U_{Y,−,H,i}}))) ,

where 111_{Y,+,H,i} and 111_{Y,−,H,i} are copies of 111_{Y,H} and −111_{Y,H} respectively.

Proposition 3.1. Let G be a finite nontrivial perfect group and V a real G-module with V^{G} = 0. Let Y denote S(ℝ⊕V), α (∈ Ω(G)) the element appearing in Lemma 2.1, and ω (∈ ω_{G}^{0}(Y)) the element appearing in Lemma 2.3. Furthermore let A be a set of subgroups of G and φ_{+}, φ_{−} : A → ℤ_{≥0} maps such that

[G/G] − α = ∑_{H∈A}(φ_{+}(H) − φ_{−}(H))[G/H].

Then there exists a G-framed map fff = (f, b) consisting of a G-map f : X → Y := S(ℝ ⊕ V) and a G-vector bundle isomorphism

b : T (X) ⊕ ε_{X}(ℝ^{m+1}) → f^{∗}T (Y) ⊕ ε_{X}(ℝ^{m+1}),

and H-framed cobordisms (F_{H}, B_{H}) between 111_{Y,H} and Res_{H}^{G}fff, consisting of H-maps

F_{H} : (W_{H}, ∂W_{H}) → (I × Y, ∂(I × Y))

and H-vector bundle isomorphisms

B_{H} : T (W_{H}) ⊕ ε_{W_{H}}(ℝ^{m+1}) → F_{H}^{∗}T (I × Y) ⊕ ε_{W_{H}}(ℝ^{m+1})

respectively, for all proper subgroups H ⊊ G such that (1) fff|_{f^{−1}(U_{Y,−})} is the stabilization of 111_{Y,G}|_{U_{Y,−}},

(2) fff|_{f^{−1}(U_{Y,+})} is the stabilization of fff|_{φ_{+},φ_{−}}. Here m is some (large) integer.

Proof. Define Z_{+} and Z_{−} as the subsets {y_{+}} and {y_{−}} of Y respectively, and set

\begin{aligned}γ_{+} & = ∑_{H∈A}(φ_{+}(H) − φ_{−}(H))[G/H], \\ γ_{−} & = (1 − 0)[G/G], \end{aligned}

where Y^{G} = {y_{+}, y_{−}}. It follows immediately from Lemma 2.3 that ω|_{Z_{+}} = γ_{+} and ω|_{Z_{−}} = γ_{−}. By [20, Theorem 4.4], we obtain a G-normal map (f, b′) consisting of a G-map f : X → Y and a G-vector bundle isomorphism

b′ : T (X) ⊕ ε_{X}(V′) → f^{∗}T (Y) ⊕ ε_{X}(V′),

and, for each H ⊊ G, an H-normal cobordism (F_{H}, B′_{H}), between Res_{H}^{G}(id_{Y} , id_{T (Y)⊕ε_{Y} (V′)}) and Res_{H}^{G}(f, b), consisting of an H-map F_{H} : (W_{H}, ∂W_{H}) → (I × Y, ∂(I × Y)) and an H-vector bundle isomorphism

B′_{H} : T (W_{H}) ⊕ ε_{W_{H}}(V′) → F_{H}^{∗}T (I × Y) ⊕ ε_{W_{H}}(V′),

where V′ is some real G-module. Using [20, Theorem 3.6], one can replace V′ by ℝ^{m+1} for sufficiently large m, and obtain a G-vector bundle isomorphism b : T (X) ⊕ ε_{X}(ℝ^{m+1}) → f^{∗}T (Y) ⊕ ε_{X}(ℝ^{m+1}) and H-vector bundle isomorphisms B_{H} : T (W_{H}) ⊕ ε_{W_{H}}(ℝ^{m+1}) → F_{H}^{∗}T (I × Y) ⊕ ε_{W_{H}}(ℝ^{m+1}) from b′ and B′_{H} above. By [20, Theorem 4.4], we can choose the G-framed map fff := (f, b) so as to satisfy the conditions (1) and (2) above. □

We remark that for the G-map f above and any subgroup H of G such that dim V^{H} ≥ 1, the restriction f^{H} : X^{H} → Y^{H} is of degree 1.

4. Proof of Theorem 1.1

In the current section, we assume throughout that G = A_{5}. We begin by recalling basic properties of A_{5}, the alternating group of degree 5. For n = 2, 3 and 5, choose cyclic subgroups C_{n} of A_{5} of order n so that C_{2} ⊂ N_{A_{5}}(C_{3}) and C_{2} ⊂ N_{A_{5}}(C_{5}) (see [27, p.339]). Since N_{A_{5}}(C_{n}) is dihedral of order 2n, we denote N_{A_{5}}(C_{n}) by D_{2n}. Moreover, N_{A_{5}}(D_{4}) is isomorphic to the alternating group of degree 4 and hence we denote N_{A_{5}}(D_{4}) by A_{4}. Any subgroup of A_{5} is conjugate to one of the subgroups {e}, C_{n}, D_{2n}, A_{4} or A_{5}, where n = 2, 3 or 5. The group A_{5} has precisely five (mutually nonisomorphic) irreducible real representations, and the dimensions of them are 1, 3, 3, 4 and 5. The two irreducible real A_{5}-representaions of dimension 3 are Galois conjugate to each other. Let ℝ, U(3), U(4) and U(5) be irreducible real A_{5}-representation spaces (with G-invariant inner product) of dimension 1, 3, 4 and 5, respectively. The dimensions of the H-fixed point set W^{H} of W = U(3), U(4) and U(5) are as in Table 1.

H  {e} C_{2} C_{3} C_{5} D_{4} D_{6} D_{10} A_{4} A_{5}
dim U(3)^{H} 3 1 1 1 0 0 0 0 0
dim U(4)^{H} 4 2 2 0 1 1 0 1 0
dim U(5)^{H} 5 3 1 1 2 1 1 0 0

Table 1

Define a real A_{5}-module V by

(4.1)  V = U(3) ⊕ U(5)

and set Y = S(ℝ ⊕ V). Then, the dimensions of the H-fixed point sets Y^{H}, H ⊂ A_{5}, are as in Table 2.

\begin{matrix}H & \{e\} & C_{2} & C_{3} & C_{5} & D_{4} & D_{6} & D_{10} & A_{4} & A_{5} \\ dim Y^{H} & 8 & 4 & 2 & 2 & 2 & 1 & 1 & 0 & 0 \end{matrix}

Table 2

It follows that

Iso(A_{5}, Y) = S(A_{5}) ∖ (A_{4}),

where Iso(A_{5}, Y) and S(A_{5}) denote respectively the set of all isotropy subgroups of Y in G and the set of all subgroups of G.

Next, recall that the formula

∑_{g∈G} |(G/H)^{g}| = |G|

is valid for any finite group G and any subgroup H of G. Using the formula, we obtain Table 3, whose entries are the numbers of elements of (G/H)^{g} when G = A_{5}.

H  {e} C_{2} C_{3} C_{5} D_{6} D_{10} A_{4}
ord(g) = 5 0 0 0 2 0 1 0
ord(g) = 3 0 0 2 0 1 0 2
ord(g) = 2 0 2 0 0 2 2 1
g = e  60 30 20 12 10 6 5

Table 3

It follows from the table that in the case G = A_{5}, the idempotent α in Lemma 2.1 is explicitly given by

(4.2)  α = [G/G] − [G/A_{4}] − [G/D_{10}] − [G/D_{6}] + [G/C_{3}] + 2[G/C_{2}] − [G].

Now set

A = {{e}, C_{2}, C_{3}, D_{6}, D_{10}, A_{4}}.

From the formula (4.2), it follows that

[A_{5}/A_{5}] − α = ∑_{H∈A}(φ_{+}(H) − φ_{−}(H))[A_{5}/H],

for the maps φ_{+}, φ_{−} : A → ℤ_{≥0} defined in Table 4 below.

\begin{matrix}H & \{e\} & C_{2} & C_{3} & D_{6} & D_{10} & A_{4} \\ φ_{+}(H) & 1 & 0 & 0 & 1 & 1 & 1 \\ φ_{−}(H) & 0 & 2 & 1 & 0 & 0 & 0 \end{matrix} Table 4

Proposition 4.1. Let G = A_{5} and let V, Y, φ_{+} and φ_{−} be as above. Then, the A_{5}-framed map fff = (f, b) and the H-framed cobordisms F F F = (F, B), H ⊊ A_{5}, in Proposition 3.1 can be chosen to satisfy the following additional properties:

(1) For each subgroup H such that {e} ⊊ H ⊊ A_{5},

the map f^{H} : X^{H} → Y^{H}

is a homotopy equivalence.

(2) For each subgroup H = D_{6}, D_{10}, A_{4}, and for any nontrivial subgroup K of H,

the map F_{H}^{K} : W_{H}^{K} → I × Y^{K}

is an integral homology equivalence.

We shall prove the proposition above in Section 5.

The next remark follows immediately from Proposition 3.1.

Remark 4.2. The map f in Proposition 4.1 has the following properties:

(1) f^{−1}(p_{+})^{A_{5}} = ∅,

(2) f^{−1}(p_{+})^{A_{4}} = {x_{+,A_{4}}} (one point),

(3) Iso(G, f^{−1}(U_{Y,+})) = S(G) ∖ (A_{5}), and

(4) Iso(G, f^{−1}(U_{Y,−})) = S(G) ∖ (A_{4}).

Let fff = (f, b) and F F F_{H} = (F_{H}, B_{H}), where H ⊊ A_{5}, be the A_{5}-framed map and the Hframed cobordisms described in Proposition 4.1. Note that dim Y^{H} = 4 if and only if H is a subgroup of order 2, and that dim Y^{K} ≠ 3 for any subgroup K. Since f^{C_{2}} : X^{C_{2}} → Y^{C_{2}} is a homotopy equivalence, X^{C_{2}} is a 4-dimensional homotopy sphere and is connected. Define the doubly parametrized form ring A A A (cf. [4, p.278]) by

A A A = (ℤ, A_{5}, ∅, A_{5}(2), λ, w_{Y}),

where

A_{5}(2) = {g ∈ A_{5} | g^{2} = e, g ≠ e},

λ = (−1)^{4} and w_{Y} : G → {1, −1} is the trivial homomorphism. Then, by [4, Theorem 7.3], the obstruction σ(fff) (to convert f : X → Y to a homotopy equivalence by equivariant surgery on the free part of X) lies in the abelian group W_{8}(A A A, A_{5}(2)) (= W_{8}(ℤ, A_{5}, ∅, A_{5}(2), A_{5}(2)), according to the notation in [12]).

Applying the framed cobordism invariance theorem [23, Theorem 3.2] to the H-framed cobordism F F F_{H}, we conclude σ(Res_{H}^{G}fff) = 0 for each H ⊊ A_{5}. Since Res_{H}^{G}σ(fff) = σ(Res_{H}^{G}fff), Lemma 2.4 implies σ(fff) = 0. Thus, we can perform equivariant surgery on the free part of X so that the resulting map f′ : X′ → Y is a homotopy equivalence. Then, X′ is a homotopy sphere with exactly one G-fixed point. Taking the equivariant connected sum of the right number of copies of X′, we obtain a smooth action of G on the 8-dimensional standard sphere with exactly one fixed point. This is explained in [13, Proposition 1.3] by letting the F there denote the set of Sylow subgroups of A_{5}. □

5. Reflection Method and Proof of Proposition 4.1

We return in this section to the general situation where G is any finite group. We develop a certain way of performing equivariant surgery which we call the reflection method and apply it to prove 4.1.

Let fff and F F F_{H}, H ⊊ G, be as in Proposition 3.1.

For an H-invariant subset Z of W_{H}, a compact H-submanifold N of W_{H} is called a product H-cobordism neighborhood of Z with respect to Ψ if N is an H-neighborhood of Z and Ψ : N → I × (X ∩ N) is an H-diffeomorphism such that

(1) Ψ(X ∩ N) = {0} × (X ∩ N),

(2) the restriction

Ψ|_{X∩N} : X ∩ N → {0} × (X ∩ N)

coincides with the canonical map, and

(3) Ψ^{−1}({1} × (X ∩ N)) = Y ∩ N.

Definition 5.1. Let fff be a G-framed map, H a subgroup of G, F F F_{H} an H-framed cobordism between fff and another G-framed map f′, and K ≠ {e} a subgroup of H. We say that the pair (fff,F F F_{H}) is adjusted for (H, K) if

(1) f^{K} : X^{K} → Y^{K} is a homotopy equivalence, and

(2) F_{H}^{K} : W_{H}^{K} → I × Y^{K} is an integral homology equivalence.

We provide next a method which enables us to adjust a pair (fff,F F F_{H}) as in (5.1), for (H, K) under the assumption that N_{G}(K) ⊂ H. The following notation

\begin{matrix}X^{>K} = \{x ∈ X | G_{x} ⊋ K\} and \\ W_{H}^{>K} = \{w ∈ W_{H} | H_{w} ⊋ K\} \end{matrix}

will be used where G_{x} and H_{w} denote the isotropy subgroups of G at x and H at w, respectively.

Lemma 5.2 (Reflection Method). Let H and K ≠ {e} be proper subgroups of G such that N_{G}(K) ⊂ H, and let fff and F F F_{H} be as in Proposition 3.1. Suppose there is a product H-cobordism neighborhood N_{>K} of HX^{>K} ∪ HW_{H}^{>K} in W_{H} with respect to an Hdiffeomorphism Ψ_{>K} : N_{>K} → I × (X ∩ N_{>K}). Then one can perform G-surgery of fff and H-surgery of F F F_{H} of isotropy type (K) so that the resulting maps fff′ and F F F ′_{H} satisfy the conditions:

(1) f′^{K} : X′^{K} → Y^{K} is a homotopy equivalence,

(2) F′_{H}^{K} : W′_{H}^{K} → I × Y^{K} is a homotopy equivalence, and

(3) there exists a closed H-regular neighborhood U_{K} of HW′_{H}^{K} in W′_{H} such that U_{K} ⊃ N_{>K} and U_{K} is a product H-cobordism neighborhood of HW′_{K}^{K} with respect to an H-diffeomorphism ψ_{K} : U_{K} → I × (X′ ∩ U_{K}) with ψ_{K}|_{N_{>K}} = Ψ_{>K}.

Proof. This follows from [17, Theorem 4.2] and its proof. □

Proof of Proposition 4.1. Let G = A_{5} and H ⊊ A_{5}. Let fff and F F F_{H} denote respectively an A_{5}-framed map and H-framed cobordism obtained by Proposition 3.1 where φ_{+} and φ_{−} are specified as in Proposition 4.1. Since each maximal subgroup of A_{5} is conjugate to A_{4}, D_{6} or D_{10}, it suffices to treat the cases H = A_{4}, D_{6} and D_{10}.

Step 1 : (H, K) = (A_{5}, A_{4}). By construction, X^{A_{4}} has precisely two points and f^{A_{4}} : X^{A_{4}} → Y^{A_{4}} is a homotopy equivalence. It is clear that W_{A_{4}}^{A_{4}} has two components diffeomorphic to [0, 1] and the others are diffeomorphic to S^{1}. Perform 1-dimensional A_{4}- surgery of F F F_{A_{4}} to remove all components diffeomorphic to S^{1} and thereby obtain a new F F F_{A_{4}} such that F_{A_{4}}^{A_{4}} : W_{A_{4}}^{A_{4}} → I × Y^{A_{4}} is a homotopy equivalence. Thus the new W_{A_{4}}^{A_{4}} is diffeomorphic to I × X^{A_{4}}. Hence the new pair (fff,F F F_{A_{4}}) is adjusted for (A_{4}, A_{4}). Moreover there exists a product A_{4}-cobordism neighborhood U_{A_{4}} of W_{A_{4}}^{A_{4}} in W_{A_{4}} with respect to some A_{4}-diffeomorphism ψ_{A_{4}} : U_{A_{4}} → I × (X ∩ U_{A_{4}}).

Step 2 : (H, K) = (D_{6}, D_{6}). Note that X^{>D_{6}} = X^{A_{5}} = f^{−1}(y_{−}), and W_{D_{6}}^{>D_{6}} = ∅. There exists a product D_{6}-cobordism neighborhood N_{>D_{6}} of f^{−1}(y_{−}) (= D_{6}X^{>D_{6}} ∪ D_{6}W_{D_{6}}^{>D_{6}}) in W_{D_{6}} with respect to some D_{6}-diffeomorphism Ψ_{>D_{6}} : N_{>D_{6}} → I × (X ∩ N_{>D_{6}}). Since N_{A_{5}}(D_{6}) = D_{6}, we can apply the Reflection Method above. Doing this, we obtain a pair (fff,F F F_{D_{6}}) which is adjusted for (D_{6}, D_{6}). Moreover there exists a product D_{6}-cobordism neighborhood U_{D_{6}} in W_{D_{6}} with respect to some D_{6}-diffeomorphism φ_{D_{6}} : U_{D_{6}} → I ×(X ∩ U_{D_{6}}).

Step 3 : (H, K) = (D_{10}, D_{10}). This is similar to Step 2.

Step 4 : (H, K) = (A_{4}, D_{4}). Note that X^{>D_{4}} = X^{A_{4}} and W_{A_{4}}^{>D_{4}} = W_{A_{4}}^{A_{4}}. Recall that in Step 1, we obtained a product A_{4}-cobordsim neighborhood N_{>D_{4}} = U_{A_{4}} of W_{A_{4}}^{A_{4}} (= A_{4}X^{>D_{4}} ∪ A_{4}W_{A_{4}}^{>D_{4}}) in W_{A_{4}} with respect to Ψ_{>D_{4}} = ψ_{A_{4}}. Since N_{A_{5}}(D_{4}) = A_{4}, we can apply the Reflection Method. Doing this, we obtain a pair (fff,F F F_{A_{4}}) which is adjusted for (A_{4}, D_{4}). Moreover W_{A_{4}}^{D_{4}} has a product A_{4}-cobordism neighborhood U_{D_{4}} in W_{A_{4}} with respect to some A_{4}-diffeomorphism ψ_{D_{4}} : U_{D_{4}} → I × (X ∩ U_{D_{4}}).

Step 5 : (H, K) = (D_{10}, C_{5}). Note that X^{>C_{5}} = X^{D_{10}} and W_{D_{10}}^{>C_{5}} = W_{D_{10}}^{D_{10}}. Recall that in Step 3, we obtained a product D_{10}-cobordism neighborhood N_{>C_{5}} = U_{D_{10}} of W_{D_{10}}^{D_{10}} (= D_{10}X^{>C_{5}} ∪ D_{10}W_{D_{10}}^{>C_{5}}) in W_{D_{10}} with respect to some Ψ_{>C_{5}} = ψ_{D_{10}} : U_{D_{10}} → I × (X ∩ U_{D_{10}}). Since N_{A_{5}}(C_{5}) = D_{10}, we can apply the Reflection Method and obtain a pair (fff,F F F_{D_{10}}) which is adjusted for (D_{10}, C_{5}). Moreover W_{D_{10}}^{C_{5}} has a product D_{10}- cobordism neighborhood U_{C_{5}} in W_{D_{10}} with respect to some D_{10}-diffeomorphim ψ_{C_{5}} : U_{C_{5}} → I × (X ∩ U_{C_{5}}).

Step 6 : (H, K) = (D_{6}, C_{3}). Note that W_{D_{6}}^{>C_{3}} = W_{D_{6}}^{D_{6}}, but X^{>C_{3}} ⊋ X^{D_{6}}. Recall that in Step 2, we obtained a product D_{6}-cobordism neighborhood U_{D_{6}} of W_{D_{6}}^{>C_{3}} in W_{D_{6}}. There are exactly four subgroups of A_{5} properly containing C_{3}. They are A_{5}, D_{6}, and two subgroups isomorphic to A_{4}, say A(1) and A(2). Thus

X^{>C_{3}} = X^{D_{6}} ∪ X^{A(1)} ∪ X^{A(2)}.

Recall that dim X^{C_{3}} = 2 and dim X^{A(i)} = 0, hence dim W_{D_{6}} = 3 and

dim(X^{>C_{3}} ∖ W_{D_{6}}^{>C_{3}}) = 0.

Without loss of generality, we can assume that F_{D_{6}}^{C_{3}} is 1-connected, consequently W_{D_{6}}^{C_{3}} is connected. Then it is easy to find a product D_{6}-cobordism neighborhood N_{>C_{3}} of X^{>C_{3}} ∪ W_{D_{6}}^{>C_{3}} in W_{D_{6}} with respect to some D_{6}-diffeomorphism Ψ_{>C_{3}} : N_{>C_{3}} → I × (X ∩ N_{>C_{3}}). Since N_{A_{5}}(C_{3}) = D_{6}, we can apply the Reflection Method and obtain a pair (fff,F F F_{D_{6}}) which is adjusted for (D_{6}, C_{3}). Moreover there is a product D_{6}-cobordism neighborhood U_{C_{3}} of W_{D_{6}}^{C_{3}} in W_{D_{6}} with respect to some D_{6}-diffeomorphism ψ_{C_{3}} : U_{C_{3}} → I × (X ∩ U_{C_{3}}).

The next step needs some terminology. Definition 5.3. Let W be a manifold with boundary ∂W. A submanifold A of W is called a neat submanifold of W if the following holds:

(1) ∂A = A ∩ ∂W.

(2) There exists a collar neighborhood C of ∂W in W such that C ≅^{φ} ∂W × I with

respect to some diffeomorphism φ, and A ∩ C ≅^{φ|_{A∩C}} ∂A × I.

Step 7 : (H, K) = (A_{4}, C_{2}). Note that W_{A_{4}}^{>C_{2}} = W_{A_{4}}^{D_{4}}, but X^{>C_{2}} ⊋ X^{D_{4}}. There are seven subgroups of A_{5} which properly contain D_{4}. They are A_{5}, A_{4}, D_{4}, two distinct subgroups isomorphic to D_{6}, say D(1) and D(2), and two distinct subgroups isomorphic to D_{10}, say D(3) and D(4). Thus we have

X^{>C_{2}} = X^{D_{4}} ∪ ⋃_{i=1}^{4} X^{D(i)}.

Recall that dim X^{D(i)} = 1, dim X^{D_{4}} = 2 and dim X^{C_{2}} = 4. Hence dim W_{A_{4}}^{D_{4}} = 3 and dim W_{A_{4}}^{C_{2}} = 5. Without loss of generality, we can suppose that F_{A_{4}}^{C_{2}} is 2-connected. Consequently W_{A_{4}}^{C_{2}} is 1-connected.

Below we shall find a product A_{4}-cobordism neighborhood N_{>C_{2}} of A_{4}X^{>C_{2}} ∪ W_{A_{4}}^{D_{4}} (= A_{4}X^{>C_{2}} ∪ A_{4}W_{A_{4}}^{>C_{2}}). After this is done, since N_{A_{5}}(C_{2}) = D_{4} ⊂ A_{4}, we can apply the Reflection Method and obtain a pair (fff,F F F_{A_{4}}) which is adjusted for (A_{4}, C_{2}). Moreover there is a product A_{4}-cobordism neighborhood U_{C_{2}} of A_{4}W_{A_{4}}^{C_{2}} in W_{A_{4}} with respect to some A_{4}-diffeomorphism ψ_{C_{2}} : U_{C_{2}} → I × (X ∩ U_{C_{2}}).

Let U_{D_{4}} be a product A_{4}-cobordism neighborhood of W_{A_{4}}^{D_{4}} in W_{A_{4}} with respect to an A_{4}- diffeomorphims ψ_{D_{4}} obtained in Step 4. By definition, U_{D_{4}} is an A_{4}-tubular neighborhood of W_{A_{4}}^{D_{4}} in W_{A_{4}}. Set

∂_{0}(U_{D_{4}}) = U_{D_{4}} ∩ Closure(W_{A_{4}} ∖ U_{D_{4}}), and
Int_{0}(U_{N}) = U_{N} ∖ ∂_{0}(U_{D_{4}}).

The restriction

ψ_{D_{4}}^{−1}|_{I×(X^{>C_{2}}∩∂_{0}(U_{N}))} : I × (X^{>C_{2}} ∩ ∂_{0}(U_{N})) → W_{A_{4}}^{C_{2}} ∖ U_{D_{4}}

is an A_{4}-embedding. It is easy to see that X^{>C_{2}} ∖ Int_{0}(U_{D_{4}}) has four components each of which is diffeomorphic to [0, 1]. They are

I(i) = X^{D(i)} ∖ Int_{0}(U_{D_{4}})

for i = 1, 2, 3, 4. Consider the inclusion

ι_{i} : (I(i), ∂I(i)) → (W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}}), ∂_{0}(U_{D_{4}}))

for i = 1, 3. Since the map

π_{1}(Y^{C_{2}} ∖ Int_{0}(U_{D_{4}})) → π_{1}(W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}}))

induced by the inclusion is surjective, there exists a map

η_{i} : I × I(i) → W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}})

satisfying the following properties:

(1) η_{i} is an embedding and Im(η_{i}) is a neat submanifold of ∂W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}}),

(2) η_{i}|_{{0}×I(i)} : {0} × I(i) → W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}}) coincides with ι_{i},

(3) η_{i}({1} × I(i)) ⊂ Y^{C_{2}} ∖ Int_{0}(U_{D_{4}}), and

(4) the restriction of η_{i} to I × ∂I(i) coincides with the restriction of ψ_{D_{4}}^{−1}.

Furthermore, putting η_{1} and η_{3} in general position, we may suppose that η_{1}, hη_{1}, η_{3} and hη_{3} are mutually disjoint for some h ∈ D_{4} ∖ C_{2}. Then, extending the maps η_{1} and η_{3}, we get a D_{4}-embedding

η : I × ∐_{i=1}^{4} I(i) → W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}})

whose image is a neat submanifold of W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}}). Clearly

A_{4}X^{>C_{2}} ∖ X^{D_{4}} = ∐{gD_{4}(X^{>C_{2}} ∖ X^{D_{4}}) | gD_{4} ∈ A_{4}/D_{4}}.

Thus the map

A_{4} ×_{D_{4}} η : I × (A_{4} ×_{D_{4}} ∐_{i=1}^{4} I(i)) → A_{4}W_{A_{4}}^{C_{2}} ∖ Int_{0}(U_{D_{4}})

is an A_{4}-embedding. Moreover, if U′ is a closed A_{4}-tubular neighborhood of Im(A_{4} ×_{D_{4}} η) in W_{A_{4}} ∖ Int_{0}(U_{D_{4}}), then the union

N_{>C_{2}} := U_{D_{4}} ∪ U′

is a product A_{4}-cobordism neighborhood of A_{4}X^{>C_{2}} ∪ A_{4}W_{A_{4}}^{>C_{2}} in W_{A_{4}}.

Step 8 : (H, K) = (A_{4}, C_{3}). By Step 6, f^{C_{3}} is a homotopy equivalence. Thus it suffices to modify F_{A_{4}} by A_{4}-surgery of isotropy type (C_{3}) (relative to the boundary) so that F_{A_{4}}^{C_{3}} becomes an integral homology equivalence. Note that N_{A_{4}}(C_{3}) = C_{3}, dim W_{A_{4}}^{C_{3}} = 3 and dim W_{A_{4}}^{>C_{3}} = 1. Thus by [24, Theorem 1.1], the equivariant surgery obstruction σ(F F F _{A_{4}}^{C_{3}}) lies in Wall’s group L_{3}^{h}(ℤ[C_{3}/C_{3}], triv). But the group is trivial and hence the obstruction vanishes. Thus we can adjust (fff,F F F_{A_{4}}) for (A_{4}, C_{3}).

Step 9 : (H, K) = (D_{6}, C_{2}). By Step 7, f^{C_{2}} is a homotopy equivalence. Thus, it suffices to modify F_{D_{6}} by D_{6}-surgery of isotropy type (C_{2}) (relative to the boundary) so that F_{D_{6}}^{C_{2}} becomes an integral homology equivalence. Note that N_{D_{6}}(C_{2}) = C_{2}, dim W_{D_{6}}^{C_{2}} = 5 and dim W_{D_{6}}^{>C_{2}} = 2. Thus by [18, Theorem A], the equivariant surgery obstruction σ(F F F _{D_{6}}^{C_{2}}) lies in Wall’s group L_{5}^{h}(ℤ[C_{2}/C_{2}], triv). But the group is trivial and hence the obstruction vanishes. Thus we can adjust (fff,F F F_{D_{6}}) for (D_{6}, C_{2}).

Step 10 : (H, K) = (D_{10}, C_{2}). This is similar to Step 9.

This completes the proof of Proposition 4.1. □

Remark 5.4. We proved in [19] and [2], respectively, that S^{6} and S^{7} have smooth one fixed point actions of A_{5}, using the triviality of certain obstruction groups, namely that W_{6}(ℤ[A_{5}], ΓA_{5}(2); triv) = 0 and W_{7}(ℤ[A_{5}], ΓA_{5}(2); triv) = 0, respectively. But replacing the real A_{5}-module V = U(4) ⊕ U(4) in (4.1) by V = U(3) ⊕ U(3) and V = U(3) ⊕ U(4), respectively, the arguments of the present paper provide alternative proofs of the results above without employing the triviality of obstruction groups.

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Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany

Department of Environmental and Mathematical Sciences Faculty of Environmental Science and Technology, Okayama University, Okayama, 700-8530 Japan

  E-mail address: bak@mathematik.uni-bielefeld.de
  E-mail address: morimoto@ems.okayama-u.ac.jp