ON FIXED POINT DATA OF SMOOTH ACTIONS ON SPHERES

Masaharu Morimoto

We report results obtained jointly with K. Pawałowski.

There are two fundamental questions about smooth actions on manifolds. Let G be a finite group and M a manifold.

Question 1. Which manifolds F can be the G-fixed point sets of G-actions on M, i.e. M^{G} = F?

Question 2. Which G-vector bundles ν over F can be the G-tubular neighborhoods (i.e. G-normal bundles) of F = M^{G} in M?

If ν can be realized as a subset of M in the way above, we say that (F, ν) occurs as the G-fixed point data in M. If for a real G-module W with W^{G} = 0, (F, ν ⊕ε_{M}^{W}) occurs as the G-fixed point data in M then we say that (F, ν) stably occurs as the G-fixed point data in M. These questions were studied by B. Oliver [O2] in the case where G is not of prime power order and M is a disk or a Euclidean space. The topic of the current talk is the case where G is an Oliver group and M is a sphere.

Let G be a finite group not of prime power order. A G-action on M is called P-proper if M^{P} ⊋ M^{G} for any Sylow subgroup P of G. There are necessary conditions for (F, ν) to stably occur as the G-fixed point data of a P-proper Gaction on a sphere.

(F1) (Oliver Condition) χ(F) ≡ χ(M) mod n_{G} (where n_{G} is the integer called Oliver’s number [O1]).

(B1) (Product Bundle Condition) τ_{F} ⊕ ν = 0 in K̃O(F).

(B2) (Smith Condition) For each prime p and any Sylow p-subgroup P of G, τ_{F} ⊕ ν = 0 in K̃O_{P} (F)_{(p)}.

By Oliver [O2], Conditions (F1), (B1) and (B2) are also necessary–sufficient conditions for (F, ν) to stably occur as the G-fixed point data in a disk. By [O1], n_{G} is equal to 1 if and only if there are no normal series P ◁H ◁G such that |P | = p^{s}, H/P is cyclic, and |G/H| = q^{t} (s, t ≥ 0). A group G with n_{G} = 1 is called an Oliver group. Clearly any nonsolvable group is an Oliver group. A nilpotent group is an Oliver group if and only if it has at least three noncyclic Sylow subgroups. In the case where G is an Oliver group, Condition (F1) provides no restriction.

We begin the preparation for our sufficient conditions. For a finite group G and a prime p, let G^{p} denote the minimal normal subgroup of G such that G/G^{p} is of p-power order (possibly G^{p} = G). Let L(G) denote the set of all subgroups H of G such that H ⊇ G^{p} for some prime p. Let P(G) denote the set of all subgroups P of G such that |P | is a prime power (possibly |P | = 1). A G-action on M is said to be (P, L)-proper if the action is P-proper and if any connected component of X^{H} (H ∈ L(G)) does not contain a connected component of M^{G} as a proper subset. If G is an Oliver group then the G-action on

V (G) = (ℝ[G] − ℝ) − ⨁_{p\|G|}(ℝ[G/G^{p}] − ℝ)

is (P, L)-proper ([LM]). A finite group G is said to be admissible if there is a real G-module V such that dim V^{P} > 2 dim V^{H} for any P ∈ P(G) and any H ≤ G with H ⊋ P, and dim V^{H} = 0 for any H ∈ L(G).

Theorem (M.M.–M. Yanagihara [MY1–2]). Let G be an Oliver group. If G^{2} = G or G^{p} ≠ G for at least 2 distinct odd primes then G is admissible. In particular, an Oliver group G is admissible in each case: G is nilpotent; G is perfect.

The symmetric group of degree 5 is not admissible.

K. H. Dovermann–M. Herzog recently proved that S_{n} (n ≧ 6) are admissible.

Our main result is:

Theorem A. Let G be an admissible Oliver group (resp. an Oliver group) . Let F be a closed manifold (resp. a finite discrete space) and let ν be a real G-vector bundle over F such that dim ν^{H} = 0 whenever H ∈ L(G). Then the following (1)–(3) are equivalent:

(1) (F, ν) stably occurs as the G-fixed point data of a (P, L)-proper G-action on a sphere.

(3) (F, ν) stably occurs as the G-fixed point data in a disk.

(3) τ_{M} ⊕ ν satisfies (B1)–(B2).

A finite group not of prime power order belongs to exactly one of the following six classes ([O2]):

A: G has a dihedral subquotient of order 2n for a composite integer n.

B: G ∉ A and G has a composite order element conjugate to its inverse.

C: G ∉ A ∪ B, G has a composite order element and the Sylow 2-subgroups are not normal in G.

C_{2}: G has a composite order element and the Sylow 2-subgroup is normal in G.

D: G has no elements of composite order and the Sylow 2-subgroups are not normal in G.

D_{2}: G has no elements of composite order and the Sylow 2-subgroup is normal in G.

Corollary B. Let G be a nontrivial perfect group and F a closed manifold. Then F occurs as the G-fixed point set of a P-proper G-action on a sphere if and only if F occurs as the G-fixed point set in a disk (in other words,

G ∈ A: there is no restriction.

G ∈ B: c_{ℝ}([τ_{F}]) ∈ c_{ℍ}({ \atop KSp}(F)) + Tor(K̃(F))

G ∈ C: [τ_{F}] ∈ r_{ℂ}(K̃(F)) + Tor(K̃O(F))

G ∈ D: [τ_{F}] ∈ Tor(K̃O(F)). )

Theorem C. Let G be a nilpotent Oliver group and F a closed manifold. Then the following (1)–(3) are equivalent.

(1) F occurs as the G-fixed point set of a P-proper G-action on a sphere.

(2) τ_{F} is stably complex.

(3) F occurs as the G-fixed point set of a G-action on a disk.

Our basic methods are:

(1) An extension of the method of equivariant bundles in [O2] (with modifications).

(2) The equivariant thickening of [P].

(3) The equivariant surgery results of [M1–2].

References

[BM] Bak, A. and Morimoto, M., Equivariant surgery with middle dimensional singular sets. I, Forum Math. 8 (1996), 267-302.

[LM] Laitinen, E. and Morimoto. M., Finite groups with smooth one fixed point actions on spheres, Reports Dept. Math. Univ. Helsinki no. 25 (1993).

[M1] Morimoto, M., Equivariant surgery theory: Construction of equivariant normal maps, Publ. Res. Inst. Math. Sci. Kyoto Univ. 31 (1995), 145–167.

[M2] ———, Equivariant surgery theory: Deleting-inserting theorem of fixed point sets on spheres, Preprint April 1996.

[MY1] Morimoto, M. and Yanagihara, M., On gap conditions of finite group actions, Preprint January 1995.

[MY2] ———, The gap conditions for S_{5} and GAP programs, J. Fac. Env.Sci. Tech., Okayama University 1 (1996), 1–13.

[O1] Oliver, R., Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helvetici 50 (1975), 155-177.

[O2] Oliver, B., Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, to appear in Topology.

[P] Pawałowski, K., Fixed point sets of smooth group actions on disks and Euclidean spaces, Topology 28 (1989), 273–28.

Department of Environmental and Mathematical Sciences, Okayama University, Okayama, Japan

   E-mail address: morimoto@math.ems.okayama-u.ac.jp