Generalized Rochlin invariants of fixed point sets

Abstract

The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine. Given a semifree differentiable S-action on a closed mod 2 homology sphere M with fixed point set F , generalized Rochlin invariants are definable for both M and F , and one result of this paper states that these two invariants are equal. This yields restrictions on the types of semifree differentiable S-actions that some homology spheres can support and the fixed point sets of actions on homology (8k + 7)-spheres. An action of a group G on a space X is said to be semifree if for each x ∈ X either X is fixed under every element of G or else x is not fixed by any element of G except the identity. During the nineteen sixties and seventies it became apparent that the techniques of differential topology had numerous applications to differentiable actions of compact Lie groups (cf. [Bro2], [BP], [RS]). In particular, these and previously developed techniques yielded considerable information on semifree differentiable actions of S and S on spheres. One result was a complete description of the homeomorphism types of the possible fixed point sets. Specifically, these are all closed manifolds with the same integral homology as a sphere of some appropriate dimension (see [HH, Ch. V,§4]). On the other hand, questions about the diffeomorphism types of the fixed point sets are more difficult to answer. In this paper we shall prove a result (Theorem B below) that complements previous work on the smooth realization question; this is a special case of a more general result (Theorem A) relating the diffeomorphism type of the fixed point set to the diffeomorphism type of the ambient manifold. Although evidence suggests that an analog of Theorem B holds for semifree S-actions (see Proposition 3.2 and [Sc4]), the proof of such an analog seems likely to require additional input. The proofs of Theorems A and B involve a higher dimensional analog of the well known Rochlin invariant for closed homology 3-spheres (e.g., see [EK], where it is called the μ-invariant).