AN INTEGER INVARIANT OF A GROUP ACTION (TOPOLOGY, FIBRATIONS, TRANSFER)

Restrictions imposed on the topology of a space X by the action of a group G are investigated via an invariant recently defined by Gottlieb. Gottlieb\u27s trace divides many classical integers associated with the pair (G,X) such as Euler characteristics of invariant subspaces, Lefschetz numbers of equivariant self maps, and characteristic numbers. The necessary definitions and the fundamental results of Gottlieb are given in Section 1. In Section 2 we are interested in the behavior of the trace when the action is restricted to a subgroup. We show that for a compact connected Lie group G the trace does not change when the action is restricted to the normalizer of a maximal torus; for a finite group the trace is the product of the traces of Sylow p-subgroups. As an application we extend a theorem of Browder and Katz to the non-free case. To analyze the trace on an invariant subspace in Section 3 we work in the category of finite G-cw complexes with G a compact Lie group. The obstruction to removing an equivariant cell is, without changing the trace, is shown to lie in the cohomology of the isotropy subgroup of the cell. Thus we establish that the trace of an action depends only on the singular set. Another consequence is that the trace is completely determined by the family of isotropy subgroups in low dimensions. For a finite group G, the greatest common divisor of orbit sizes is divisible by the trace. In general equality holds only in low dimensions (Section 3) or when G is an elementary abelian p-group acting smoothly on a compact manifold (a result of W. Browder). In the appendix we prove this greatest common divisor divides the Lefschetz number of an equivariant self map. This leads to several Borsuk-Ulam type results