学术科研-Geometry & Topology Seminar

Converse of Smith Theory

Abstract

Suppose G is a finite group, and f is a map from a CW complex F to the fixed point of a G-CW complex Y. Is it possible to extend F to a finite G-CW complex X satisfying X^G = F, and extend f to a G-map g: X→ Y, such that g is a homotopy equivalence after forgetting the G-action?

In case Y is a single point, the problem becomes whether a given finite CW complex F is the fixed point of a G-action on a finite contractible CW complex. In 1942, P.A. Smith showed that the fixed point of a p-group action on a finite Z_p-acyclic complex is still Z_p-acyclic. In 1971, Lowell Jones proved a converse for semi-free cyclic group action on finite contractible X. In 1975, Robert Oliver proved that, for general action on finite contractible X, if the order of G is not prime power, then the only obstruction is the Euler characteristic of F.

We extend the classical results of Lowell Jones and Robert Oliver to the general setting. For semi-free action, we encounter a finiteness type obstruction. For general action by group of not prime power order, the obstruction is the Euler characteristics over components of Y^G. We calculate such obstructions for various examples.

This is a joint work with Sylvain Cappell and Shmuel Weinberger.