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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 XG = F, and extend f to a G-map gX→ 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 Zp-acyclic complex is still Zp-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 YG. We calculate such obstructions for various examples.

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