The G-signature Theorem on Witt spaces

Abstract

Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed point set associated to g is normally non-singular, we prove a formula for Sign(g,X), the G-signature of X computed at g, thus extending to Witt G-pseudomanifolds the fundamental result proved by Atiyah, Segal and Singer on smooth compact G-manifolds. Along the way, we give a detailed study of the fixed point set of a Thom-Mather G-space X and our main result in this direction is a sufficient condition ensuring that the fixed point set associated to G is included in X in a normally non-singular manner. This latter result provides many examples where our formula applies.