The G-signature Theorem on Witt spaces

TL;DR

This paper extends the Atiyah-Segal-Singer G-signature formula to Witt G-pseudomanifolds using intersection cohomology, providing a new tool for analyzing fixed point sets under group actions.

Contribution

It introduces a formula for the G-signature at an element g on Witt G-pseudomanifolds, generalizing known results from smooth manifolds.

Findings

Derived a formula for Sign(g,X) under fixed point set conditions

Provided conditions ensuring fixed point sets are normally non-singular

Offered numerous examples where the formula applies

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.