The Fixed Point Theorem in Equivariant Cohomology

Abstract

In this paper we study the ${S^1}$-equivariant de Rham cohomology of infinite dimensional ${S^1}$-manifolds. Our main example is the free loop space $LX$ where $X$ is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology $h_T^*$ which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces $LX$. This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the $\hat A$-polynomial of $X$ arises as an equivariant characteristic class, in the theory $h_T^*$, of the normal bundle to $X$, considered as the space of constant loops, in $LX$.