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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The fixed point theorem in equivariant cohomology


Authors: J. D. S. Jones and S. B. Petrack
Journal: Trans. Amer. Math. Soc. 322 (1990), 35-49
MSC: Primary 58A10; Secondary 55N35, 55N91, 58A12
MathSciNet review: 1010411
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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$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1010411-X
PII: S 0002-9947(1990)1010411-X
Article copyright: © Copyright 1990 American Mathematical Society