The fixed point theorem in equivariant cohomology
The fixed point theorem in equivariant cohomology
复制标题
等变上同调中的不动点定理
DOI：
10.1090/s0002-9947-1990-1010411-x
复制
发表时间：
1990
影响因子：
1.3
通讯作者：
S. Petrack
中科院分区：
文献类型：
--
作者：
J. D. Jones;S. Petrack
本文研究了无限维S流形的S -等变de Rham上同调。我们主要的例子是自由环空间LX，其中X是一个有限维流形，圆由旋转环作用。构造了一种新的等变上同调形式h*，它与有限维空间中常用的周期等变上同调形式一致，并证明了经典不动点定理在环空间LX上的一个合适的类比。这为研究循环空间上的微分形式提供了一个上同框架，我们将这些方法应用于Witten [16]， Atiyah[2]和Bismut[5]的工作中出现的各种问题。特别地，继Atiyah在[2]中的论述之后，我们证明了X的a -多项式在理论h*中，作为LX中常循环空间中X的正规束的一个等变特征类。
In this paper we study the S -equivariant de Rham cohomology of infinite dimensional S -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* 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 A-polynomial of X arises as an equivariant characteristic class, in the theory h*, of the normal bundle to X, considered as the space of constant loops, in LX .