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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equivariant bundles and cohomology

Author: A. Kozlowski
Journal: Trans. Amer. Math. Soc. 296 (1986), 181-190
MSC: Primary 55N25; Secondary 55R10
MathSciNet review: 837806
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Abstract: Let $G$ be a topological group, $A$ an abelian topological group on which $G$ acts continuously and $X$ a $G$-space. We define "equivariant cohomology groups" of $X$ with coefficients in $A$, $H_G^i(X;A)$, for $i \geq 0$ which generalize Graeme Segal’s continuous cohomology of the topological group $G$ with coefficients in $A$. In particular we have $H_G^1(X;A) \simeq$ equivalence classes of principal $(G,A)$-bundles over $X$. We show that when $G$ is a compact Lie group and $A$ an abelian Lie group we have for $i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)$ where $\tau A$ is the sheaf of germs of sections of the bundle $(X \times EG \times A)/G \to (X \times EG)/G$. For $i = 1$ and the trivial action of $G$ on $A$ this is a theorem of Lashof, May and Segal.

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Keywords: Equivariant cohomology, equivariant bundle, quasi-abelian category
Article copyright: © Copyright 1986 American Mathematical Society