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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
DOI: https://doi.org/10.1090/S0002-9947-1986-0837806-8
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0837806-8
Keywords: Equivariant cohomology, equivariant bundle, quasi-abelian category
Article copyright: © Copyright 1986 American Mathematical Society

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