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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Equivariant bundles and cohomology
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by A. Kozlowski PDF
Trans. Amer. Math. Soc. 296 (1986), 181-190 Request permission

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
  • © Copyright 1986 American Mathematical Society
  • 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