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Cohomology of Flat Principal Bundles

Part of: Global differential geometry Fiber spaces and bundles

Published online by Cambridge University Press:  22 May 2018

Yanghyun Byun  and
Joohee Kim
Yanghyun Byun
Affiliation:
Department of Mathematics, Hanyang University, Seoul, Korea ([email protected]; [email protected])
Joohee Kim
Affiliation:
Department of Mathematics, Hanyang University, Seoul, Korea ([email protected]; [email protected])
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Abstract

We invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H*dR(G) itself. Then, we show that when a flat connection A exists on a principal G-bundle P, we may construct a homomorphism EA: H*dR(G)→H*dR(P), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we show that that both the flat principal bundle and its adjoint bundle have the real coefficient cohomology isomorphic to that of the trivial bundle.

Keywords

flat principal bundlede Rham cohomologyadjoint bundle

MSC classification

Primary: 53C05: Connections, general theory
Secondary: 55R20: Spectral sequences and homology of fiber spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 869 - 877
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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