Cohomology of Flat Principal Bundles

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),...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 2018-08, Vol.61 (3), p.869-877
Main Authors: Byun, Yanghyun, Kim, Joohee
Format: Article
Language:English
Subjects:
Bundling
Homology
Homomorphisms
Lie groups
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Summary: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.
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091517000475