Cohomology of Flat Principal Bundles

Yanghyun Byun, Joohee Kim

Research output: Contribution to journalArticleResearchpeer-review

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.

Original languageEnglish
Pages (from-to)869-877
Number of pages9
JournalProceedings of the Edinburgh Mathematical Society
Volume61
Issue number3
DOIs
StatePublished - 2018 Aug 1

Fingerprint

Principal Bundle
Cohomology
Bundle
Isomorphic
De Rham Cohomology
Flat Connection
Analytic group
Homomorphism
Corollary
Trivial
Algebra
Invariant
Coefficient
Theorem

Keywords

  • adjoint bundle
  • de Rham cohomology
  • flat principal bundle

Cite this

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Cohomology of Flat Principal Bundles. / Byun, Yanghyun; Kim, Joohee.

In: Proceedings of the Edinburgh Mathematical Society, Vol. 61, No. 3, 01.08.2018, p. 869-877.

Research output: Contribution to journalArticleResearchpeer-review

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