### 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 E_{A}: 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 language | English |
---|---|

Pages (from-to) | 869-877 |

Number of pages | 9 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 61 |

Issue number | 3 |

DOIs | |

State | Published - 2018 Aug 1 |

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### Keywords

- adjoint bundle
- de Rham cohomology
- flat principal bundle

### Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*61*(3), 869-877. https://doi.org/10.1017/S0013091517000475

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*Proceedings of the Edinburgh Mathematical Society*, vol. 61, no. 3, pp. 869-877. https://doi.org/10.1017/S0013091517000475

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

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Cohomology of Flat Principal Bundles

AU - Byun, Yanghyun

AU - Kim, Joohee

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - adjoint bundle

KW - de Rham cohomology

KW - flat principal bundle

UR - http://www.scopus.com/inward/record.url?scp=85047225769&partnerID=8YFLogxK

U2 - 10.1017/S0013091517000475

DO - 10.1017/S0013091517000475

M3 - Article

VL - 61

SP - 869

EP - 877

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 3

ER -