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COHOMOLOGY OF FLAT PRINCIPAL BUNDLES

Authors
Byun, YanghyunKim, Joohee
Issue Date
Aug-2018
Publisher
CAMBRIDGE UNIV PRESS
Keywords
flat principal bundle; de Rham cohomology; adjoint bundle
Citation
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, v.61, no.3, pp.869 - 877
Indexed
SCIE
SCOPUS
Journal Title
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY
Volume
61
Number
3
Start Page
869
End Page
877
URI
https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/149553
DOI
10.1017/S0013091517000475
ISSN
0013-0915
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 T, 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-Ilirsch 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.
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