Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-20T03:36:08.467Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bleecker, D., Gauge theory and variational principles (Addison–Wesley, 1981).Google Scholar
2.Booss, B. and Bleecker, D., Topology and analysis: the Atiyah–Singer index formula and gauge-theoretic physics (Springer–Verlag, New York, 1985).Google Scholar
3.Bott, R. and Tu, L. W., Differential forms in algebraic topology (Springer–Verlag, New York, 1982).Google Scholar
4.Byun, Y. and Kim, J., Cohomology of flat bundles and a Chern-Simons functional, J. Geom. Phys. 111 (2017), 82–93.CrossRefGoogle Scholar
5.Chevalley, C. and Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.Google Scholar
6.Donaldson, S. and Kronheimer, P., The geometry of four-manifolds (Oxford University Press, New York, 1990).Google Scholar
7.Dostoglou, S. and Salamon, D., Instanton homology and symplectic fixed points, In Symplectic geometry (ed. Salamon, D.), LMS Lecture Note Series, Volume 192, pp. 57–93 (Cambridge University Press, 1993).Google Scholar
8.Dostoglou, S. and Salamon, D., Self-dual instantons and holomorphic curves, Ann. Math. 139 (1994), 581–640.Google Scholar
9.Hatcher, A., Algebraic topology (Cambridge University Press, 2002).Google Scholar
10.Mackenzie, K., Classification of principal bundles and Lie groupoids with prescribed gauge group bundle, J. Pure Appl. Algebra 58 (1989), 181–208.Google Scholar
11.Milnor, J., On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1957), 215–223.Google Scholar
12.Wehrheim, K., Energy identity for anti-self-dual instantons on ℂ × Σ, Math. Res. Lett. 13(1) (2006), 161–166.Google Scholar