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Cohomology of Flat Principal Bundles

Published online by Cambridge University Press:  22 May 2018

Yanghyun Byun
Affiliation:
Department of Mathematics, Hanyang University, Seoul, Korea (yhbyun@hanyang.ac.kr; kjh0423@hanyang.ac.kr)
Joohee Kim
Affiliation:
Department of Mathematics, Hanyang University, Seoul, Korea (yhbyun@hanyang.ac.kr; kjh0423@hanyang.ac.kr)

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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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), 8293.CrossRefGoogle Scholar
5.Chevalley, C. and Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85124.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. 5793 (Cambridge University Press, 1993).Google Scholar
8.Dostoglou, S. and Salamon, D., Self-dual instantons and holomorphic curves, Ann. Math. 139 (1994), 581640.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), 181208.Google Scholar
11.Milnor, J., On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1957), 215223.Google Scholar
12.Wehrheim, K., Energy identity for anti-self-dual instantons on ℂ × Σ, Math. Res. Lett. 13(1) (2006), 161166.Google Scholar