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Some remarks on conjugacy classes of bundle gauge groups. (English) Zbl 0854.55013

Let \(G\) be a topological group. If two principal \(G\)-bundles are equivalent then their gauge groups are conjugate. The author studies the question of when conjugacy of gauge groups implies that the bundles are equivalent. First, she gives an example of two bundles whose gauge groups are isomorphic but not conjugate. Then, under certain conditions on \(G\) and on local trivializations of the bundles, she proves that two principal bundles over a finite polyhedron \(B\), whose gauge groups are conjugate, are equivalent iff the pullback bundles obtained by the inclusion of the 3-skeleton in \(B\) are equivalent. The paper concludes with a discussion of some special cases, in particular, with examples of bundles over real projective spaces.

MSC:

55R10 Fiber bundles in algebraic topology
57S17 Finite transformation groups
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References:

[1] S.T. Hu The Equivalence of Fibre Bundles , Annals of Math. , 53 n. 2 ( 1951 ), 256 - 276 . MR 39999 | Zbl 0042.41701 · Zbl 0042.41701 · doi:10.2307/1969542
[2] C. Morgan , R.A. Piccinini Conjugacy Classes of Groups of Bundle Automorphisms , Manuscripta Math. , 63 ( 1989 ), 233 - 244 . Article | MR 980575 | Zbl 0684.55016 · Zbl 0684.55016 · doi:10.1007/BF01168874
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