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On the existence of equivariant embeddings of principal bundles into vector bundles


Authors: Vagn Lundsgaard Hansen and Jesper Michael Møller
Journal: Proc. Amer. Math. Soc. 88 (1983), 157-161
MSC: Primary 57M12; Secondary 55R25, 57Q35, 57S17
DOI: https://doi.org/10.1090/S0002-9939-1983-0691299-5
MathSciNet review: 691299
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a finite group and let $ X$ be, say, a connected CW-complex of dimension $ k \geqslant 1$. Let $ \pi :E \to X$ be a principal $ G$-bundle and $ p:V \to X$ an $ m$-dimensional $ G$-vector-bundle with trivial action of $ G$ on $ X$. By an equivariant embedding of $ \pi $ into $ p$ we understand an equivariant embedding $ h:E \to V$ commuting with projections. We prove a general embedding theorem, a main special case of which is the following

Theorem. If $ k < m$ and if the action of $ G$ on $ V$ is free outside the zero section for $ p$, then any principal $ G$-bundle $ \pi :E \to X$ can be embedded equivariantly into $ p:V \to X$.


References [Enhancements On Off] (What's this?)

  • [1] A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. (2) 78 (1963), 223-255. MR 0155330 (27:5264)
  • [2] P. F. Duvall and L. S. Husch, Embedding finite covering spaces into bundles, Proc. 1979 Topology Conf. (Ohio Univ., Athens, Ohio, 1979), Topology Proc. 4 (1979), 361-370. MR 598280 (82g:55016)
  • [3] K. Grove, Center of mass and $ G$-local triviality of $ G$-bundles, Proc. Amer. Math. Soc. 54 (1976), 352-354. MR 0394712 (52:15511)
  • [4] V. L. Hansen, Embedding finite covering spaces into trivial bundles, Math. Ann. 236 (1978), 239-243. MR 0494121 (58:13051)
  • [5] D. Husemoller, Fibre bundles, 2nd ed., Graduate Texts in Math., vol. 20, Springer-Verlag, Berlin and New York, 1966.
  • [6] K. J. Prevot, Imbedding smooth involutions in trivial bundles, Pacific J. Math. 89 (1980), 163-168. MR 596927 (82a:55018)
  • [7] G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Math., vol. 61, Springer-Verlag, Berlin and New York, 1978. MR 516508 (80b:55001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0691299-5
Keywords: Principal $ G$-bundle, $ G$-vector-bundle, equivariant embeddings
Article copyright: © Copyright 1983 American Mathematical Society

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