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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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: 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$.


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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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