Equivariant embeddings of principal -bundles into complex vector bundles |
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Authors: | Jesper Michael Møller |
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Affiliation: | Mathematical Institute, University of Copenhagen, Universitetsparken 5, DK-2100 København Ø, Denmark |
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Abstract: | Let π: E→X be a principal n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and . We compute the primary obstruction to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle εm. We use these criteria for a discussion of the classification of the equivalence classes of principal -bundles that admit equivariant embeddings into εm. Finally, we offer an example of a principal -bundle that admit an ordinary but not an equivariant embedding into ε1. |
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Keywords: | Primary 57M12 Secondary 57S17 |
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