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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Spherical bundles adapted to a $ G$-fibration

Author: J. P. E. Hodgson
Journal: Trans. Amer. Math. Soc. 263 (1981), 355-361
MSC: Primary 55R25; Secondary 57Q50
MathSciNet review: 594413
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Abstract: A spherical fibration $ p:E \to B$ is said to be adapted to a $ G$-fibration $ \pi :E \to E/G$ if there is a fibration $ q:E/G \to B$ with fibre the quotient of a sphere by a free $ G$-action and such that the composition $ q \circ \pi = p$. In this paper it is shown that for spherical bundles in the PL, TOP or Homotopy categories that are adapted to $ {Z_2}$- or $ {S^1}$-fibrations there is a procedure analogous to the splitting principle for vector bundles that enables one to define characteristic classes for these fibrations and to relate them to the usual characteristic classes. The methods are applied to show that a spherical fibration over a $ 4$-connected base which is adapted to an $ {S^1}$-fibration admits a PL structure.

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Keywords: Block bundle, spherical fibration, projectivisation, characteristic classes, transversality obstructions
Article copyright: © Copyright 1981 American Mathematical Society