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A continuous version of the Borsuk-Ulam theorem

Author: Jan Jaworowski
Journal: Proc. Amer. Math. Soc. 82 (1981), 112-114
MSC: Primary 55R25; Secondary 55R20
MathSciNet review: 603612
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Abstract: Let $ p:E \to B$ be an $ n$-sphere bundle, $ q:V \to B$ be an $ {{\mathbf{R}}^n}$-bundle and $ f:E \to V$ be a fibre preserving map over a paracompact space $ B$. Let $ \overline p :\overline E \to B$ be the projectivized bundle obtained from $ p$ by the antipodal identification and let $ {\overline A _f}$ be the subset of $ \overline E $ consisting of pairs $ \{ e, - e\} $ such that $ fe = f( - e)$. If the cohomology dimension $ d$ of $ B$ is finite then the map $ (\bar{p} \vert \overline{A}_f)^*$ is injective for a continuous cohomology theory $ {H^*}$. Moreover, if the $ j$th Stiefel-Whitney class of $ q$ is zero for $ 1 \leqslant j \leqslant r$ then $ (\bar{p} \vert \overline{A}_f)^*$ is injective in degrees $ i \geqslant d - r$. If all the Stiefel-Whitney classes of $ q$ are zero then $ (\bar{p} \vert \overline{A}_f)^*$ is injective in every degree.

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

  • [1] K. Borsuk, Drei Sätze über die $ n$-dimensionale Euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
  • [2] J. E. Connett, On the cohomology of the fixed-point sets and coincidence-point sets, Indiana Univ. Math. J. 24 (1974-75), 627-634. MR 0365553 (51:1805)
  • [3] D. Husemoller, Fibre bundles, McGraw-Hill, New York, 1966. MR 0229247 (37:4821)

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Keywords: Sphere bundle, vector space bundle, antipodal map, involution, fibre preserving map, Stiefel-Whitney class
Article copyright: © Copyright 1981 American Mathematical Society

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