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Proceedings of the American Mathematical Society

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

 

 

Vietoris-Begle theorem and spectra


Authors: Jerzy Dydak and George Kozlowski
Journal: Proc. Amer. Math. Soc. 113 (1991), 587-592
MSC: Primary 55N05; Secondary 55N20, 55P20
MathSciNet review: 1046999
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Abstract: The following generalization of the Vietoris-Begle Theorem is proved: Suppose $ {\left\{ {{E_k}} \right\}_{k \geq 1}}$ is a CW spectrum and $ f:X' \to X$ is a closed surjective map of paracompact Hausdorff spaces such that $ \operatorname{Ind} X = m < \infty $. If $ {f^*}:{E^k}(x) \to {E^k}({f^{ - 1}}(x))$ is an isomorphism for all $ x \in X$ and $ k = {m_0},{m_0} + 1, \ldots ,{m_0} + m$, then $ {f^*}:{E^n}(X) \to {E^n}(X')$ is an isomorphism and $ {f^*}:{E^{n + 1}}(X) \to {E^{n + 1}}(X')$ is a monomorphism for $ n = {m_0} + m$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1046999-9
Keywords: Vietoris-Begle theorem, spectra, cohomology
Article copyright: © Copyright 1991 American Mathematical Society