Cohomology fibrations
type: Item
instance of: scholarly article
title: Cohomology fibrations
zbMATH Open document ID: 1001.55017
DOI: 10.1016/S0166-8641(01)00137-7
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(f:X\to Y\) be a proper map of metric spaces, that is, inverse images of compact sets are compact. Let \(R\) be a commutative ring and suppose there is a sequence of \(R\)-submodules \(M_k\) of the Čech cohomology modules \(H^k(X;R)\) for all \(k\geq 0\) such that the homomorphism induced by inclusion takes \(M_k\) isomorphically onto \(H^k(f^{-1}(y);R)\) for \(y\in Y\). It is shown that, if all the \(R\)-modules \(M_k\) are finitely generated, then \(f\) resembles a locally trivial bundle in a suitable cohomological sense. More precisely, given \(y_0\in Y\) and \(m\geq 0\) there is a neighborhood \(U\) of \(y_0\) such that, for any compact subset \(A\) of \(U\) and \(y\in A\), the module \(H^m(f^{-1}(A);R)\) is isomorphic to \(H^m(A\times f^{-1}(y);R)\) by an isomorphism with appropriate properties.
Mathematics Subject Classification ID: 55R05
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 1787554
zbMATH Keywords: Čech cohomology
zbMATH Keywords: bundle
author: J. Dydak
reviewed by: Robert F. Brown
MaRDI profile type: Publication
cites work: Q5541099
cites work: Q3768026
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full work available at URL: s0166-8641(01)00137-7
OpenAlex ID: W4213334704
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description: scientific article; zbMATH DE number 1787554