Cohomology fibrations
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.
scientific article; zbMATH DE number 1787554
scientific article