On the cohomology of an isolating block and its invariant part (Q1022819)

A compact subset \(B\) of the phase space of a flow is called an isolating block provided its boundary consists of the set of exit points \(B^-\), the set of entrance points \(B^+\), and the set of points which slide on the boundary from \(B^-\) to \(B^+\). Denote by \(a^-\) the set of points in \(B^-\) for which the whole positive semi-trajectory is contained in \(B\) and denote by \(I(B)\) the invariant part of \(B\). If \(S\) is invariant and \(S=I(B)\) we call \(B\) an isolating block for \(S\). Let \(H^\ast\) denote the Alexander-Spanier cohomology. The main theorem asserts that if \(B\) is an isolating block for \(S\) and \(a^-\) has a compact neighborhood \(Y\) in \(\partial B\), \(Y\subset B^-\), such that the homomorphism \(H^\ast (Y)\to H^\ast(a^-)\) induced by the inclusion is an isomorphism then there exists an isolating block \(W\subset B\) for \(S\) such that the homomorphism \(H^\ast(W) \to H^\ast(S)\) induced by the inclusion is an isomorphism. The theorem is applied in the proof of the following result: if \(S\) is an isolated invariant set for a flow on a \(3\)-manifold and \(H^\ast(S)\) is finitely generated then there exists an isolating block \(B\) for \(S\) such that \(B\) is a topological \(3\)-manifold with boundary and the inclusion induces an isomorphism \(H^\ast(B)\to H^\ast(S)\). The paper contains also other results on cohomology of isolating blocks and their subsets.