On the cohomology of an isolating block and its invariant part
type: Item
instance of: scholarly article
title: On the cohomology of an isolating block and its invariant part
zbMATH Open document ID: 1183.37027
author: Anna Gierzkiewicz
author: Klaudiusz Wójcik
published in: Topological Methods in Nonlinear Analysis
publication date: 2009-06-23T00:00:00Z
review text: 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.
reviewed by: Roman Srzednicki
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 5567743
zbMATH Keywords: isolating block
zbMATH Keywords: Conley index
zbMATH Keywords: Alexander-Spanier cohomology
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5567743