The second van-Kampen theorem for topological spaces
type: Item
instance of: scholarly article
title: The second van-Kampen theorem for topological spaces
zbMATH Open document ID: 0829.55010
DOI: 10.1016/0166-8641(94)90020-5
author: Andreas Zastrow
published in: Topology and its Applications
publication date: 1996-01-15T00:00:00Z
review text: The van Kampen theorem in its usual form computes the fundamental group of a union of two open subspaces from knowledge of the two fundamental groups and that of the intersection. Local conditions are needed to ensure validity. Another form describes the situation where a space is obtained by identifying two homeomorphic subspaces, again with local conditions. This is less well known, although given as an exercise on pages 321 and 322 of \textit{R. Brown} [Topology: a geometric account of general topology, homotopy types and the fundamental groupoid (1988; Zbl 0655.55001)] where it is suggested that it be derived from the author's groupoid form of the usual van Kampen theorem. This paper examines this result in some detail using conventional techniques of algebraic topology and discussing counterexamples to the result when local conditions are omitted. Interesting subsidiary results on collapsing `\(I\)-bundles' are given that may also be of purely geometric use.
reviewed by: Timothy Porter
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 681622
zbMATH Keywords: collapsing \(I\)-bundles
zbMATH Keywords: van Kampen theorem
zbMATH Keywords: fundamental group
zbMATH Keywords: groupoid
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)90020-5
OpenAlex ID: W1987900911
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description: scientific article; zbMATH DE number 681622