The second van-Kampen theorem for topological spaces
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.
scientific article; zbMATH DE number 681622
scientific article