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Van Kampen’s theorem for locally sectionable maps. (English) Zbl 1492.18013

The paper generalizes the following Van Kampen theorem for unions of non-connected spaces to the context where families of subspaces of the base space \(B\) are replaced with a large space \(E\) equipped with a locally sectionable continuous map \( p: E \rightarrow B\).
Theorem 1.1. [R. Brown and A. R. Salleh, Arch. Math. 42, 85–88 (1984; Zbl 0521.57002)]. Let \((B_\lambda)_{\lambda\in \Lambda}\) be a family of subspaces of \(B\) such that the interiors of the sets \((B_\lambda)_{\lambda\in \Lambda}\) cover \(B\), and let \(S\) be a subset of \(B\). Suppose \(S\) meets each path-component of each one-fold, two-fold, and each three-fold intersection of distinct members of the family \((B_\lambda)_{\lambda\in \Lambda}\). Then there is a coequalizer diagram in the category of groupoids.
In Section 2, they prove a more general theorem, and then, after various additional remarks in Section 3, they briefly discuss a possibility of deducing it from Theorem 1.1 in Section 4; for instance such deduction is obviously possible when all points of \(B\) are taken as base points.

MSC:

18F60 Categories of topological spaces and continuous mappings
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
55Q05 Homotopy groups, general; sets of homotopy classes
55R99 Fiber spaces and bundles in algebraic topology

Citations:

Zbl 0521.57002
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References:

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[14] 64RONALD BROWN, GEORGE JANELIDZE, AND GEORGE PESCHKE
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[16] Ronald Brown: School of Computer Science, Bangor University, Bangor, United Kingdom
[17] George Janelidze: Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7700, South Africa · Zbl 1375.18008
[18] George Peschke: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada · Zbl 0852.55016
[19] Email:ronnie.profbrown@btinternet.com george.janelidze@uct.ac.za george.peschke@ualberta.ca
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