Homotopy decompositions of polyhedra into products with the second factor \(S^1\)
type: Item
instance of: scholarly article
title: Homotopy decompositions of polyhedra into products with the second factor \(S^1\)
zbMATH Open document ID: 1202.55002
author: Danuta Kołodziejczyk
published in: Topology and its Applications
publication date: 2010-10-29T00:00:00Z
review text: The homotopy type of \(X\) and \(Y\) is called a \textsl{factor of the homotopy type of} \(X\times Y\). The general problem is: Does there exist a polyhedron with infinitely different Homotopy factors? The author gives the following partial results: {\parindent6mm \begin{itemize}\item[(1)] Let \(X_1 \times S^1 \simeq X_2 \times S^1 \simeq P,\;\;P\) a finite polyhedron, \(\pi_1(X_i)\) finite, then \(X_1 \simeq X_2\). \item[(2)] Suppose \(\pi_1(P)\) of a finite polyhedron is abelian, then there are only finitely many different homotopy types of \(X\) such that \(X \times S^1 \simeq P\). \item[(3)] The same holds whenever the fundamental group of \(P\) is nilpotent and torsion-free. \item[(4)] The same assertion as in (3) whenever \(S^1\) is replaced by an Eilenberg-MacLane complex \(K(G,1)\) with arbitrary group \(G\). \end{itemize}} Furthermore the author 1) formulates more open problems of the same kind, concerning spaces as well as groups; and 2) presents counterexamples asserting e.g., that in (3) there can appear more than one factor of different homotopy type.
reviewed by: Friedrich Wilhelm Bauer
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 54C56
zbMATH DE Number: 5808691
zbMATH Keywords: polyhedron
zbMATH Keywords: CW-complex
zbMATH Keywords: ANR
zbMATH Keywords: FANR
zbMATH Keywords: homotopy type
zbMATH Keywords: shape
zbMATH Keywords: direct factor
zbMATH Keywords: nilpotent group
MaRDI profile type: Publication
full work available at URL: j.topol.2010.07.019
OpenAlex ID: W2031203217
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DOI: 10.1016/J.TOPOL.2010.07.019
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description: scientific article; zbMATH DE number 5808691