Homotopy decompositions of polyhedra into products with the second factor \(S^1\)
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.
scientific article; zbMATH DE number 5808691
scientific article