Homotopy decompositions of polyhedra into products with the second factor S1

As the main results of this paper we prove that for every polyhedron P with abelian or torsion-free nilpotent fundamental group there are only finitely many different homotopy types of Xi such that Xi×S1≃P. The same holds for any finite K(G,1) with nilpotent fundamental group in place of S1. The problem, if there exists a polyhedron with infinitely many direct factors of different homotopy types (K. Borsuk, 1970) [2] is still unsolved, even if we assume the second factor to be S1.