Homotopy decompositions of polyhedra into products with the second factor

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 X i such that X i × S 1 ≃ P . The same holds for any finite K ( G , 1 ) with nilpotent fundamental group in place of S 1 . 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 S 1 .