Cartesian powers of shapes of FANR's and polyhedra

Abstract

We show that a finite polyhedron P of nontrivial homotopy type cannot be homotopy equivalent to a proper Cartesian factor of itself. In other words, P≃P×K implies that K≃1. Consequently, if P is a finite polyhedron such that Pn≃P, for some integer n≥2, then the homotopy type of P is trivial. This answers positively the problem [1, Problem (34.10)] stated by K. Borsuk in 1971 (in an equivalent, shape-theoretical formulation) and a question from Borsuk's monograph “Theory of Shape” [2, Problem (7.13), p. 142].We also prove that if (X,x)∈FANR, then Shn(X,x)=Sh(X,x), for some integer n≥2, implies that Sh(X)=1. This resolves positively Problem (5.6) from [7] (1981). Similarly as for polyhedra, first we show that if X∈FANR and Sh(X)≠1, then X cannot have the shape of a proper Cartesian factor of itself.