SELF-CLOSENESS NUMBERS OF PRODUCT SPACES

The self-closeness number of a CW-complex is a homotopy invariant defined by the minimal number n such that every selfmap of X which induces automorphisms on the first n homotopy groups of X is a homotopy equivalence. In this article we study the self-closeness numbers of finite Cartesian products, and prove that under certain conditions (called reducibility), the self-closeness number of product spaces is equal to the maximum of the self-closeness numbers of the factors. A series of criteria for the reducibility are investigated, and the results are used to determine self-closeness numbers of product spaces of some special spaces, such as Moore spaces, Eilenberg-MacLane spaces or atomic spaces.