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.

Keywords

self-homotopy equivalence, self-closeness number, product space, reducibility

2010 Mathematics Subject Classification

55P10, 55Q05

Full Text (PDF format)

Received 26 March 2022

Received revised 5 May 2022

Accepted 5 May 2022

Published 12 April 2023