Self-closeness numbers of product spaces

TL;DR

This paper investigates the self-closeness numbers of product spaces, establishing conditions under which these numbers equal the maximum of the factors' numbers, with applications to specific spaces like Moore and Eilenberg-MacLane spaces.

Contribution

It introduces the concept of reducibility for product spaces and provides criteria to determine when the self-closeness number of a product equals the maximum of its factors.

Findings

Self-closeness number of product spaces can be determined by maximum of factors under reducibility.

Criteria for reducibility are established and applied to special spaces.

Results facilitate computation of self-closeness numbers for complex spaces.

Abstract

The self-closeness number of a CW-complex is a homotopy invariant defined by the minimal number $n$ such that every self-maps 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 equals to the maximum of 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.