Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

Self-closeness numbers of product spaces

Pages: 249 – 264

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a13


Pengcheng Li (Department of Mathematics, School of Sciences, Great Bay University, Dongguan, China)


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.


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

2010 Mathematics Subject Classification

55P10, 55Q05

Received 26 March 2022

Received revised 5 May 2022

Accepted 5 May 2022

Published 12 April 2023