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# Homology, Homotopy and Applications

## Volume 25 (2023)

### Number 2

### Self-closeness numbers of non-simply-connected spaces

Pages: 21 – 36

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n2.a2

#### Author

#### Abstract

The self-closeness number of a connected CW complex is the least integer $n$ such that any of its self-maps inducing an isomorphism in $\pi_\ast$ for $\ast \leqslant n$ is a homotopy equivalence. We prove that under a mild condition, the self-closeness number of a non-simply-connected finite complex coincides with that of its universal cover whenever the universal cover is a finite $\mathrm{H}_0$-space or a finite co-$\mathrm{H}_0$-space. We give several interesting examples to which the result applies.

#### Keywords

self-closeness number, group of self-equivalences, $\mathrm{H}_0$-space, co-$\mathrm{H}_0$-space, $p$-universal space

#### 2010 Mathematics Subject Classification

55P10, 55S37

Received 20 January 2022

Received revised 26 May 2022

Accepted 1 June 2022

Published 27 September 2023