Self-closeness numbers of non-simply-connected spaces

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

Full Text (PDF format)

Received 20 January 2022

Received revised 26 May 2022

Accepted 1 June 2022

Published 27 September 2023