(Co)homology self-closeness numbers of simply-connected spaces

The (co)homology self-closeness number of a simply-connected based CW-complex $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions $\leqslant k$ is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology self-closeness numbers of spaces in certain cofibrations, define the $\operatorname{mod} p$ (co)homology self-closeness number of simply-connected $p$-local spaces with finitely generated homologies and study some properties of the $(\operatorname{mod} p)$ (co)homology self-closeness numbers.

Keywords

self-homotopy equivalence, self-closeness number, cofibration

2010 Mathematics Subject Classification

55P05, 55P10

Full Text (PDF format)

Received 9 November 2019

Received revised 1 February 2020

Accepted 3 February 2020

Published 5 August 2020