(Co)Homology Self-closeness Numbers of Simply-connected Spaces

TL;DR

This paper introduces and compares (co)homology self-closeness numbers for simply-connected spaces, exploring their properties, invariance, and relations within cofibrations and p-local spaces, advancing understanding of homotopy invariants.

Contribution

It defines the mod p (co)homology self-closeness number for p-local spaces and investigates their properties and relationships, extending the concept to new classes of spaces.

Findings

(Co)homology self-closeness numbers are homotopy invariants.

Comparison of these numbers in cofibrations reveals structural insights.

Properties of mod p (co)homology self-closeness numbers are established.

Abstract

The (co)homology self-closeness number of a simply-connected based CW-complexes $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions$\leq k$ 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 mod $p$ (co)homology self-closeness number of simply-connected $p$-local spaces with finitely generated homologies and study some properties of the (mod $p$) (co)homology self-closeness numbers.