Self-Closeness Number of Non-Simply-Connected Spaces

TL;DR

This paper investigates the self-closeness number of non-simply-connected spaces, providing conditions under which it equals that of their universal covers, advancing understanding of homotopy equivalences in algebraic topology.

Contribution

It introduces new conditions relating the self-closeness numbers of non-simply-connected spaces to their universal covers, extending previous results in homotopy theory.

Findings

Established conditions for equality of self-closeness numbers between a space and its universal cover

Analyzed the self-closeness number for specific classes of non-simply-connected spaces

Enhanced understanding of homotopy equivalences in the context of covering spaces

Abstract

The self-closeness number $N\mathcal{E}(X)$ of a space $X$ is the least integer $k$ such that any self-map is a homotopy equivalence whenever it is an isomorphism in the $n$-th homotopy group for each $n\le k$. We discuss the self-closeness numbers of certain non-simply-connected $X$ in this paper. As a result, we give conditions for $X$ such that $N\mathcal{E}(X)=N\mathcal{E}(\widetilde{X})$, where $\widetilde{X}$ is the universal covering space of $X$.