Certain maps preserving self-homotopy equivalences

TL;DR

This paper investigates maps between spaces that preserve the structure of self-homotopy equivalences, introducing new classes of such maps and an equivalence relation based on their induced effects on homotopy groups, with examples from spheres and Lie groups.

Contribution

It defines and studies $ extit{E}$-maps and co-$ extit{E}$-maps, and introduces an $ extit{E}$-equivalence relation for rationalized spaces, expanding understanding of homotopy self-equivalence preservation.

Findings

Examples involving spheres, Lie groups, and homogeneous spaces using Sullivan models.

Introduction of an $ extit{E}$-equivalence relation for rationalized spaces.

Characterization of maps that induce homomorphisms between self-homotopy equivalence groups.

Abstract

Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy equivalences for a connected CW complex $X$. We observe two classes of maps $\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps $X\to Y$ that induce the homomorphisms $\mathcal{E}(X)\to \mathcal{E}( Y)$ and $\mathcal{E}(Y)\to \mathcal{E}(X)$, respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an $\mathcal{E}$-equivalence relation between rationalized spaces $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$ as a geometric realization of an isomorphism $\mathcal{E}(X_{\mathbb{Q}})\cong \mathcal{E}(Y_{\mathbb{Q}})$.