Certain maps preserving self-homotopy equivalences

Abstract

Let \(\mathcal {E}(X)\) be the group of homotopy classes of self homotopy equivalences for a connected CW complex X. We consider two classes of maps, \(\mathcal {E}\)-maps and co-\(\mathcal {E}\)-maps. They are defined as the maps \(X\rightarrow Y\) that induce homomorphisms \(\mathcal {E}(X)\rightarrow \mathcal {E}( Y)\) and \(\mathcal {E}(Y)\rightarrow \mathcal {E}(X)\), 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 \(\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})\).