# Certain maps preserving self-homotopy equivalences

Research paper by Jin-ho Lee, Toshihiro Yamaguchi

Indexed on: 28 Oct '16Published on: 14 Oct '16Published in: Journal of Homotopy and Related Structures

#### Abstract

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 $$\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})$$ .AbstractLet $$\mathcal {E}(X)$$ 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 $$\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})$$ . $$\mathcal {E}(X)$$ $$\mathcal {E}(X)$$X $$\mathcal {E}$$ $$\mathcal {E}$$ $$\mathcal {E}$$ $$\mathcal {E}$$ $$X\rightarrow Y$$ $$X\rightarrow Y$$ $$\mathcal {E}(X)\rightarrow \mathcal {E}( Y)$$ $$\mathcal {E}(X)\rightarrow \mathcal {E}( Y)$$ $$\mathcal {E}(Y)\rightarrow \mathcal {E}(X)$$ $$\mathcal {E}(Y)\rightarrow \mathcal {E}(X)$$ $$\mathcal {E}$$ $$\mathcal {E}$$ $$X_{{\mathbb Q}}$$ $$X_{{\mathbb Q}}$$ $$Y_{{\mathbb Q}}$$ $$Y_{{\mathbb Q}}$$ $$\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})$$ $$\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})$$