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