# Certain maps preserving self-homotopy equivalences

Research paper by **Jin-ho Lee, Toshihiro Yamaguchi**

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

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

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