Rational homotopy type of free and pointed mapping spaces between spheres

Abstract

Denote by $map(X, Y )$ (respec. $map ^{\ast}(X, Y )$) the space of free (respec. pointed) maps from $X$ to $Y$ . Whenever $X$ is a finite $CW$-complex and $Y$ is a nilpotent $CW$-complex of finite type over $Q$, then any path component of both map $(X, Y )$ and $map ^{\ast} (X, Y ) $ are nilpotent $CW$-complexes of finite type over $Q$ and in particular, it can be rationalized in the classical sense. From the Sullivan approach to rational homotopy theory , and based in the fundamental work of Haefliger, there is a standard procedure to obtain Sullivan models of the path components $map _{f}(X, Y )$ and $map ^{\ast}_{f} (X, Y ) $ of $map(X, Y ) $ and $map ^{\ast}(X, Y ) $ respectively, containing the map $f : X → Y $. In this note, we show the advantage of this procedure and use it repeatedly to explicitly describe the rational homotopy type of free and pointed mapping spaces between sphere.