Rational self-homotopy equivalences and Whitehead exact sequence

Abstract

For a simply connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalence of $X$ and let ${\mathcal{E}}_{♯}(X)$ be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group $\frac{\mathcal{E}(X)}{{\mathcal{E}}_{♯}(X)}$ can be identified with a subgroup of $Aut({\pi }_{\ast }(X))$. The aim of this work is to determine this subgroup for rational spaces. We construct the Whitehead exact sequence associated with the minimal Sullivan model of $X$ which allows us to define the subgroup $Coh.Aut(\mathrm{Hom}({\pi }_{\ast }(X),\mathbb{Q}))$ of self-coherent automorphisms of the graded vector space $\mathrm{Hom}({\pi }_{\ast }(X),\mathbb{Q})$. As a consequence we establish that $\mathcal{E}(X)/{\mathcal{E}}_{♯}(X)\cong Coh.Aut(\mathrm{Hom}({\pi }_{\ast }(X),\mathbb{Q}))$. In addition, by computing the group $Coh.Aut(\mathrm{Hom}({\pi }_{\ast }(X),\mathbb{Q}))$, we give examples of rational spaces that have few self-homotopy equivalences.

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