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 $\mathcal{E}_{\sharp}(X)$ be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group $\frac{\mathcal{E}(X)}{\mathcal{E}_{\sharp}(X)}$ can be identified with a subgroup of $Aut(\pi_{*}(X))$. 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 $\mathrm{Coh.Aut}(\mathrm{Hom}\big(\pi_{*}(X),\Bbb Q)\big)$ of self-coherent automorphisms of the graded vector space $\mathrm{Hom}(\pi_*(X),\Bbb Q)$. As a consequence we establish that $\mathcal{E}(X) / \mathcal{E}_{\sharp}(X) \cong \mathrm{Coh.Aut} (\mathrm{Hom}(\pi_*(X),\Bbb Q))$. In addition, by computing the group $\mathrm{Coh.Aut}\big(\mathrm{Hom}(\pi_{*}(X),\Bbb Q)\big)$, we give examples of rational spaces that have few self-homotopy equivalences.