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.
Categories: math.AT
