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Rational self-homotopy equivalences and Whitehead exact sequence

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Mahmoud Benkhalifa

For a simply connected CW-complex $X$, let $\mathcal{E}(X)$ denotethe group of homotopy classes of self-homotopy equivalence of $X$and let $\mathcal{E}_{\sharp}(X)$ be its subgroup of homotopyclasses 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 aimof this work is to determine this subgroup for rational spaces. Weconstruct the Whitehead exact sequence associated with the minimalSullivan model of $X$ which allows us to define the subgroup$\mathrm{Coh.Aut}(\mathrm{Hom}\big(\pi_{*}(X),\Bbb Q)\big)$ ofself-coherent automorphisms of the gradedvector space $\mathrm{Hom}(\pi_{*}(X),\Bbb Q)$. As a consequence weestablish that$\frac{\mathcal{E}(X)}{\mathcal{E}_{\sharp}(X)}\cong\mathrm{Coh.Aut}\big(\mathrm{Hom}(\pi_{*}(X),\BbbQ)\big)$. In addition, by computing the group$\mathrm{Coh.Aut}\big(\mathrm{Hom}(\pi_{*}(X),\Bbb Q)\big)$, we giveexamples of rational spaces that have few self-homotopyequivalences.

Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 111-121