Rational self-homotopy equivalences and Whitehead exact sequence

2009/05/09 by Mahmoud Benkhalifa, Benkhalifa, Mahmoud
Mathematics · #55P62 #55Q05 #55S35 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55P62 #msc:55Q05 #msc:55S35

paper · pdf · doi:10.48550/arxiv.0905.1396

arxiv created 2009/05/09 · openalex publication_date 2009/05/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a simply connected CW-complex X, let E(X) denote the group of homotopy classes of self-homotopy equivalence of X and let E_\sharp(X) be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group \fracE(X)E_\sharp(X) can be identified with a subgroup of Aut(π_*(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(Hom(π_*(X),\Bbb Q)) of self-coherent automorphisms of the graded vector space Hom(π_*(X),\Bbb Q). As a consequence we establish that E(X) / E_\sharp(X) ≅ Coh.Aut (Hom(π_*(X),\Bbb Q)). In addition, by computing the group Coh.Aut(Hom(π_*(X),\Bbb Q)), we give examples of rational spaces that have few self-homotopy equivalences.

Citations

• On the classification problem of the quasi-isomorphism classes of free chain algebras

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