Adams-Hilton model and the group of self-homotopy equivalences of a simply connected cw-complex

2019/09/08 by Mahmoud Benkhalifa, Benkhalifa, Mahmoud
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT

paper · pdf · doi:10.48550/arxiv.1909.03473

arxiv created 2019/09/08 · arxiv updated 2019/09/10

Abstract

Let R be a principal ideal domain (PID). For a simply connected CW-complex X of dimension n, let Y be a space obtained by attaching cells of dimension q to X, q>n, and let A(Y) denote an Adams-Hilton model of Y. If \mathcal E(A(Y)) denotes the group of homotopy self-equivalences of A(Y) and \mathcal E_*(A(Y)) its subgroup of the elements inducing the identity on H_*( Y,R), then we construct two short exact sequences: \underseti⊕ H_q(ΩX,R)\rightarrowtail E(A(Y))\overset \twoheadrightarrowΓ^q_n , \underseti⊕ H_q(ΩX,R) \rightarrowtail \E_*(A(Y))\overset \twoheadrightarrowΠ^q_n where i=rank H_q(Y,X;R), Γ^q_n is a subgroup of \aut(Hom(H_q( Y,X;R))× \E(A(X)) and Π^q_n is a subgroup of \mathcal E_*(A(X)).

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