The group of self-homotopy equivalences of A n 2-polyhedra

Abstract

Abstract Let X be a finite type A n 2 {A_{n}^{2}} -polyhedron, n ≥ 2 {n\geq 2} . In this paper, we study the quotient group ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) {\mathcal{E}(X)/\mathcal{E}_{*}(X)} , where ℰ ⁢ ( X ) {\mathcal{E}(X)} is the group of self-homotopy equivalences of X and ℰ * ⁢ ( X ) {\mathcal{E}_{*}(X)} the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. We show that not every group can be realised as ℰ ⁢ ( X ) {\mathcal{E}(X)} or ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) {\mathcal{E}(X)/\mathcal{E}_{*}(X)} for X an A n 2 {A_{n}^{2}} -polyhedron, n ≥ 3 {n\geq 3} , and specific results are obtained for n = 2 {n=2} .