HOMOTOPY SELF-EQUIVALENCE GROUPS OF UNIONS OF SPACES: INCLUDING MOORE-SPACES

Abstract

The group ϵ(M^m(A) v Mⁿ(π)) of homotopy self-equivalence classes of two Moore spaces is faithfully represented onto a (multiplicative) group of matrices for n≥m≥3. We consider, in this note, related representations of ϵ(M^m(Λ)vMⁿ(π)), for finitely generated Λ and π in the case where n≥4, and also where n=3 if ext(Λ, π)=0. The representation onto a matrix group, similar to that in the case above, is not, in general, valid. We show however that ϵ(M²(Λ)vMⁿ(π)) is represented onto ϵ(M²(Λ))× ϵ(Mⁿ(π) in this case, and that this representation determines an isomorphism with an iterated semi-direct product ϵ(M²(Λ)v Mⁿ(π)) ≅ {(Mⁿ(π), M²(Λ))⋊ ext(π Λ ⊗ π)} ⋊ (ϵ(M²(Λ)) × ϵ (Mⁿ(π)).