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

Abstract

The group ?(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 ?(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 ?(M²(Λ)vMⁿ(π)) is represented onto ?(M²(Λ))× ?(Mⁿ(π) in this case, and that this representation determines an isomorphism with an iterated semi-direct product ?(M²(Λ)v Mⁿ(π)) ? {(Mⁿ(π), M²(Λ))? ext(π Λ ? π)} ? (?(M²(Λ)) × ? (Mⁿ(π)).

More generally we review, and-extend, the theory of the representation of the (generalized) near ring (XvY,XvY) onto the matrix (generalized) near-ring (XvY, XxY) where appropriate, in the case where X and Y are h-coloops; and we deduce results for the representation of ?(XvY, XvY). Some of the results published previously in the case of simply-connected CW co-h-spaces, extend to the case where X and Y are path-connected h-coloops one of which is well-pointed. We note the obstructions to the existence of a homomorphic section, and consider a number of special cases which occur when some of the groups are trivial.