On the homotopy category of Moore spaces and the cohomology of the category of abelian groups

Abstract

The homotopy category of Moore spaces in degree 2 represents a nontrivial cohomology class in the cohomology of the category of abelian groups. We describe various properties of this class. We use James–Hopf invariants to obtain explicitly the image category under the functor chain complex of the loop space. An abelian group A determines the Moore space M(A) = M(A, 2) which up to homotopy equivalence is the unique simply connected CW-space X with homology groups H2X = A and HiX = 0 for i > 2. Since M(A) can be chosen to be a suspension, the set of homotopy classes [M(A),M(B)] is a group which is part of a classical central extension of groups (1) Ext(A,ΓB) 1⁄2 [M(A),M(B)] 3 Hom(A,B) due to Barratt. It is known that (1) in general is not split, for example [M(Z/2),M(Z/2)] = Z/4. We are not interested here in this additive structure of the sets [M(A),M(B)] but in the multiplicative structure given by the composition of maps, in particular in the extension of groups (2) Ext(A,ΓA) 1⁄2 E(M(A)) 3 Aut(A), where E(M(A)) is the group of homotopy equivalences of the space M(A). The extension (2) determines the cohomology class (3) {E(M(A))} ∈ H2(Aut(A),Ext(A,ΓA)). Though the group E(M(A)) is defined in an “easy” range of homotopy theory the cohomology class (3) is not yet computed for all abelian groups A. In this paper we prove a nice algebraic formula for the class (3) if A is a product of cyclic groups and we show that {E(M(A))} is trivial if 1991 Mathematics Subject Classification: 55E05, 55E25, 55J.