Two functors from Abelian groups to perfect groups

Abstract

The past decade or so has seen considerable development of the idea, originating with Kan and Thurston [12], of modelling a given sequence of abelian groups by the homology of another group. At the heart of such procedures lies the notion of an acyclic group, one having the same homology as the trivial group. (In this note all homology is taken to have trivial integer coefficients.) At some stage one seeks to embed a given group in a convenient acyclic group. Particularly striking in this context is the result of Baumslag, Dyer and Heller that when the given group G is abelian there exists an acyclic group of which G is the centre [l Theorem 7.11. Unfortunately the proof gives hardly any idea of what such an acyclic group must look like. Indeed, the proof is arguably ‘back-to-front’: it combines the existence of an Eilenberg-MacLane space of type K(G, 2) with the main result of [12] in order to deduce the existence of the acyclic group. Here we stay more within the realm of group theory and give an explicit construction of an acyclic group with centre G. This construction involves the perfect locally nilpotent groups of McLain. Reversing the viewpoint of [l] gives a consequent construction of a K(G, 2) space, differing from the usual model. We also obtain an explicit perfect group having G as its Schur multiplier; again, the existence, but not direct construction, of such a group is implied by [2 Theorem H]. An attraction of the constructions given below is their functorial nature. To be specific, let Ab, AC and Perf refer respectively to the categories of abelian, acyclic and perfect groups and their group homomorphisms. (Of course, since the first homology of a group is its abelianisation, AC is a subcategory of Perf .) In Section 2 we define