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The paper constructs an "exotic" algebraic 2-complex over the generalized quaternion group of order 28, with the boundary maps given by explicit matrices over the group ring. This result depends on showing that a certain ideal of the group ring is stably free but not free. As it is not known whether the complex constructed here is geometrically realizable, this example is proposed as a suitable test object in the investigation of an open problem of C.T.C. Wall, now referred to as the D(2)-problem. AMS Classification 57M20; 55P15, 19A13Keywords Algebraic 2-complex, Wall's D(2)-problem, geometric realization of algebraic 2-complexes, homotopy classification of 2-complexes, generalized quaternion groups, partial projective resolution, stably free nonfree moduleThe main topic of this paper is the construction of an "exotic" algebraic 2-complex over Q 28 , the generalized quaternion group of order 28. The result provides significantly more detail than mere existence proofs, as the boundary maps are given by explicit matrices over the group ring (Section 4). This example should serve as a suitable test object in the investigation of an open problem of C.T.C. Wall [15, p.57], now [9] referred to as the D(2)-problem. D(2)-problem Suppose X is a finite three-dimensional connected CW-complex (with universal cover X ) such that H 3 ( X, Z) = 0 and H 3 (X, B) = 0 for all local coefficient systems B on X . Is X homotopy equivalent to a finite 2-complex?F.E.A. Johnson [7], [9] has shown that for finite base groups this question has an affirmative answer if, and only if, every algebraic 2-complex is geometrically realizable.The work of K.W. Gruenberg and P.A. Linnell [5, (2.6)] on minimal resolutions of lattices shows that stably equivalent lattices over ZQ 32 , despite stringent

Abstract.The Schur multiplicator Hfi of a (finite) metacyclic group G is computed with the help of the Lyndon spectral sequence. The order of Hfi is a useful invariant of G. A metacyclic group with vanishing Schur multiplicator can be presented with two generators and two relators. A simple description of the totality of these groups is given and all such /»-groups are classified.

We consider spines of spherical space forms; i.e., spines of closed oriented 3-manifolds whose universal cover is the 3-sphere. We give sufficient conditions for such spines to be homotopy or simple homotopy equivalent to 2-complexes with the same fundamental group G and minimal Euler characteristic 1. If the group ring ZG satisfies stably-free cancellation, then any such 2-complex is homotopy equivalent to a spine of a 3-manifold. If K,(ZG) is represented by units and K is homotopy equivalent to a spine X, then K and X are simple homotopy equivalent. We exhibit several infinite families of non-abelian groups G for which these conditions apply.