Classifications of 2-complexes whose finite fundamental group is that of a 3-manifold

Abstract

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.

Stable modules and Wall's D(2)-problem

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“…The question of which finite groups are 2-tame is discussed in [16], [17]. By a change of emphasis, the result of Beyl, Latiolais and Waller [1], classifying geometrical homotopy types over fundamental groups of certain 3-manifolds, can also be interpreted in this light. In particular, cyclic groups, dihedral groups of order 4n + 2, the binary Euclidean groups T * , O * , I * , the quaternion groups Q (8), Q (16) and Q(8n + 4) (n ≥ 1) are all known to be 2-tame, whereas over the quaternion groups Q(2 n ) (n ≥ 5), there exist algebraic 2-complexes for which no geometrical realization is as yet known.…”