Abstract
The D(2)-problem is to determine whether for a three-dimensional complex X, the vanishing of 3-dimensional cohomology, in all coe-cients, is enough to guarantee that X is homotopically two-dimensional. We show that for flnite complexes with flnite fundamental group, a positive solution to the D(2)-problem is obtained precisely when all stably free algebraic 2- complexes are geometrically realizable. The proof makes very strong use of techniques which apply to flnite fundamental groups but not more generally; in particular, Yoneda's Theorem that, for flnite groups, group cohomology is representable by stable modules of flnite type, and also the Swan{Jacobinski Cancellation Theorem for such stable modules.
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