Commentarii Mathematici Helvetici
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Published online: 2003-03-31
Stable modules and Wall's D(2)-problemF.E.A. Johnson (1) University College London, UK
The D(2)-problem is to determine whether for a three-dimensional complex X, the vanishing of 3-dimensional cohomology, in all coefficients, is enough to guarantee that X is homotopically two-dimensional. We show that for finite complexes with finite 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 finite fundamental groups but not more generally; in particular, Yoneda's Theorem that, for finite groups, group cohomology is representable by stable modules of finite type, and also the Swan-Jacobinski Cancellation Theorem for such stable modules.
Keywords: Cohomologically 2-dimensional 3-complex, algebraic 2-complex, stable module, k-invariant, geometrical realization
Johnson F.E.A.: Stable modules and Wall's D(2)-problem. Comment. Math. Helv. 78 (2003), 18-44. doi: 10.1007/s000140300001