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

### F.E.A. Johnson

University College London, UK

## Abstract

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.

## Cite this article

F.E.A. Johnson, Stable modules and Wall's D(2)-problem. Comment. Math. Helv. 78 (2003), no. 1, pp. 18–44

DOI 10.1007/S000140300001