Stable modules and Wall's D(2)-problem
Wall's D(2) problem asks whether a finite three dimensional cell complex such that \(H^{3}(X;{\mathcal B}) = 0\) for all local coefficient systems \(\mathcal B\) on \(X\) is homotopy equivalent to a finite \(2\)-dimensional complex. For finitely presented groups \(G\) one defines an algebraic \(2\)-complex over \(G\) to be an exact sequence \({\mathbb E}\) of modules and homomorphisms over \({\mathbb Z}[G]\) of the form: \[ {\mathbb E} = ( 0 \rightarrow J \rightarrow E_{2} \rightarrow E_{1} \rightarrow E_{0} \rightarrow {\mathbb Z} \rightarrow 0) \] where each \(E_{r}\) is finitely generated over \({\mathbb Z}[G]\). \textbf{Alg}\(_{G}\) denotes the category of such algebraic 2-complexes. The paper shows that for finite groups \(G\), the \(D(2)\) property holds if and only if each algebraic \(2\)-complex \(E \in \)\textbf{Alg}\(_{G}\) admits a geometric realization by a finite based \(2\)-complex. The proof relies on techniques that apply to fundamental groups, the Yoneda lemma, and the Swan-Jacobinski cancellation theorem.
scientific article; zbMATH DE number 1990084