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Generalised Swan modules and the D(2) problem. (English) Zbl 1097.57005
For cyclic groups \(G\) it has been known for some time that algebraic two-complexes over \(G\) can be realized up to chain homotopy type by two-dimensional CW-complexes with fundamental group \(G\). The present paper establishes this result in the case \(G=\mathbb Z/n\times \mathbb Z\). The question is strongly related to Wall’s D(2) property, which is also proven for \(\mathbb Z\times \mathbb Z\). That not all groups may behave that “nicely” is indicated by reference to works of F. E. A. Johnson as well as of F. R. Beyl and N. Waller [Algebr. Geom. Topol. 5, 899–910 (2005; Zbl 1084.57003)].

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
20J05 Homological methods in group theory
55P10 Homotopy equivalences in algebraic topology
55Q05 Homotopy groups, general; sets of homotopy classes
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References:
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