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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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