Abelian coverings (Q1816715)

Let \(\overline K\) be an Abelian covering of a finite simplicial complex \(K\) with \(\mathbb{Z}^q\) as group of deck transformations. Let \(F\) be the quotient field of \(\mathbb{R}[\mathbb{Z}^q]\). Denote by \(C_*(\overline K)\) the simplicial \(\mathbb{R}[\mathbb{Z}^q]\)-chain complex of \(\overline K\). Then we obtain an \(F\)-cochain complex \(\hom_{\mathbb{R}[\mathbb{Z}^q]}(C_*(\overline K),F)\). It is shown that its \(p\)th Betti number (taken over the field \(F\)) agrees with the combinatorial \(L^2\)-Betti number of \(\overline X\), i.e. the von Neumann dimension of the reduced \(L^2\)-cohomology of the cochain complex of Hilbert modules \(\hom_{\mathbb{R}[\mathbb{Z}^q]}(C_*(\overline K,l^2(\mathbb{Z}^q))\).