Finite domination and Novikov rings, by Andrew A. Ranicki

The main result of the paper is that for any ring $A$ a finite f.g. free $A[z,z^{-1}]$-module chain complex $C$ is $A$-module chain equivalent to a finite f.g. projective $A$-module chain complex if and only if $H_*(A((z))\otimes_{A[z]}C)=H_*(A((z^{-1}))\otimes_{A[z]}C)=0$ with $A((z)),A((z^{-1}))$ the Novikov rings of formal power series. The result has the following application to topology: the infinite cyclic cover $\overline{X}=\widetilde{X}/\pi$ of a finite $CW$ complex $X$ with universal cover $\widetilde{X}$ and fundamental group $\pi_1(X)=\pi \times Z$ is finitely dominated if and only if $H_*(X;Z[\pi]((z)))=H_*(X;Z[\pi]((z^{-1})))=0$.

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