Novikov cohomology, finite domination, and cohomological dimension

Abstract

We introduce the $\Sigma^*$-invariant of a group of finite type, which is defined to be the subset of non-zero characters $\chi \in \mathrm H^1(G;\mathbb R)$ with vanishing associated top-dimensional Novikov cohomology. We prove an analogue of Sikorav's Theorem for this invariant, namely that $\mathrm{cd}(\ker \chi) = \mathrm{cd}(G) - 1$ if and only if $\pm \chi \in \Sigma^*(G)$ for integral characters $\chi$. This implies that cohomological dimension drop is an open property among integral characters. We also study the cohomological dimension of arbitrary co-Abelian subgroups. The techniques yield a short new proof of Ranicki's criterion for finite domination of infinite cyclic covers, and in a different direction, we prove that the algebra of affiliated operators $\mathcal U(G)$ of a RFRS group $G$ has weak dimension at most one if and only if $G$ is an iterated (cyclic or finite) extension of a free group.