Novikov cohomology, finite domination, and cohomological dimension

Abstract

We introduce the ${\Sigma }^{\ast }$-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 $\pm \chi \in {\Sigma }^{\ast }(G)$ 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.

26 pages