Novikov cohomology, finite domination, and cohomological dimension
Sam P. Fisher
TL;DR
The paper introduces the Σ^*-invariant of finite-type groups as the set of nonzero characters χ for which the top-dimensional Novikov cohomology vanishes, establishing a Sikorav-type criterion that interiorizes cohomological-dimension drops: for integral χ, cd(ker χ) = cd(G) − 1 iff ±χ ∈ Σ^*(G). It develops a robust framework using twisted Laurent series and Novikov rings to prove Ranicki’s finite-domination criterion in cohomology and to generalize Sikorav’s theorem via long exact sequences rather than cycle-by-cycle analysis. It extends the theory to higher Σ^*_m invariants, proves openness results, and derives consequences for coabelian kernels, PD-groups, and RFRS groups, including a precise equivalence between weak dimension of affiliated operator algebras and iterated extensions of free groups. Finally, it demonstrates that vanishing top-dimensional Novikov cohomology does not in general force kernel-cd drops, by providing one-relator and BS(2,3) counterexamples that separate cohomology from homology data in this Novikov setting.
Abstract
We introduce the $Σ^*$-invariant of a group of finite type, which is defined to be the subset of non-zero characters $χ\in \mathrm H^1(G;\mathbb R)$ with vanishing associated top-dimensional Novikov cohomology. We prove an analogue of Sikorav's Theorem for this invariant, namely that $\mathrm{cd}(\ker χ) = \mathrm{cd}(G) - 1$ if and only if $\pm χ\in Σ^*(G)$ for integral characters $χ$. 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.