Novikov cohomology, finite domination, and cohomological dimension
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912652836470784 |
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| author | Fisher, Sam P. |
| author_facet | Fisher, Sam P. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14796 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Novikov cohomology, finite domination, and cohomological dimension Fisher, Sam P. Group Theory 20F65, 20J05 (Primary) 16S34, 20C07 (Secondary) 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. |
| title | Novikov cohomology, finite domination, and cohomological dimension |
| topic | Group Theory 20F65, 20J05 (Primary) 16S34, 20C07 (Secondary) |
| url | https://arxiv.org/abs/2510.14796 |