Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917258201137152 |
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| author | Cha, Jae Choon Lim, Geunho |
| author_facet | Cha, Jae Choon Lim, Geunho |
| contents | We obtain a solution to a bordism version of Gromov's linearity problem over a large family of acyclic groups, for manifolds with arbitrary dimension. Every group embeds into some acyclic group in this family. Thus, the linear bordism problem has an affirmative solution over a possibly enlarged acyclic group. Our result holds in both PL and smooth categories, and for both oriented and unoriented cases. In the PL case, our results hold without assuming bounded local geometry. As an application, we prove that there is a universal linear bound for the Cheeger-Gromov $L^2$ $ρ$-invariants of PL $(4k-1)$-manifolds associated with arbitrary regular covers. We also show that the minimum number of simplices in a PL triangulation of $(4k-1)$-manifolds with a fixed simple homotopy type is unbounded if the fundamental group has nontrivial torsion. The proof of our main results builds on quantitative algebraic and geometric techniques over the simplicial classifying spaces of groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11885 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants Cha, Jae Choon Lim, Geunho Geometric Topology Algebraic Topology Metric Geometry 53C23, 57Q20, 55U10 We obtain a solution to a bordism version of Gromov's linearity problem over a large family of acyclic groups, for manifolds with arbitrary dimension. Every group embeds into some acyclic group in this family. Thus, the linear bordism problem has an affirmative solution over a possibly enlarged acyclic group. Our result holds in both PL and smooth categories, and for both oriented and unoriented cases. In the PL case, our results hold without assuming bounded local geometry. As an application, we prove that there is a universal linear bound for the Cheeger-Gromov $L^2$ $ρ$-invariants of PL $(4k-1)$-manifolds associated with arbitrary regular covers. We also show that the minimum number of simplices in a PL triangulation of $(4k-1)$-manifolds with a fixed simple homotopy type is unbounded if the fundamental group has nontrivial torsion. The proof of our main results builds on quantitative algebraic and geometric techniques over the simplicial classifying spaces of groups. |
| title | Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants |
| topic | Geometric Topology Algebraic Topology Metric Geometry 53C23, 57Q20, 55U10 |
| url | https://arxiv.org/abs/2404.11885 |