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Quantitative bordism over
acyclic groups and Cheeger–Gromov $\rho$-invariants
Jae Choon Cha and Geunho Lim |
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Geometry & Topology 30 (2026) 1899–1929
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Abstract |
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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 -invariants of PL -manifolds associated with arbitrary regular covers. We also show that the minimum number of simplices in a PL triangulation of -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. |
Keywords
quantitative bordism, Cheeger-Gromov rho-invariant
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Mathematical Subject Classification
Primary: 53C23, 57Q20, 55U10
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References |
Publication
Received: 1 July 2024
Revised: 24 September 2025
Accepted: 5 November 2025
Published: 18 July 2026
Proposed: Urs Lang
Seconded: David Gabai, Mladen Bestvina
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Authors |
| Jae Choon Cha | |
| Department of Mathematics, Center
for Research in Topology Pohang University of Science and Technology Pohang South Korea |
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| Geunho Lim | |
| Institut de Mathématiques de
Jussieu-Paris Rive Gauche, CNRS Sorbonne Université, Université Paris Cité Paris France |
Kangnam University Yongin-si South Korea |
| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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