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Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants

Jae Choon Cha, Geunho Lim

TL;DR

The paper develops a quantitative bordism theory over acyclic groups, proving a linear bound for bordisms to trivial ends when the target group is extended to a BDH acyclic group, and deduces a universal linear bound for Cheeger-Gromov L2 ρ-invariants of PL (4k−1)-manifolds under arbitrary regular covers. Central to the approach are two innovations: (i) a quantitative transversality framework for simplicial-cell complexes that yields inverse-image submanifolds and cobordisms with linearly controlled complexity, and (ii) a BDH-based chain-homotopy calculus that converts algebraic data into explicit bordisms with linear complexity. These results yield new complexity bounds, enable almost-linear smooth null-bordisms, and provide lower and upper bounds for ρ-invariants that connect topological complexity to analytic invariants. The work also shows that linear bordism bounds fail in dimension 1 and establishes unbounded complexity within fixed simple homotopy types in the presence of torsion, highlighting the nuanced role of dimension and group structure in quantitative bordism. Overall, the paper offers a robust topological toolkit for controlling bordism and ρ-invariants quantitatively across dimensions, with implications for manifold complexity and geometric topology.

Abstract

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.

Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants

TL;DR

The paper develops a quantitative bordism theory over acyclic groups, proving a linear bound for bordisms to trivial ends when the target group is extended to a BDH acyclic group, and deduces a universal linear bound for Cheeger-Gromov L2 ρ-invariants of PL (4k−1)-manifolds under arbitrary regular covers. Central to the approach are two innovations: (i) a quantitative transversality framework for simplicial-cell complexes that yields inverse-image submanifolds and cobordisms with linearly controlled complexity, and (ii) a BDH-based chain-homotopy calculus that converts algebraic data into explicit bordisms with linear complexity. These results yield new complexity bounds, enable almost-linear smooth null-bordisms, and provide lower and upper bounds for ρ-invariants that connect topological complexity to analytic invariants. The work also shows that linear bordism bounds fail in dimension 1 and establishes unbounded complexity within fixed simple homotopy types in the presence of torsion, highlighting the nuanced role of dimension and group structure in quantitative bordism. Overall, the paper offers a robust topological toolkit for controlling bordism and ρ-invariants quantitatively across dimensions, with implications for manifold complexity and geometric topology.

Abstract

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.
Paper Structure (16 sections, 18 theorems, 33 equations, 3 figures)

This paper contains 16 sections, 18 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Linear bordism over acyclic groups
  3. Linear bound for Cheeger-Gromov L2 rho-invariants
  4. Submanifolds and cobordism from simplicial-cellular transversality
  5. Simplicial-cell complexes and quantitative subdivision
  6. Submanifolds from simplicial-cellular transversality
  7. Cobordism from simplicial-cellular transversality
  8. Baumslag-Dyer-Heller acyclic groups and quantitative chain homotopy
  9. Quantitative PL bordism
  10. Proof of Theorem \ref{['theorem:main-quantitative-bordism']}
  11. Construction of bordism from a chain null-homotopy
  12. Quantitative smooth bordism
  13. Linear bounds for Cheeger-Gromov rho-invariants
  14. Applications to complexity estimates and stable complexity
  15. Unbounded complexity within a fixed simple homotopy type
  16. ...and 1 more sections

Key Result

Theorem 1

Let $M$ be a closed triangulated PL $n$-manifold over the acyclic group $\mathcal{A}(G)$, where $G$ is an arbitrary group. Then there exists a triangulated PL bordism $W$ over $\mathcal{A}(G)$ from $M$ to a trivial end such that $\Delta(W) \leq C(n) \cdot \Delta(M)$ where $C(n)$ is a constant depend

Figures (3)

  • Figure 1: An illustration of the assertion. Here $n=3$, $p=1$, $\sigma=[e_0,e_1]$, $A=[v_0,v_1]$, and $f$ is a simplicial map sending vertices as indicated. The inverse image $Y_\sigma \cap \operatorname{st}(A) = f^{-1}(\hat{\sigma}) \cap \operatorname{st}(A)$ is the blue-shaded 2-complex, and $E_A = Y_\sigma \cap A$ is a single point in this example. In $\operatorname{st}(A)$, $Y_\sigma \cap \operatorname{st}(A)$ is isotopic rel $E_A$ to the join $E_A\cdot \operatorname{lk}(A)$, which is the red-shaded 2-complex.
  • Figure 2: A top-dimensional subdivision of $\Delta^p$ for $p=2$.
  • Figure 3: The bordism $Z=V \mathbin{\mathop{\cup}\limits_{\partial_+V = \partial_-U}} U$

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 2
  • Remark 1.8
  • Theorem 3
  • Remark 1.9
  • ...and 38 more