Homology and homotopy complexity in negative curvature

TL;DR

This paper investigates the complexity of homology and homotopy in negatively curved manifolds, providing bounds on torsion homology in various dimensions and analyzing convergence properties in hyperbolic 3-manifolds.

Contribution

It extends Gromov's classical theorem by establishing bounds on torsion homology in all dimensions except 3, and explores the unbounded nature of torsion in 3D hyperbolic manifolds.

Findings

Linear bounds for torsion homology in dimensions ≠ 3

Unbounded torsion homology in 3D hyperbolic manifolds

Effective estimates for the number of manifolds up to homotopy or homeomorphism

Abstract

Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to $\mathbb{H}^3$ in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in $[0,\infty]$. In dimension $d\geq 4$ a somewhat precise estimate is given for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension $d\ge 5$ up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest.