Expansion and torsion homology of 3-manifolds

TL;DR

This paper constructs 3-manifolds that are good expanders in all dimensions and demonstrates that such expanders must have complex torsion homology, with applications to topological overlap problems.

Contribution

It introduces the first examples of 3-manifolds that are good expanders in all dimensions and links expansion properties to torsion homology complexity.

Findings

Constructed 3-manifolds that are good expanders in all dimensions.

Showed that expanders must have significant torsion homology.

Provided applications to topological overlap problems.

Abstract

A Riemannian manifold is a called a good rational expander in dimension $i$ if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume. We construct 3-manifolds which are good expanders in all dimensions. On the other hand, we show that expanders must be topologically complicated: they must have lots of torsion homology. We also give some applications to topological overlap problems, constructing examples of 3-manifolds with large width over $\mathbb R^2$.