Quantitative aspects of acyclicity

TL;DR

This paper investigates the Cheeger constant of complexes, providing bounds, discussing expansion properties, and applying probabilistic and non-Abelian methods to understand their acyclicity and expansion characteristics.

Contribution

It introduces new methods for bounding the Cheeger constant and cosystolic norm, and explores expansion in pseudomanifolds, geometric lattices, and random complexes.

Findings

Bounds on cosystolic norm and Cheeger constant

Expansion properties of pseudomanifolds and geometric lattices

Probabilistic upper bounds and non-Abelian expansion applications

Abstract

We study several aspects of the $k$-th Cheeger constant of a complex X, a parameter that quantifies the distance of $X$ from a complex $Y$ with nontrivial $k$-th cohomology over $\mathbb{Z}_2$. Our results include general methods for bounding the cosystolic norm of a cochain and for bounding the Cheeger constant of a complex, a discussion of expansion of pseudomanifolds and geometric lattices, probabilistic upper bounds on Cheeger constants, and application of non-Abelian expansion to random complexes.