Coarse homological invariants of metric spaces

TL;DR

This paper introduces coarse topological invariants for metric spaces, establishing their properties, invariance under coarse equivalence, and characterizing specific spaces like quasi-trees, with implications for group theory and topology.

Contribution

It defines coarse cohomological invariants, proves their invariance under coarse equivalence, and characterizes unbounded quasi-trees and higher-dimensional analogues of classical theorems.

Findings

Coarse cohomological dimension coincides with group cohomological dimension for finitely generated groups.

Coarse cohomological dimension is monotone under coarse embeddings.

Unbounded quasi-trees have coarse cohomological dimension one.

Abstract

Inspired by group cohomology, we define several coarse topological invariants of metric spaces. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group, then the coarse cohomological dimension of G as a metric space coincides with the cohomological dimension of $G$ as a group whenever the latter is finite. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one. A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or $\infty$. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and $H^k(G,FG)=0$ for k<n, then $\dim H^n(G,FG)$=0,1 or $\infty$, significantly extending a result of Farrell from 1975. Moreover, in the case $\dim H^n(G,FG)=1$, then G must be a coarse Poincar\'e duality group. We prove an analogous result for metric spaces.