Coarse homology of leaves

TL;DR

This paper explores the coarse homology of leaves in foliations of compact manifolds, demonstrating that many leaves have non-finitely generated coarse homology and constructing non-leaves with trivial coarse homology.

Contribution

It proves that leaves can have non-finitely generated coarse homology and improves existing non-leaf constructions to produce non-leaves with trivial coarse homology.

Findings

Many leaves have non-finitely generated coarse homology.

Existence of non-leaves with trivial coarse homology.

Enhanced construction methods for non-leaves.

Abstract

We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to be finitely generated. We show that this is not the case by proving that there exist many leaves with non-finitely generated coarse homology. Moreover, we improve Schweitzer's non-leaf construction and produce non-leaves with trivial coarse homology.