On the topology of leaves of singular Riemannian foliations

TL;DR

This paper investigates the topological properties of leaves in closed singular Riemannian foliations, revealing conditions under which leaves are finitely covered by nilpotent spaces and describing their fundamental groups.

Contribution

It establishes new results relating the topology of leaves to the fundamental group of the ambient manifold in singular Riemannian foliations.

Findings

Leaves are finitely covered by nilpotent spaces if the manifold is simply connected.

The fundamental group of generic leaves is characterized.

Leaves have virtually nilpotent fundamental groups if the manifold's fundamental group is virtually nilpotent.

Abstract

In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation $(M,\fol)$. If $M$ is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If $M$ has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well.