Hausdorff leaf spaces for codim-1 foliations

TL;DR

This paper investigates the topology of Hausdorff leaf spaces for codim-1 foliations, showing they are isometric to finite metric graphs and establishing conditions for convergence of warped foliations.

Contribution

It characterizes the Hausdorff leaf space for all codim-1 foliations on compact manifolds as finite metric graphs and constructs manifolds for any such graph.

Findings

HLS for codim-1 foliations is isometric to finite connected metric graphs

Constructs manifolds with prescribed HLS as a given metric graph

Provides necessary and sufficient conditions for warped foliations to converge to HLS

Abstract

The topology of the Hausdorff leaf spaces (HLS) for a codim-1 foliation is the main topic of this paper. At the beginning, the connection between the Hausdorff leaf space and a warped foliations is examined. Next, the author describes the HLS for all basic constructions of foliations such as transversal and tangential gluing, spinning, turbulization, and suspension. Finally, it is shown that the HLS for any codim-1 foliation on a compact Riemannian manifold is isometric to a finite connected metric graph. In addition, the author proves that for any finite connected metric graph G there exists a compact foliated Riemannian manifold (M,F,g) with codim-1 foliation such that the Hausdorff leaf space for F is isometric to G. Finally, the necessary and sufficient condition for warped foliations of codim-1 to converge to HLS(F) is given.