Proof of the Verjovsky Conjecture

TL;DR

This paper proves the Verjovsky conjecture, establishing that all codimension-one Anosov flows on high-dimensional manifolds are topologically equivalent to suspensions of hyperbolic toral automorphisms, with implications for flow integrability.

Contribution

The paper provides a proof of the Verjovsky conjecture and extends it to show conditions under which stable and unstable bundles are jointly integrable for volume-preserving flows.

Findings

All codimension-one Anosov flows in dimension > 3 are topologically equivalent to suspensions of hyperbolic toral automorphisms.

A time change can ensure joint integrability of stable and unstable bundles in volume-preserving flows.

The conjecture is derived from a more general result on flow integrability.

Abstract

In this paper we present a proof of the Verjovsky conjecture: Every codimension-one Anosov flow on a manifold of dimension greater than three is topologically equivalent to the suspension of a hyperbolic toral automorphism. In fact, the conjecture is derived from possible more general result that says that for every codimension-one volume-preserving Anosov flow on a manifold of dimension greater than three, a suitable time change guarantees that the stable and unstable sub-bundles are then jointly integrable.