Quotient manifolds of flows

TL;DR

This paper explores which manifolds can be obtained as quotients of flows of vector fields, showing that all countably presented groups and various homology types can be realized, including exotic and unsmoothable manifolds.

Contribution

It demonstrates that any epimorphism of countably presented groups can be realized as a quotient of a flow, and it constructs manifolds with flexible homology and exotic properties as flow quotients.

Findings

Every countably presented group epimorphism is realized by a flow quotient.

Manifolds with all even (and some odd) intersection pairings are realizable as flow quotients.

Existence of topologically equivalent but smoothly inequivalent flows with specified quotient manifolds.

Abstract

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least 2 arise in this manner. Most Euclidean spaces of dimensions 5 and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For m at least 2r>2, there is a topological flow on (R^{2r+1}-(8 points))xR^m that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.