Quotient manifolds of flows

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.