Orbit Spaces of Gradient Vector Fields

TL;DR

This paper investigates the topological structure of orbit spaces generated by generalized gradient vector fields of Morse functions, revealing their local contractibility and homotopy properties despite being non-Hausdorff.

Contribution

It establishes that these orbit spaces are locally contractible and that the quotient map is a weak homotopy equivalence with path lifting, advancing understanding of their topology.

Findings

Orbit spaces are locally contractible.

Quotient maps are weak homotopy equivalences.

Orbit spaces have a structured topological nature despite non-Hausdorffness.

Abstract

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting property.