Orbit Spaces of Gradient Vector Fields

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.

💡 Research Summary

The paper investigates the orbit spaces arising from generalized gradient vector fields (f‑gradients) associated with Morse functions on smooth manifolds. Although such orbit spaces are often non‑Hausdorff, the authors demonstrate that they possess a surprisingly regular local topology and retain much of the homotopical information of the original manifold.

First, the authors recall the definition of an f‑gradient: a smooth vector field v on a manifold M satisfies (2.1) v(f)>0 off the critical set S(f) and (2.2) each critical point is a non‑degenerate minimum of v(f). By multiplying v with a suitable positive smooth function λ, one can always arrange that the resulting field λv is complete without changing the orbit relation.

The central result, Theorem 3.1, states that the orbit space M/∼ is locally contractible. The proof splits into two cases. For a non‑critical point p, the level set f⁻¹(f(p)) is a smooth (m‑1)‑dimensional submanifold transverse to the flow. Using the flow Φ_t, a local diffeomorphism H: ℝ×ℝ^{m‑1}→M is constructed, whose image U(p) is an open, q‑saturated neighbourhood of the orbit