Weak homotopy equivalence of the Stein-to-Weinstein map

Prove that the map W that assigns to each Stein domain structure (J, φ) on a fixed compact smooth manifold W with boundary the associated Weinstein domain W(J, φ) = (ωφ, Xφ, φ), where ωφ = −d(dφ ◦ J) and Xφ is the gradient of φ with respect to the metric gφ defined by gφ(·,·) = ωφ(·, J·), is a weak homotopy equivalence from the space Stein of Stein domain structures on W to the space Weinstein of Weinstein domain structures on W.

Background

The note introduces a new notion of gradient-like vector field characterized by dφ = g(X,·) for some positive (not necessarily symmetric) (2,0)-tensor g. Using this, a Weinstein structure on a manifold V is defined as a triple (ω, X, φ) where ω is symplectic, X is Liouville for ω, and (X, φ) satisfies the new gradient-like condition. This expands the class of admissible functions beyond generalized Morse while preserving a robust deformation theory.

Fixing a compact manifold W with boundary and restricting to functions whose maximum level set is ∂W, the spaces Stein and Weinstein denote, respectively, the spaces of Stein and Weinstein domain structures on W. The map W: Stein → Weinstein is defined by associating to a Stein structure (J, φ) the Weinstein structure (ωφ, Xφ, φ) described in Example 5(a).

With the new definition, the projection πW: Weinstein → Functions is shown to be a microfibration (Corollary 7), and Theorem 9 (restating [2, Theorem 1.1] in this framework) asserts that W induces an isomorphism on π0 and a surjection on π1. Conjecture 10, rephrasing [2, Conjecture 1.4], strengthens this to the claim that W is a weak homotopy equivalence, i.e., it induces isomorphisms on all homotopy groups.