From Morse Functions to Lefschetz Fibrations on Cotangent Bundles
Emmanuel Giroux
TL;DR
The paper constructs an explicit complex-valued symplectic Lefschetz fibration $h=f+ig: T^*M\to \mathbb{C}$ extending a Morse function $\varphi$ on a closed manifold $M$, with an adapted Morse–Smale gradient $\nu$ and imaginary part $g(p,q)=\langle p,\nu(q)\rangle$. It develops a coarse complexification $h^0$ and a rearrangement of critical values to produce a globally defined fibration whose regular fibers are Weinstein manifolds, and proves homotopy uniqueness and equivariance under an antipodal involution and complex conjugation. The work further analyzes the Lefschetz fiber $F_u=h^{-1}(u)$, describing its Weinstein structure, the vanishing cycles tied to the Morse data, and the role of the boundary double $SN^*(\nu)$ in understanding the fiber topology. This provides a concrete, computable bridge between Morse theory and symplectic Lefschetz geometry on cotangent bundles, with explicit descriptions of parallel transport, Lagrangian surgeries, and the surgery-incidence relations encoded by the vanishing cycles. The results yield a principled way to realize and study Weinstein structures arising from Morse data, enabling explicit topological and symplectic invariants of the Lefschetz fiber.
Abstract
We prove that, for any Morse function on a compact manifold and any adapted gradient satisfying the Morse-Smale condition, there is a homotopically unique complex-valued symplectic Lefschetz fibration on the cotangent bundle whose restriction to the zero-section is the given function, whose imaginary part is the evaluation of covectors on the gradient, and which is equivariant under the actions of the fiberwise antipodal involution and the complex conjugation. Then we study the topology and symplectic geometry of the regular fibers of this fibration, which are well-defined Weinstein manifolds.