Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 027, 32 pages      arXiv:2401.05734      https://doi.org/10.3842/SIGMA.2025.027

The Morse-Smale Property of the Thurston Spine

Ingrid Irmer
International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, P.R. China

Received February 07, 2024, in final form April 10, 2025; Published online April 23, 2025

Abstract
The Thurston spine consists of the subset of Teichmüller space at which the set of shortest curves, the systoles, cuts the surface into polygons. The systole function is a topological Morse function on Teichmüller space. This paper studies the local properties of the Thurston spine, and the smooth pieces out of which it is constructed. Some of these local properties are shown to have global consequences, for example that the Thurston spine satisfies properties defined in terms of the systole function analogous to that of Morse-Smale complexes of (smooth) Morse functions on compact manifolds with boundary.

Key words: Thurson spine; moduli space; handle decomposition; systole; topological Morse function; unstable manifold; mapping class group.

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