The Morse-Smale Property of the Thurston Spine

Abstract

The Thurston spine consists of the subset of Teichm\"uller space at which the set of shortest curves, the systoles, cuts the surface into polygons. The systole function is a topological Morse function on Teichm\"uller 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.