  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 shows that the Thurston spine satisfies a property defined in terms of
the systole function analogous to that of Morse-Smale complexes of (smooth)
Morse functions on compact manifolds with boundary.
