The Morse-Smale Property of the Thurston Spine

TL;DR

This work shows that the Thurston spine $\mathcal{P}_{g}$ inside Teichmüller space $\mathcal{T}_{g}$ exhibits Morse–Smale–type behavior for the piecewise-smooth systole function $f_{\mathrm{sys}}$. By developing tangent-cone and cone-of-increase machinery, it proves that critical points lie in $\mathcal{P}_{g}$ and that $\mathcal{P}_{g}$ contains stable unstable-manifold-like cells, with locally top-dimensional strata being balanced. The results connect Thurston’s deformation flow to an equivariant handle decomposition and provide a robust local-to-global picture for the systole-based stratification, including explicit analysis around Schmutz’s genus-2 critical point. The framework hinges on minimal filling sets, loci $E(C,d)$, eutaxy concepts, and a Lipschitz-Lipschitz path structure that generalizes Morse theory to the Teichmüller setting. Overall, the paper clarifies how the piecewise smooth geometry of $\mathcal{T}_{g}$ organizes into a Morse–Smale-like complex guided by the systole function.

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.

Figures (6)

• Figure 1: Thurston's Lipschitz map construction, where $m$ is shown in black and $A$ in blue in the upper leftmost figure.
• Figure 4: Inner versus outer loci. In the figure on the right, the locus should be regarded as coming out of the page. A filling set of curves that intersect pairwise at most once must have cardinality at least four. This figure represents a lower-dimensional analogue, where $\mathrm{Sys}(\{c_{i}\})^{\infty}$ denotes a point in the completion of $\mathcal{T}_{g}$ with respect to the Weil--Petersson metric at which the curve $c_{i}$ has been pinched.
• Figure 8: The level sets of the systole function passing through the critical point are shown in green. A level set at higher value of $f_{\mathrm{sys}}$ is shown in violet. The red arrows show the direction in which $f_{\mathrm{sys}}$ is increasing. The simplices $\sigma_{1}$ and $\sigma_{2}$ are shown in blue.
• Figure 9: The intersection of level sets near a boundary point $p$ of $f_{\mathrm{sys}}$. The vector $v_{\{c_{1}, c_{2}, c_{3}}$ shows a direction in which $f_{\mathrm{sys}}$ is increasing away from $p$ to second order.
• Figure 11: The set of 6 systoles at the critical point in the example are shown on the right. The left side of the figure shows a fundamental domain with edges lying along the systoles, and with numbers indicating the glung maps (not the labels of the curves). This figure is taken from Steinberg.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Corollary 1.4
• Definition 2.1: systole stratum
• Proposition 2.2: Thurston, Bers
• Definition 2.3: $\mathrm{Min}(C)$
• Definition 2.4: modified version of SchmutzVoronoi
• Lemma 2.5: consequence of SchmutzMorse
• Lemma 3.1