Morse theory on moduli of curves

TL;DR

This work introduces a novel Morse-theoretic lens on the moduli space $\overline{\mathcal{M}}_{g,n}$ by constructing $sys_T$—a family of Morse functions defined as $sys_T(X)=-T\log\sum_{\\gamma\text{ s.c.g.}} e^{-\\frac{1}{T} l_{\\gamma}(X)}$—which converge to the systole as $T\to0^+$. The authors develop a robust Teichmüller-geometric framework, including the extension to the Deligne–Mumford boundary, a fan decomposition in the major subspace, and detailed analysis of critical points via eutacticity, yielding a complete Morse-theoretic picture on $\overline{\mathcal{M}}_{g,n}$. They prove the $sys_T$-Morse property, establish gradient-flow dynamics within strata, and show that the resulting Morse handles provide a natural cell decomposition for all $(g,n)$, with concrete applications to the cohomology of moduli spaces. The results give explicit, computable topological data (e.g., low-degree homology) and a path toward universal cell decompositions, advancing our geometric understanding of the global structure of moduli spaces. The framework also connects with known systole theory and builds a bridge between hyperbolic geometry and Morse theory on moduli spaces.}

Abstract

We provide a new approach to studying the moduli space of curves via Morse theory and hyperbolic geometry, by introducing a family of Morse functions on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked point, from the Teichmüller theoretic perspective. They are weighted exponential averages of the lengths of all simple closed geodesics. These Morse functions behave well with respect to the Deligne-Mumford stratification of $\overline{\mathcal{M}}_{g,n}$. The critical points can be characterized by a combinatorial property named eutacticity, and the Morse index can be computed accordingly. Also, the Weil-Petersson gradient flow of the Morse functions is well defined on $\overline{\mathcal{M}}_{g,n}$, which can be used to build the Morse theory. These functions might be the first explicit examples of Morse functions on $\overline{\mathcal{M}}_{g,n}$, and the Morse handle decomposition gives rise to the first example of a natural cell decomposition of $\overline{\mathcal{M}}_{g,n}$ in theory, that works for all pairs $(g,n)$.

Figures (13)

• Figure 1: Organization of the paper by types of points in $\overline{\mathcal{M}}_{g,n}$
• Figure 2: In this example, $\{v_1,v_2\}$ is biased, $\{v_3,v_4,v_5\}$ is eutactic, and $\{v_1,v_2,v_3,v_4,v_5\}$ is semi-eutactic
• Figure 3: Fan decomposition about $\{v_1,v_2,v_3,v_4\}\subset\mathbb R^2$
• Figure 4: The angle between the two vectors is at most $\frac{\pi}{2}-2\theta_0$
• Figure 5: Two positions of $\tau$: almost perpendicular to the major subspace or not

Theorems & Definitions (94)

• Definition
• Remark 1.1
• Remark 1.2
• Theorem 2.1: Wolpert wolpert1987geodesic
• Theorem 2.2: Huber huber1959analytischen
• Theorem 2.3: Wolpert wolpert2008behavior, wolpert2009weil
• Theorem 2.4: Wolpert wolpert2008behavior, wolpert2009weil
• Theorem 2.5: Wolpert wolpert2009extension
• Definition 3.1
• Definition 3.2: Topological Morse function, cf. morse1959topologically