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Stability phenomena in Deligne-Mumford compactifications via Morse theory

Changjie Chen

TL;DR

This work analyzes the rational homology of the Deligne–Mumford compactification $\overline{\mathcal M}_{g,n}$ through the Morse theory of the sys_T functions, linking short geodesic data to boundary strata. It shows that in low degrees, homology is supported on the boundary in the stable range, and establishes finite generation and stability of $H_k$ across all genera and marked points by attaching stable-graph strata of dimension 0, providing explicit generators from a finite set of critical points. The approach unifies combinatorial, geometric, and Morse-theoretic techniques, recovering Harer-style stability and giving a concrete geometric construction of homology generators via boundary attachings. The results illuminate how the low-degree and stable homology of $\overline{\mathcal M}_{g,n}$ is controlled by boundary combinatorics and Morse-theoretic data, with potential implications for mapping-class-group stability and moduli-space topology.

Abstract

We study the rational homology of the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$ of the moduli space of stable curves via a family of Morse functions, the $\text{sys}_T$ functions, which encode geometric information about short geodesics on hyperbolic surfaces. Exploiting the Morse-theoretic properties of $\text{sys}_T$, including the existence of an index gap and the behavior of critical points near boundary strata, we prove that in low degrees the homology of $\overline{\mathcal M}_{g,n}$ is supported entirely on the boundary $\partial \mathcal M_{g,n}$. Furthermore, we establish finite generation and stability phenomena for the rational homology across all genera and numbers of marked points. Using stable graphs and explicit attaching maps, we show that for each fixed degree $k$, a finite set of critical points generates all $k$-th homology classes via attaching stable graphs of stratum dimension $0$. This result recovers previously known stability in the genus direction, such as Tosteson's theorem, and provides a concrete, geometric construction of homology generators. Our approach unifies combinatorial, geometric, and Morse-theoretic techniques to give a comprehensive picture of the low-degree and stable homology of $\overline{\mathcal M}_{g,n}$.

Stability phenomena in Deligne-Mumford compactifications via Morse theory

TL;DR

This work analyzes the rational homology of the Deligne–Mumford compactification through the Morse theory of the sys_T functions, linking short geodesic data to boundary strata. It shows that in low degrees, homology is supported on the boundary in the stable range, and establishes finite generation and stability of across all genera and marked points by attaching stable-graph strata of dimension 0, providing explicit generators from a finite set of critical points. The approach unifies combinatorial, geometric, and Morse-theoretic techniques, recovering Harer-style stability and giving a concrete geometric construction of homology generators via boundary attachings. The results illuminate how the low-degree and stable homology of is controlled by boundary combinatorics and Morse-theoretic data, with potential implications for mapping-class-group stability and moduli-space topology.

Abstract

We study the rational homology of the Deligne-Mumford compactification of the moduli space of stable curves via a family of Morse functions, the functions, which encode geometric information about short geodesics on hyperbolic surfaces. Exploiting the Morse-theoretic properties of , including the existence of an index gap and the behavior of critical points near boundary strata, we prove that in low degrees the homology of is supported entirely on the boundary . Furthermore, we establish finite generation and stability phenomena for the rational homology across all genera and numbers of marked points. Using stable graphs and explicit attaching maps, we show that for each fixed degree , a finite set of critical points generates all -th homology classes via attaching stable graphs of stratum dimension . This result recovers previously known stability in the genus direction, such as Tosteson's theorem, and provides a concrete, geometric construction of homology generators. Our approach unifies combinatorial, geometric, and Morse-theoretic techniques to give a comprehensive picture of the low-degree and stable homology of .

Paper Structure

This paper contains 9 sections, 15 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Stable graphs and the $\mathop{\mathrm{sys_T}}\nolimits$ functions
  3. Stable graph and Deligne-Mumford compactification
  4. The Morse functions
  5. Subtleties in using the Weil--Petersson metric for Morse theory
  6. Weil--Petersson metric near the boundary
  7. Gradient systems
  8. Low degree homology
  9. Stability Phenomena

Key Result

Theorem 1.1

For any $k>0$, when $g+n$ is sufficiently large, in particular when $\mathop{\mathrm{ind}}\nolimits(g,n)>k+1$, the inclusion induces an isomorphism

Theorems & Definitions (29)

  • Theorem 1.1: Low-degree homology is supported on the boundary, Theorem \ref{['isomorphism of low degree homology']}
  • Theorem 1.2: Finite generation of homology over attaching, Theorem \ref{['finite generation']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: chen2023morse
  • Definition 2.5
  • Theorem 2.6: chen2023index
  • Definition 3.1: chen2023morse
  • Definition 3.2
  • ...and 19 more