The acyclicity of the complex of homologous curves

TL;DR

This paper proves that the complex of homologous curves $\mathcal{C}_{\vec{x}}(S_g)$ on a closed orientable surface $S_g$ of genus $g\ge2$ is $(g-3)$--acyclic for a primitive homology class $\vec{x}$. The authors build on Bestvina--Bux--Margalit’s contractible complex of minimizing cycles $\mathcal{B}_{\vec{x}}(S_g)$ and apply PL--Morse theory to the pair $(\mathcal{B}_{\vec{x}}(S_g), \mathcal{C}_{\vec{x}}(S_g))$ to show $\widetilde{H}_k(\mathcal{B},\mathcal{C};\mathbb{Z})=0$ for $k\le g-2$, implying the desired acyclicity since $\mathcal{B}_{\vec{x}}(S_g)$ is contractible. The paper also develops the complex of splitting curves via partitioned surfaces and analyzes its connectivity through bicellular covers and spectral sequences, with an auxiliary complex of draining cycles governing the descending links in the PL--Morse framework. Together, these tools enable a thorough analysis of the homology of the Torelli-related objects and contribute to understanding low-dimensional homology in the mapping class group setting. The results provide a robust framework for relating curve complexes, minimizing cycles, and draining cycles to derive high-connectivity and acyclicity properties relevant to Torelli group computations.

Abstract

We show that the complex of homologous curves of a closed, oriented surface of genus g is (g-3)--acyclic.

Figures (8)

• Figure 1: The curve $\delta \in \mathcal{C}_{\mathop{\mathrm{sep}}\nolimits}(\Sigma)$ but not $\mathcal{C}_{\mathop{\mathrm{split}}\nolimits}(\Sigma)$
• Figure 2: A splitting curve.
• Figure 3: A curve $\delta$ with $W(\delta) = 1$.
• Figure 4: A 2-cell in $\mathcal{C}_{\vec{x}}(S_g)$.
• Figure 5: Two vertices $M$ and $N$ in the complex of draining cycles.

Theorems & Definitions (36)

• Theorem A
• Lemma 1.1
• Lemma 2.1
• proof : Proof of Lemma \ref{['bicellularlemma']}
• Proposition 3.1
• Lemma 3.2: Harerstability
• Lemma 3.3
• Lemma 3.4
• proof : Proof of Lemma \ref{['splspecdeclink']}
• proof : Proof of Lemma \ref{['downconvlemma']}