Weight two compactly supported cohomology of moduli spaces of curves

TL;DR

The paper proves that the weight $2$ piece of the compactly supported cohomology of the moduli spaces ${\mathcal M}_{g,n}$ is computable as the cohomology of a graph complex $X_{g,n}$, closely related to the Getzler–Kapranov graph framework. It constructs a zig-zag of quasi-isomorphisms linking the weight-$2$ complex to the GK complexes and to a resolution $\widetilde{\mathsf{GK}}^2_{g,n}$, then to $X_{g,n}$, allowing explicit computations and structural understanding of $\operatorname{gr}_2 H_c^\bullet({\mathcal M}_{g,n})$. For $n=0$, it expresses $\operatorname{gr}_2 H_c^k({\mathcal M}_g)$ in terms of weight $0$ data from ${\mathcal M}_{g',1}$ and ${\mathcal M}_{g',2}$, yielding new infinite families of nonvanishing unstable cohomology, including the first odd-degree examples, with dimensions growing at least exponentially in $g$ for several indices. The work also develops a filtration by the number of vertices, introduces a quasi-isomorphic subcomplex $X^{\star}_{g,n}$, and provides explicit presentations and resolutions of $H^2(\overline{\mathcal M}_{0,n})$, enabling systematic generation of weight-$2$ classes and injections from known weight-$0$ data. Overall, the paper advances a concrete, graph-theoretic framework for discovering and controlling unstable cohomology in the moduli of curves, with connections to embedding-calculus graph complexes.

Abstract

We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves $M_{g,n}$ and show that this can be computed as the cohomology of a graph complex that is closely related to graph complexes arising in the study of embedding spaces. For $n = 0$, we express this cohomology in terms of the weight zero compactly supported cohomology of $M_{g',n'}$ for $g' \leq g$ and $n' \leq 2$, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on $M_g$, including the first such families in odd degrees. In particular, we show that the dimension of $H^{4g-k}(M_g)$ grows at least exponentially with $g$, for $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Definition 2.2