The fifteenth cohomology group of moduli spaces of stable curves

Abstract

The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let $S_{\omega }^{\ast }({\overline{\mathcal{M}}}_{g,n})\subset H^{\ast }({\overline{\mathcal{M}}}_{g,n})$ denote the semi-tautological extension generated by $H^{11}({\overline{\mathcal{M}}}_{1,11})$, that is, the smallest system of subrings of $H^{\ast }({\overline{\mathcal{M}}}_{g,n})$ containing $H^{11}({\overline{\mathcal{M}}}_{1,11})$ and the tautological subrings, which is also closed under pullbacks along forgetful maps and pushforwards along gluing maps. Recently, it has been shown that $H^{11}({\overline{\mathcal{M}}}_{g,n})=S_{\omega }^{11}({\overline{\mathcal{M}}}_{g,n})$ and $H^{13}({\overline{\mathcal{M}}}_{g,n})=S_{\omega }^{13}({\overline{\mathcal{M}}}_{g,n})$ for all $g$ and $n$. Meanwhile, $H^{15}({\overline{\mathcal{M}}}_{1,n})\ne S_{\omega }^{15}({\overline{\mathcal{M}}}_{1,n})$ for all $n\ge 15$. Nevertheless, we prove that $H^{15}({\overline{\mathcal{M}}}_{2,n})=S_{\omega }^{15}({\overline{\mathcal{M}}}_{2,n})$ for all $n$. Moreover, we determine all relations in $S_{\omega }^{15}({\overline{\mathcal{M}}}_{g,n})$ for all $g\ge 2$ and give explicit bases for this vector space in terms of decorated graphs. Conditional upon the conjectured dimension of $H^{15}({\overline{\mathcal{M}}}_{3,14})$, we also prove that $H^{15}({\overline{\mathcal{M}}}_{3,n})=S_{\omega }^{15}({\overline{\mathcal{M}}}_{3,n})$ for all $n$. As an application, we prove $H^{\ast }({\overline{\mathcal{M}}}_{g,n})=R{H}^{\ast }({\overline{\mathcal{M}}}_{g,n})$ for several new small pairs $(g,n)$.

arXiv comment37 pages, 1 figure, 4 tables