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 $S^*_ω(\overline{\mathcal M}_{g,n}) \subset H^*(\overline{\mathcal M}_{g,n})$ denote the semi-tautological extension generated by $H^{11}(\overline{\mathcal M}_{1,11})$, that is, the smallest system of subrings of $H^*(\overline{\mathcal M}_{g,n})$ 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 $H^{11}(\overline{\mathcal M}_{g,n}) = S^{11}_ω(\overline{\mathcal M}_{g,n})$ and $H^{13}(\overline{\mathcal M}_{g,n}) = S^{13}_ω(\overline{\mathcal M}_{g,n})$ for all $g$ and $n$. Meanwhile, $H^{15}(\overline{\mathcal M}_{1,n}) \neq S_ω^{15}(\overline{\mathcal M}_{1,n})$ for all $n \geq 15$. Nevertheless, we prove that $H^{15}(\overline{\mathcal M}_{2,n}) = S^{15}_ω(\overline{\mathcal M}_{2,n})$ for all $n$. Moreover, we determine all relations in $S_ω^{15}(\overline{\mathcal M}_{g,n})$ for all $g \geq 2$ 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 $H^{15}(\overline{\mathcal M}_{3,n}) = S^{15}_ω(\overline{\mathcal M}_{3,n})$ for all $n$. As an application, we prove $H^*(\overline{\mathcal M}_{g,n}) = RH^*(\overline{\mathcal M}_{g,n})$ for several new small pairs $(g,n)$.