Low Morse index for the systole function

Abstract: We show that for any $k\ge0$, all critical points for the systole function and the $\text{sys}T$ functions in $\mathcal{M}{g,n}$ have index at least $k$, for all but finitely many pairs $(g,n)$. This implies that the lowest Morse index can be bounded from below by a positive number that goes to infinity as the dimension of the moduli space increases. We conclude by Morse theory that all the rational (co)homology of the Deligne-Mumford compactification $\overline{\mathcal{M}}{g,n}$ of degree at most $k$ comes from the boundary $\partial\mathcal{M}{g,n}$ for all but finitely many pairs $(g,n)$. In particular, we show that there exists a universal constant $C>0$, such that the lowest Morse index is at least $C\log\log(-\chi+1)$. Developing the theorems, we also obtain an explicit upper bound on the cardinality of a $j$-system on a $(g,n)$-surface.