Index gap of the systole function

Abstract

It is known that the systole function is topologically Morse on the moduli space ${\mathcal{M}}_{g,n}$ and the ${\text{sys}}_T$ functions are $C^2$-Morse on the Deligne-Mumford compactification ${\overline{\mathcal{M}}}_{g,n}$. In this paper, We show that these Morse functions admit an index gap on ${\mathcal{M}}_{g,n}$. Specifically, there exists a universal constant $C>0$ such that any critical point in ${\mathcal{M}}_{g,n}$ has Morse index at least $C\log \log (g+n)$. This implies by Morse theory that the low degree homology of the Deligne-Mumford compactification ${\overline{\mathcal{M}}}_{g,n}$ comes from the boundary $\partial {\mathcal{M}}_{g,n}$.

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