Index gap of the systole function

TL;DR

This paper proves that the systole function's Morse index on moduli space grows at least logarithmically with genus and punctures, revealing a topological gap and implications for homology from boundary components.

Contribution

It establishes a universal lower bound on the Morse index of the systole function on moduli space, showing an index gap related to genus and punctures.

Findings

Morse index of critical points grows at least logarithmically with genus and punctures.

Low degree homology of the compactification is derived from the boundary.

The systole function is topologically Morse with an index gap.

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 $\partial\mathcal M_{g,n}$.