Connectivity of complexes of separating curves

TL;DR

This paper proves the high connectivity of separated curve complexes on surfaces, with implications for the topology of moduli spaces, by establishing new connectivity properties for both closed and punctured surfaces.

Contribution

It introduces new connectivity results for separated curve complexes on surfaces, extending understanding of their topological properties and implications for moduli space topology.

Findings

Separated curve complex of genus g is (g-3)-connected

Connectivity properties for punctured surfaces with partitioned removed sets

Implications for algebraic topology of moduli spaces

Abstract

We prove that the separated curve complex of a closed orientable surface of genus g is (g-3)-connected. We also obtain a connectivity property for a separated curve complex of the open surface that is obtained by removing a finite set from a closed one, but it is then assumed that the removed set is endowed with a partition and that the separating curves respect that partition. These connectivity statements have implications for the algebraic topology of the moduli space of curves.