A note on the connectivity of certain complexes associated to surfaces

TL;DR

This paper presents simplified proofs of connectivity properties for various complexes related to surfaces, including new results on complexes involving splitting curves and homologous curves, with implications for surface topology.

Contribution

It introduces a trick that simplifies proofs of connectivity and simple connectivity for several complexes associated with surfaces, including new complexes involving splitting and homologous curves.

Findings

Complexes of curves, separating curves, nonseparating curves, pants, and cut systems are connected for high genus.

The complex of separating curves is simply connected for genus at least 4.

New complexes involving splitting and homologous curves are connected, with the latter related to homology relations.

Abstract

This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves, nonseparating curves, pants, and cut systems are all connected for genus $g \gg 0$. We also prove that two new complexes are connected : one involves curves which split a genus $2g$ surface into two genus $g$ pieces, and the other involves curves which are homologous to a fixed curve. The connectivity of the latter complex can be interpreted as saying the ``homology'' relation on the surface is (for $g \geq 3$) generated by ``embedded/disjoint homologies''. We finally prove that the complex of separating curves is simply connected for $g \geq 4$.