A note on the connectivity of certain complexes associated to surfaces

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 $g \geq 4$.