Subject: curves on surfaces Date: Sun, 11 Feb 2001 13:40:35 -0500 From: Allen Hatcher To: Donald Davis > What is the maximum number of disjoint, >pairwise non-homotopic, non-null-homotopic simple closed >curves that can be placed on an orientable surface of >genus g? The number is 3g - 3 if g > 1 and the surface is closed, or 3g - 3 + b if the surface has b boundary circles. A collection of curves is maximal if and only if it decomposes the surface into pairs of pants; this follows from the classification of surfaces. So by Euler characteristic considerations the number is the same for all maximal collections, and one can compute it by looking at any particular maximal collection. This is a very classical result, surely known to Dehn, probably also Mobius, Riemann, Gauss, ... Low-dimensional topologists quote the result freely without mentioning a reference. In fact I don't know an explicit reference. Allen Hatcher