CHARACTERIZING HOMOTOPY OF SYSTEMS OF CURVES ON A COMPACT SURFACE BY CROSSING NUMBERS

Let C-1,..., C-k and C-1',..., C-k' be closed curves on a compact surf ace S. We characterize (in terms of counting crossings) when there exi sts a permutation pi of {1,..., k} such that C-pi(i)' is freely homoto pic to C-i or C-i(-1), for each i = 1,..., k.