The complex of curves on non-orientable surfaces

Let &$F{M) c 0>(.R + —0) denote the projectivized space of measured foliations on a compact surface M with negative Euler characteristic, as studied by Thurston [3], and let ^ ^ ( M) denote the subspace consisting of those foliations in which each boundary component is a leaf (containing at least one singularity). If M is the sum of g tori, d disks and p projective planes, then 0>PQ{M) s S6s+3p+:w"7 and &&{M) is the join of &&0(M) to a (d- 1)-simplex. There is a subcomplex Xo in ^^0{M) whose (n — l)-simplices consist of foliations obtained by "enlarging " n disjoint simple, closed, connected curves Cl 5..., Cn, none of which bounds a disk, or is boundary parallel, and no two of which bound an annulus. A subcomplex X of ^^(M) can be defined in much the same way, except that we allow proper arcs as well as simple closed curves. The complexes Xo and X have an interesting structure in their own right; since they are preserved under diffeomorphism, their structure gives geometric insight into the structure of automorphisms of M. Unfortunately, their structure is quite complex, since Xo and X are rarely locally finite