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 S " 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. Floyd and Hatcher [2] have constructed all the low-dimensional examples (and one of dimension 5) in the cases where M is orientable. The purpose of the present paper is two-fold. First, we show that when M is not orientable, Xo has some surprising qualities not found when M is orientable. Secondly, we construct the complexes for those remaining cases in which the foliation space is of dimension one or two. The case of the 3-punctured RP is particularly interesting and is the subject of §3. In this case Xo is the complex obtained from the tetrahedron by repeated star subdivision of all the faces but no edges. In the resulting complex the vertices represent all the 1-sided curves, and an interior point of each edge all the 2-sided curves. I would like to thank David Chillingworth for his very helpful comments on the original manuscript.