The disjoint curve property


A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve property. This paper shows that in a closed, orientable three-manifold all splittings of sufficiently large genus have the disjoint curve property. From this and a solution to the generalized Waldhausen conjecture it would follow that any closed, orientable three manifold contains only finitely many full splittings.


The distance of a splitting is a notion due to Hempel. It is the shortest path in the curve complex (of the Heegaard splitting surface) between two curves which bound essential disks in opposite handlebodies. I have given three results in this line, each of the form: If M is a closed orientable three-manifold, and H is a splitting with genus g(H) > C(t), then the distance of H is at most K. Here t is the number of tetrahedra in a minimal triangulation of M.

The first result has C(t) a linear function while K = 4. This is written up in Chapter Three of my corrected thesis, Almost normal Heegaard splittings. This version contains the weakest result but is fairly straight-forward --- all necessary definitions are contained in Chapter Two of the thesis.

Completely self-contained, with a better result, is my paper The disjoint annulus property. Here K = 3, but the function C(t) is now quadratic. I doubt this will ever appear, as I have finally obtained the optimal theorem along these lines.

The optimal result is written up in the paper The disjoint curve property, and was published in GT. Here K = 2: this cannot be improved without imposing additional restrictions on the topology of M. To prove this the function C(t) is taken to be slightly super-exponential. This last paper is fairly long.

If you have any questions or corrections please contact me via email.

last touched - Jul. 2004
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