## The disjoint curve property

**Abstract:**
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.

**Comments:**
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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