Summary: Virtually Haken Dehnfilling.
D. Cooper and D.D. Long \Lambda
May 8, 1997
We show that ``most'' Dehnfillings of a nonfibered, atoroidal, Haken threemanifold with
torus boundary are virtually Haken.
Suppose that X is a compact, oriented, threemanifold with boundary a torus T : We will pick a
basis of H 1 (T ) represented by simple loops –; ¯ such that – = 0 in H 1 (X; Q): We call – a longitude
and ¯ a meridian. A slope, ff; on T is the isotopy class of essential unoriented simple closed curve.
The manifold X(ff) is the result of Dehnfilling along the slope ff: This means that a solid torus
is glued along its boundary to T so that a meridian disc of the solid torus is glued onto ff: The
manifold X is atoroidal if every Z \Theta Z subgroup of ß 1 X is conjugate into ß 1 T: The distance between
two slopes ff; fi is \Delta(ff; fi) which is the absolute value of the algebraic intersection number of the
homology classes represented by these slopes.
Theorem 1.1 Suppose that X is a compact, connected, oriented, irreducible, atoroidal threemanifold
with boundary a torus T: Suppose that S is a compact, connected, oriented, nonseparating, incom
pressible surface properly embedded in X with nonempty boundary. Suppose that S is not a fiber of
a fibration of X over the circle. Let g be the genus of S and b the number of boundary components
of S: Then there is an integer 1 Ÿ P (X; S) Ÿ 3g \Gamma 2 + b with the following property.