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Virtually Haken Dehnfilling. D. Cooper and D.D. Long \Lambda

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.
1 Results
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.


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics