Yo'av Rieck

Separating incompressible surfaces and stabilizations of Heegaard splittings

We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold M and any integer m> 0, a surgery on a link in M of at most 2m + 1 components will provide a closed, orientable, irreducible 3-manifold containing m disjoint, non-parallel, separating, incompressible surfaces of arbitrarily high genus. (2) There exists a 3-manifold M containing separating incompressible surfaces Sn of genus g(Sn) arbitrarily large, such that the amalgamation of minimal Heegaard splittings of two resulting 3-manifolds cutting along Sn can be stabilized g(Sn) − 3 times to a minimal Heegaard splitting of M .

Persistence of Heegaard structures under Dehn filling

It is well known that a Heegaard surface may destabilize after Dehn filling, reducing the genus by one or more. This phenomenon is classified according to whether or not the core of the attached solid torus is isotopic into the destabilized surface. When it is, the destabilized surface will be a Heegaard surface for infinitely many fillings, arranged along a destabilization line in the Dehn surgery space. Here we demonstrate that a destabilization line corresponds to a slope bounding an essential surface. Such slopes are known to be finite in number and therefore so is the number of destabilization lines. n nWe apply this result to study Heegaard genus. In particular we prove, using purely topological techniques, that if X is any a-cylindrical manifold, then there are an infinite number of Dehn fillings on X which produce a manifold of the same genus as X.