2016/03/27 by Thompson, Abigail
#57M27 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1603.08154
Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface increases from zero, more complicated phenomena occur. Kobayashi showed that if a 3-manifold M3 contains an essential torus T, then it contains one which can be isotoped to intersect a strongly irreducible Heegaard splitting surface F in a collection of simple closed curves which are essential in T and in F. In general there is no global bound on the number of curves in this collection. We give conditions under which a global bound can be obtained.