Tori and Heegaard splittings

TL;DR

This paper explores the interactions between tori and Heegaard splittings in 3-manifolds, providing conditions for bounding the complexity of their intersections, extending previous results on reducible and irreducible splittings.

Contribution

It establishes conditions under which a global bound on the intersection curves between essential tori and Heegaard surfaces can be achieved.

Findings

Bounded the number of intersection curves under certain conditions

Extended Kobayashi's results to more general cases

Provided new insights into the structure of 3-manifolds with tori

Abstract

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 $M^3$ 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.