Tori and Heegaard splittings. (English) Zbl 1367.57009

Thompson, Abigail

Ill. J. Math. 60, No. 1, 141-148 (2016).

Summary: In [in: Stud. in Math. 5 (Studies Modern Topol.), 39–98 (1968; Zbl 0194.24902)] W. 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. T. Kobayashi [Osaka J. Math. 24, 173–215 (1987; Zbl 0665.57010)] 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 show that given a 3-manifold \(M\), a minimal genus, strongly irreducible Heegaard surface \(F\) for \(M\), and an essential torus \(T\), we can either restrict the number of curves of intersection of \(T\) with \(F\) (to four), find a different essential surface and minimal genus Heegaard splitting with at most four essential curves of intersection, find a thinner decomposition of \(M\), or produce a small Seifert-fibered piece of \(M\).

MSC:

57M27

Invariants of knots and \(3\)-manifolds (MSC2010)

Citations:

Zbl 0194.24902; Zbl 0665.57010