High distance Heegaard splittings of 3-manifolds

J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T Kobayashi [T Kobayashi, Casson-Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553-573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] proved the existence of arbitrarily high distance Heegaard splittings. In this work we explicitly define an infinite sequence of 3-manifolds {M-n} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of M-n is at least n. Moreover, we show that pi(1) (M-n) surjects onto pi(1) (Mn-1). Hence, if we assume that M-o has nontrivial boundary then it follows that the first Betti number beta(1) (M-n) > 0 for all n >= 1. Therefore, the sequence {M-n} consists of Haken 3-manifolds for n >= 1 and hyperbolizable 3-manifolds for n >= 3. (C) 2005 Elsevier B.V. All rights reserved.