High-distance splittings of 3-manifolds

[摘要] A Heegaard splitting (S; V1, V 2) for a closed 3-manifold M is a representation M = V1 ∪S V2 where V1 and V 2 are handlebodies and S = ∂V 1 = ∂V2 = V 1 ∩ V2. The distance of a Heegaard splitting (S; V1, V2) is the length of a shortest path in the curve complex of S which connects the subcomplexes KV1 and KV2 , where KVi is the subcomplex consisting of all vertices that correspond to simple closed curves bounding disks in Vi for i = 1, 2. In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that for any n the distance of the Heegaard Splitting of Mn is at least n. Moreover, we show that pi1(Mn) surjects onto pi1(Mn-1 ). Hence, if we assume that M0 has a non-trivial boundary, i.e. first Betti number beta1( M0) > 0, then it follows that beta1( Mn) > 0 for all n ≥ 1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n ≥ 1 and hyperbolizable 3-manifolds for n ≥ 3.

[发布机构] Rice University