On the Existence of High Index Topologically Minimal Surfaces

The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist. Consider a compact, connected, two sided surface S properly embedded in a compact, orientable 3-manifold M . The disk complex Γ(S) is the simplicial complex defined as follows: Vertices of Γ(S) are isotopy classes of compressing disks for S. A collection of n such isotopy classes is an (n − 1)-simplex of Γ(S) if there are representatives of each that are pairwise disjoint. 1. Definition. If Γ(S) is non-empty then the topological index of S is the smallest n such that πn−1(Γ(S)) is non-trivial. If Γ(S) is empty then S will have topological index 0. If H has a well-defined topological index (i.e. Γ(S) = ∅ or some homotopy group of Γ(S) is non-trivial) then we will say that S is topologically minimal. Topological index was introduced by the first author as the topological analogue of the index of an unstable minimal surface [Bacc], and later used in [Baca] and [Bacb] to prove various results about Heegaard splittings of 3-manifolds. Here we show this definition is not vacuous for high index surfaces by proving the following existence result: 2. Theorem. There is a closed 3-manifold, M, with an index 1 Heegaard surface S, such that for each n, the lift of S to some n-fold cover M of M has topological index n. The manifold M of Theorem 2 is obtained by gluing together the boundary components of the complement of a link in S, which we will construct as follows: We say S ⊂ S is a bridge sphere for a knot or link L ⊂ S if L meets each of the balls bounded by S in a collection of boundary Date: September 14, 2009. Partially supported by NSF grant DMS-0906151. Partially supported by NSF MSPRF grant 0602368.