On the existence of high index topologically minimal surfaces

Abstract

Abstract. 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 π n−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. 2. Theorem. There is a closed 3-manifold, M (1), with an index 1 Heegaard surface S, such that for each n, the lift of S to some n-fold cover M (n) of M (1) has topological index n.The manifold M (1) of Theorem 2 is obtained by gluing together the boundary components of the complement of a link in S 3 , which we will construct as follows: We say S 2 ⊂ S 3 is a bridge sphere for a knot or link L ⊂ S 3 if L meets each of the balls bounded by S 2 in a collection of boundary parallel arcs. If the minimum number of such arcs is b, then we say L is a b-bridge knot/link.Throughout the paper, we will assume L ⊂ S 3 is a two component two-bridge link such that for a regular neighborhood N of L the complement M = S 3 \ N contains no essential planar surface with Euler characteristic greater than −3.By [HT85] or [BS05], such links can be constructed by choosing a sufficiently complicated braid to define the link. Let S 2 be a bridge sphere for L that realizes its bridge number, and let B ± be the balls in S 3 bounded by S 2 . Let S = (S 2 \ N ) ⊂ M and C ± = (B ± \ N ) ⊂ M . The manifold M (1) of Theorem 2 is obtained from M by

Heegaard surfaces and the distance of amalgamation

“…The proof of Theorem 1.5 is also used by Bachman in [1] to study stabilization of Heegaard splittings.…”

“…Y ) if, there is an essential curve in P t such that (1) bounds a compressing disk in X t (resp. Y t ), and (2) M 2 and bounds an embedded disk D in M 2 that is transverse to P t .…”

Stabilization, amalgamation and curves of intersection of Heegaard splittings

“…Recent work has shown that in some sense (low genus) Heegaard splittings of "generic" 3-manifolds are amalgamations (see eg Bachman [1], Bachman, Schleimer and Sedgwick [4], Lackenby [15], Li [16] and Souto [27]). Not every Heegaard splitting, however, is an amalgamation of Heegaard splittings along some essential surface.…”

Critical Heegaard Surfaces Obtained by Self-Amalgamation

“…Definition 1.1. [4] F is critical if the compressing disks for F can be partitioned into two sets C 0 and C 1 , such that (1) for each i = 0, 1, there is at least one pair of disks…”

“…Critical surfaces, which are also defined by David Bachman [1] [4], can be regarded as topological index 2 minimal surfaces [4]. Definition 1.1.…”