Topological Index Theory for Surfaces in 3-Manifolds

Abstract

The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface $H$ with topological index $n$ to meet an incompressible surface $F$ so that the sum of the indices of the components of $H\setminus N(F)$ is at most $n$. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel's {\it distance} to surfaces with topological index $\ge 2$.

25 pages, 8 figures. Final version. To appear in Geometry & Topology

References in corpus (6)

• Topological Index Theory for Surfaces in 3-Manifolds
• Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization
• Barriers to Topologically Minimal Surfaces
• Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)
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Cited by in corpus (26)

• Topological Index Theory for Surfaces in 3-Manifolds
• On critical Heegaard splittings of tunnel number two composite knot exteriors
• Heegaard splittings and open books
• Embeddability in the 3-sphere is decidable
• Normalizing Topologically Minimal Surfaces II: Disks
• Critical Heegaard surfaces obtained by self-amalgamation
• Normalizing Topologically Minimal Surfaces I: Global to Local Index
• Normalizing Topologically Minimal Surfaces III: Bounded Combinatorics
• Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization
• Barriers to Topologically Minimal Surfaces
• A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical
• One-sided and two-sided Heegaard splittings
• Bridge spheres for the unknot are topologically minimal
• Hyperbolic manifolds containing high topological index surfaces
• On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three I - the Structure Theorem
• An Infinite Family of Links with Critical Bridge Spheres
• Examples of unstabilized critical Heegaard surfaces
• On topologically minimal surfaces of high genus
• Critical Heegaard surfaces obtained by amalgamation
• On unstabilzed genus three critical Heegaard surfaces
• Locally Helical Surfaces have bounded twisting
• Combinatorial minimal surfaces in pseudomanifolds
• On keen weakly reducbile Heegaard splittings
• On the group action of $Mod(M,F)$ on the disk complex
• Generalized Heegaard splittings and the disk complex
• Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds