Topological Index Theory for Surfaces in 3-Manifolds

TL;DR

This paper introduces a topological index for surfaces in 3-manifolds, generalizing known classes and providing a new framework for understanding surface interactions and properties.

Contribution

It defines the topological index for surfaces and proves a key theorem relating surfaces with this index to incompressible surfaces, extending classical results.

Findings

Surfaces with a well-defined topological index generalize known surface classes.

A surface with topological index n can be isotoped to meet an incompressible surface with controlled index sum.

The results unify and extend many classical theorems in 3-manifold topology.

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 $\ge 2$.