Topological Index Theory for surfaces in 3-manifolds

By David Bachman. 2009

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The disk complex of a surface in a 3‐manifold is used to define its 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 HnN.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 distance to surfaces with topological index 2. 57M99

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b87140f8e02e3cfff7d932089222dfdf3cfcc406