#### Volume 14, issue 1 (2010)

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Topological Index Theory for surfaces in 3–manifolds

### David Bachman

Geometry & Topology 14 (2010) 585–609
##### Abstract

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 $H\setminus N\left(F\right)$ 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 $\ge 2$.

##### Keywords
Heegaard splitting, minimal surface
Primary: 57M99
##### Publication
Revised: 19 November 2009
Accepted: 9 November 2009
Published: 4 February 2010
Proposed: Dave Gabai
Seconded: Joan Birman, Ron Stern
##### Authors
 David Bachman Department of Mathematics Pitzer College Claremont, CA 91711 USA http://pzacad.pitzer.edu/~dbachman