Volume 14, issue 1 (2010)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 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 distance to surfaces with topological index 2.

Keywords
Heegaard splitting, minimal surface
Mathematical Subject Classification 2000
Primary: 57M99
References
Publication
Received: 12 January 2009
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