Topological Index Theory for surfaces in 3–manifolds

Bachman, David

doi:10.2140/gt.2010.14.585

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Topological Index Theory for surfaces in 3–manifolds

David Bachman

Geometry & Topology 14 (2010) 585–609

DOI: 10.2140/gt.2010.14.585

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 $\geq 2$ .

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

Volume 14, issue 1 (2010)