Volume 17, issue 6 (2017)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24, 1 issue

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
The unstabilized canonical Heegaard splitting of a mapping torus

Yanqing Zou

Algebraic & Geometric Topology 17 (2017) 3435–3448

Let S be a closed orientable surface of genus at least 2. The action of an automorphism f on the curve complex of S is an isometry. Via this isometric action on the curve complex, a translation length is defined on f. The geometry of the mapping torus Mf depends on f. As it turns out, the structure of the minimal-genus Heegaard splitting also depends on f: the canonical Heegaard splitting of Mf, constructed from two parallel copies of S, is sometimes stabilized and sometimes unstabilized. We give an example of an infinite family of automorphisms for which the canonical Heegaard splitting of the mapping torus is stabilized. Interestingly, complexity bounds on f provide insight into the stability of the canonical Heegaard splitting of  Mf. Using combinatorial techniques developed on 3–manifolds, we prove that if the translation length of f is at least 8, then the canonical Heegaard splitting of Mf is unstabilized.

Heegaard splitting, stabilization, mapping torus, translation length
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M50
Received: 23 April 2016
Revised: 14 May 2017
Accepted: 25 May 2017
Published: 4 October 2017
Yanqing Zou
Department of Mathematics
Dalian Minzu University