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The unstabilized canonical Heegaard splitting of a mapping torus

Yanqing Zou

Algebraic & Geometric Topology 17 (2017) 3435–3448
Abstract

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.

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