#### Volume 17, issue 3 (2017)

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Positive factorizations of mapping classes

### R İnanç Baykur, Naoyuki Monden and Jeremy Van Horn-Morris

Algebraic & Geometric Topology 17 (2017) 1527–1555
##### Abstract

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic $4$–manifolds and Stein fillings of contact $3$–manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers $g$ and $n$, we tell whether or not there exist genus-$g$ Lefschetz pencils with $n$ base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic $4$–manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.

##### Keywords
mapping class groups, Lefschetz fibrations, contact manifolds, symplectic manifolds
##### Mathematical Subject Classification 2010
Primary: 20F65, 53D35, 57R17
##### Publication
Received: 24 September 2015
Revised: 13 May 2016
Accepted: 7 June 2016
Published: 17 July 2017
##### Authors
 R İnanç Baykur Department of Mathematics and Statistics University of Massachusetts Lederle Graduate Research Tower 710 North Pleasant Street Amherst, MA 01003-9305 United States Naoyuki Monden Department of Engineering Science Osaka Electro-Communication University Hatsu-cho 18-8 Neyagawa 572-8530 Japan Jeremy Van Horn-Morris Department of Mathematical Sciences The University of Arkansas Fayetteville, AR 72701 United States