Algebraic & Geometric Topology

Positive factorizations of mapping classes

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

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 3 (2017), 1527-1555.

Received: 24 September 2015
Revised: 13 May 2016
Accepted: 7 June 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 57R17: Symplectic and contact topology

mapping class groups Lefschetz fibrations contact manifolds symplectic manifolds


Baykur, R İnanç; Monden, Naoyuki; Van Horn-Morris, Jeremy. Positive factorizations of mapping classes. Algebr. Geom. Topol. 17 (2017), no. 3, 1527--1555. doi:10.2140/agt.2017.17.1527.

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