## Algebraic & Geometric Topology

### Positive factorizations of mapping classes

#### 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.

#### Article information

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

Dates
Revised: 13 May 2016
Accepted: 7 June 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841404

Digital Object Identifier
doi:10.2140/agt.2017.17.1527

Mathematical Reviews number (MathSciNet)
MR3677934

Zentralblatt MATH identifier
06762595

#### Citation

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. https://projecteuclid.org/euclid.agt/1510841404

#### References

• S Akbulut, B Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001) 319–334
• S Bauer, Almost complex $4$–manifolds with vanishing first Chern class, J. Differential Geom. 79 (2008) 25–32
• R,\.I Baykur, K Hayano, Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds, Geom. Topol. 20 (2016) 2335–2395
• R,\.I Baykur, K Hayano, N Monden, Unchaining surgery and symplectic $4$–manifolds in preparation
• R,\.I Baykur, M Korkmaz, N Monden, Sections of surface bundles and Lefschetz fibrations, Trans. Amer. Math. Soc. 365 (2013) 5999–6016
• R,\.I Baykur, J Van Horn-Morris, Families of contact $3$–manifolds with arbitrarily large Stein fillings, J. Differential Geom. 101 (2015) 423–465
• R,\.I Baykur, J Van Horn-Morris, Topological complexity of symplectic $4$–manifolds and Stein fillings, J. Symplectic Geom. 14 (2016) 171–202
• E Dalyan, M Korkmaz, M Pamuk, Arbitrarily long factorizations in mapping class groups, Int. Math. Res. Not. 2015 (2015) 9400–9414
• S,K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205–236
• E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from “Proceedings of the International Congress of Mathematicians, II” (T Li, editor), Higher Ed. Press, Beijing (2002) 405–414
• K Honda, W,H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007) 427–449
• A Kaloti, Stein fillings of planar open books, preprint (2013)
• T-J Li, The Kodaira dimension of symplectic $4$–manifolds, from “Floer homology, gauge theory, and low-dimensional topology” (D,A Ellwood, P,S Ozsváth, A,I Stipsicz, Z Szabó, editors), Clay Math. Proc. 5, Amer. Math. Soc., Providence, RI (2006) 249–261
• T-J Li, Quaternionic bundles and Betti numbers of symplectic $4$–manifolds with Kodaira dimension zero, Int. Math. Res. Not. 2006 (2006) art. id. 37385, 28 pp.
• T-J Li, Symplectic $4$–manifolds with Kodaira dimension zero, J. Differential Geom. 74 (2006) 321–352
• T,J Li, A Liu, Symplectic structure on ruled surfaces and a generalized adjunction formula, Math. Res. Lett. 2 (1995) 453–471
• A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of $B\sp 4$, Invent. Math. 143 (2001) 325–348
• D Margalit, J McCammond, Geometric presentations for the pure braid group, J. Knot Theory Ramifications 18 (2009) 1–20
• O Plamenevskaya, On Legendrian surgeries between lens spaces, J. Symplectic Geom. 10 (2012) 165–181
• O Plamenevskaya, J Van Horn-Morris, Planar open books, monodromy factorizations and symplectic fillings, Geom. Topol. 14 (2010) 2077–2101
• A Putman, An infinite presentation of the Torelli group, Geom. Funct. Anal. 19 (2009) 591–643
• M-H Saitō, K-I Sakakibara, On Mordell–Weil lattices of higher genus fibrations on rational surfaces, J. Math. Kyoto Univ. 34 (1994) 859–871
• Y Sato, Canonical classes and the geography of nonminimal Lefschetz fibrations over $S\sp 2$, Pacific J. Math. 262 (2013) 191–226
• H Short, B Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. 46 (2000) 279–312
• I Smith, Geometric monodromy and the hyperbolic disc, Q. J. Math. 52 (2001) 217–228
• I Smith, Lefschetz pencils and divisors in moduli space, Geom. Topol. 5 (2001) 579–608
• A,I Stipsicz, Sections of Lefschetz fibrations and Stein fillings, Turkish J. Math. 25 (2001) 97–101
• A,I Stipsicz, Singular fibers in Lefschetz fibrations on manifolds with $b\sb 2\sp +=1$, Topology Appl. 117 (2002) 9–21
• A,I Stipsicz, On the geography of Stein fillings of certain $3$–manifolds, Michigan Math. J. 51 (2003) 327–337
• S Tanaka, On sections of hyperelliptic Lefschetz fibrations, Algebr. Geom. Topol. 12 (2012) 2259–2286
• C Wendl, Strongly fillable contact manifolds and $J$–holomorphic foliations, Duke Math. J. 151 (2010) 337–384