Geometry & Topology

Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds

R İnanç Baykur and Kenta Hayano

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We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4–manifolds as multisections, such as Seiberg–Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the 4–ball. Various problems regarding the topology of symplectic 4–manifolds, such as the smooth classification of symplectic Calabi–Yau 4–manifolds, can be translated to combinatorial problems in this manner. After producing special monodromy factorizations of Lefschetz pencils on symplectic Calabi–Yau homotopy K3 and Enriques surfaces, and introducing monodromy substitutions tailored for generating multisections, we obtain several novel applications, allowing us to construct: new counterexamples to Stipsicz’s conjecture on fiber sum indecomposable Lefschetz fibrations, nonisomorphic Lefschetz pencils of the same genera on the same new symplectic 4–manifolds, the very first examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.

Article information

Geom. Topol., Volume 20, Number 4 (2016), 2335-2395.

Received: 20 March 2015
Revised: 31 August 2015
Accepted: 5 October 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R17: Symplectic and contact topology 57R55: Differentiable structures 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx]

symplectic 4-manifold exotic 4-manifold Lefschetz fibration Lefschetz pencil multisection nonisomorphic fibration mapping class group Dehn twist factorization exotic embedding symplectic Kodaira dimension symplectic Calabi-Yau fiber sum Seiberg-Witten invariant


Baykur, R İnanç; Hayano, Kenta. Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds. Geom. Topol. 20 (2016), no. 4, 2335--2395. doi:10.2140/gt.2016.20.2335.

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  • S Akbulut, R Kirby, Branched covers of surfaces in $4$–manifolds, Math. Ann. 252 (1979/80) 111–131
  • S Akbulut, B Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001) 319–334
  • D Auroux, L Katzarkov, A degree doubling formula for braid monodromies and Lefschetz pencils, Pure Appl. Math. Q. 4 (2008) 237–318
  • S Bauer, Almost complex $4$–manifolds with vanishing first Chern class, J. Differential Geom. 79 (2008) 25–32
  • R \.I Baykur, Topology of broken Lefschetz fibrations and near-symplectic four-manifolds, Pacific J. Math. 240 (2009) 201–230
  • R \.I Baykur, Inequivalent Lefschetz fibrations and surgery equivalence of symplectic $4$–manifolds, J. Symplectic Geom. 14 (2016) 671–686
  • R \.I Baykur, Minimality and fiber sum decompositions of Lefschetz fibrations, Proc. Amer. Math. Soc. 144 (2016) 2275–2284
  • R \.I Baykur, K Hayano, Hurwitz equivalence for Lefschetz fibrations and their multisections, to appear in Contemp. Math., "Proceedings of the 13th International Workshop on Real and Complex Singularities" (2015)
  • P Bellingeri, S Gervais, Surface framed braids, Geom. Dedicata 159 (2012) 51–69
  • S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205–236
  • S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743–785
  • J G Dorfmeister, Kodaira dimension of fiber sums along spheres, Geom. Dedicata 177 (2015) 1–25
  • H Endo, Meyer's signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000) 237–257
  • H Endo, Y Z Gurtas, Lantern relations and rational blowdowns, Proc. Amer. Math. Soc. 138 (2010) 1131–1142
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • S Finashin, Knotting of algebraic curves in $\mathbb C\rm P\sp 2$, Topology 41 (2002) 47–55
  • R Fintushel, R J Stern, Rational blowdowns of smooth $4$–manifolds, J. Differential Geom. 46 (1997) 181–235
  • R Fintushel, R J Stern, Surfaces in $4$–manifolds, Math. Res. Lett. 4 (1997) 907–914
  • R Fintushel, R J Stern, Knots, links, and $4$–manifolds, Invent. Math. 134 (1998) 363–400
  • R Fintushel, R J Stern, Families of simply connected $4$–manifolds with the same Seiberg–Witten invariants, Topology 43 (2004) 1449–1467
  • R Fintushel, R J Stern, Six lectures on four $4$–manifolds, from: “Low dimensional topology”, (T S Mrowka, P S Ozsváth, editors), IAS/Park City Math. Ser. 15, Amer. Math. Soc. (2009) 265–315
  • S Friedl, S Vidussi, On the topology of symplectic Calabi–Yau $4$–manifolds, J. Topol. 6 (2013) 945–954
  • D Gay, T E Mark, Convex plumbings and Lefschetz fibrations, J. Symplectic Geom. 11 (2013) 363–375
  • R E Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995) 527–595
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)
  • I Hambleton, M Kreck, Smooth structures on algebraic surfaces with cyclic fundamental group, Invent. Math. 91 (1988) 53–59
  • I Hambleton, M Kreck, Cancellation, elliptic surfaces and the topology of certain four-manifolds, J. Reine Angew. Math. 444 (1993) 79–100
  • A Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. 89 (1980) 89–104
  • H J Kim, Modifying surfaces in $4$–manifolds by twist spinning, Geom. Topol. 10 (2006) 27–56
  • H J Kim, D Ruberman, Topological triviality of smoothly knotted surfaces in $4$–manifolds, Trans. Amer. Math. Soc. 360 (2008) 5869–5881
  • M Korkmaz, Noncomplex smooth $4$–manifolds with Lefschetz fibrations, Internat. Math. Res. Notices (2001) 115–128
  • M Korkmaz, B Ozbagci, On sections of elliptic fibrations, Michigan Math. J. 56 (2008) 77–87
  • T-J Li, Smoothly embedded spheres in symplectic $4$–manifolds, Proc. Amer. Math. Soc. 127 (1999) 609–613
  • 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
  • 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
  • G Massuyeau, A Oancea, D A Salamon, Lefschetz fibrations, intersection numbers, and representations of the framed braid group, Bull. Math. Soc. Sci. Math. Roumanie 56(104) (2013) 435–486
  • Y Matsumoto, Lefschetz fibrations of genus two–-a topological approach, from: “Topology and Teichmüller spaces”, (S Kojima, Y Matsumoto, K Saito, M Seppälä, editors), World Sci. Publ., River Edge, NJ (1996) 123–148
  • D McDuff, M Symington, Associativity properties of the symplectic sum, Math. Res. Lett. 3 (1996) 591–608
  • W Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973) 239–264
  • B Ozbagci, Signatures of Lefschetz fibrations, Pacific J. Math. 202 (2002) 99–118
  • J Park, K-H Yun, Nonisomorphic Lefschetz fibrations on knot surgery $4$–manifolds, Math. Ann. 345 (2009) 581–597
  • Y Sato, $2$–spheres of square $-1$ and the geography of genus–$2$ Lefschetz fibrations, J. Math. Sci. Univ. Tokyo 15 (2008) 461–491
  • Y Sato, Canonical classes and the geography of nonminimal Lefschetz fibrations over $S\sp 2$, Pacific J. Math. 262 (2013) 191–226
  • A Scorpan, The wild world of $4$–manifolds, Amer. Math. Soc. (2005)
  • B Siebert, G Tian, On the holomorphicity of genus two Lefschetz fibrations, Ann. of Math. 161 (2005) 959–1020
  • 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, Indecomposability of certain Lefschetz fibrations, Proc. Amer. Math. Soc. 129 (2001) 1499–1502
  • N S Sunukjian, Surfaces in $4$–manifolds: concordance, isotopy, and surgery, Int. Math. Res. Not. 2015 (2015) 7950–7978
  • C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809–822
  • C H Taubes, ${\rm SW}\Rightarrow{\rm Gr}$: from the Seiberg–Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996) 845–918
  • M Usher, Minimality and symplectic sums, Int. Math. Res. Not. 2006 (2006) Art. ID 49857