Constructing Lefschetz-type fibrations on four-manifolds

Gay, David T¹ ; Kirby, Robion²

¹ Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa
² University of California, Berkeley, CA 94720, USA

Geometry & topology, Tome 11 (2007) no. 4, pp. 2075-2115.

Résumé

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4–manifolds. These are generalizations of Lefschetz fibrations over the 2–sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1–handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4–manifolds without 3– and 4–handles, such as the homotopy 4–spheres arising from nontrivial balanced presentations of the trivial group.

Détail

DOI : 10.2140/gt.2007.11.2075

Keywords: Lefschetz fibrations, round handles, open book decompositions, Andrews–Curtis conjecture, Gluck construction, achiral, near-symplectic forms

Bibliographie

[1] S Akbulut, R Kirby, An exotic involution of $S^{4}$, Topology 18 (1979) 75

[2] S Akbulut, R Kirby, A potential smooth counterexample in dimension $4$ to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews–Curtis conjecture, Topology 24 (1985) 375

[3] S Akbulut, B Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001) 319

[4] D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043

[5] R İ Baykur, Near-symplectic broken Lefschetz fibrations and (smooth) invariants of $4$–manifolds

[6] R İ Baykur, Kähler decomposition of $4$–manifolds, Algebr. Geom. Topol. 6 (2006) 1239

[7] C L Curtis, M H Freedman, W C Hsiang, R Stong, A decomposition theorem for $h$–cobordant smooth simply-connected compact $4$–manifolds, Invent. Math. 123 (1996) 343

[8] F Ding, H Geiges, A I Stipsicz, Surgery diagrams for contact $3$–manifolds, Turkish J. Math. 28 (2004) 41

[9] S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205

[10] S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743

[11] Y Eliashberg, Classification of overtwisted contact structures on $3$–manifolds, Invent. Math. 98 (1989) 623

[12] J B Etnyre, T Fuller, Realizing $4$–manifolds as achiral Lefschetz fibrations, Int. Math. Res. Not. (2006) 21

[13] J B Etnyre, B Ozbagci, Invariants of contact structures from open books

[14] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[15] D T Gay, Open books and configurations of symplectic surfaces, Algebr. Geom. Topol. 3 (2003) 569

[16] D T Gay, R Kirby, Constructing symplectic forms on 4-manifolds which vanish on circles, Geom. Topol. 8 (2004) 743

[17] E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: "Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)", Higher Ed. Press (2002) 405

[18] R E Gompf, Killing the Akbulut–Kirby $4$–sphere, with relevance to the Andrews–Curtis and Schoenflies problems, Topology 30 (1991) 97

[19] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. $(2)$ 148 (1998) 619

[20] R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999)

[21] J Harer, Pencils of curves on $4$–manifolds, PhD thesis, University of California, Berkeley (1979)

[22] J Harer, How to construct all fibered knots and links, Topology 21 (1982) 263

[23] C Hog-Angeloni, W Metzler, The Andrews–Curtis conjecture and its generalizations, from: "Two-dimensional homotopy and combinatorial group theory", London Math. Soc. Lecture Note Ser. 197, Cambridge Univ. Press (1993) 365

[24] R Kirby, Akbulut's corks and $h$–cobordisms of smooth, simply connected $4$–manifolds, Turkish J. Math. 20 (1996) 85

[25] R Kirby, Problems in low-dimensional topology, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35

[26] F Laudenbach, V Poénaru, A note on $4$–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337

[27] W B R Lickorish, A representation of orientable combinatorial $3$–manifolds, Ann. of Math. $(2)$ 76 (1962) 531

[28] A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of $B^4$, Invent. Math. 143 (2001) 325

[29] R Matveyev, A decomposition of smooth simply-connected $h$–cobordant $4$–manifolds, J. Differential Geom. 44 (1996) 571

[30] T Perutz, Surface-fibrations, four-manifolds, and symplectic Floer homology, PhD thesis, University of London (2005)

[31] T Perutz, Lagrangian matching invariants for fibred four-manifolds. I, Geom. Topol. 11 (2007) 759

[32] T Perutz, Lagrangian matching invariants for fibred four-manifolds. II

[33] C H Taubes, Pseudoholomorphic punctured spheres in $\mathbb{R}\times(S^1\times S^2)$: moduli space parametrizations, Geom. Topol. 10 (2006) 1855

[34] C H Taubes, Pseudoholomorphic punctured spheres in $\mathbb R\times (S^1\times S^2)$: properties and existence, Geom. Topol. 10 (2006) 785

[35] I Torisu, Convex contact structures and fibered links in $3$–manifolds, Internat. Math. Res. Notices (2000) 441

[36] M Usher, The Gromov invariant and the Donaldson–Smith standard surface count, Geom. Topol. 8 (2004) 565