×

Constructing Lefschetz-type fibrations on four-manifolds. (English) Zbl 1135.57009

In 1999, S. Donaldson discovered that symplectic 4-manifolds have Lefschetz pencil structures. In 2005, Auroux, Donaldson and Katzarkov showed that a near-symplectic 4-manifold is a broken Lefschetz fibration (BLF). In 2006, Etnyre and Fuller showed that a 4-manifold connected sum with a 2-sphere bundle over \(S^2\) is an achiral Lefschetz fibration (ALF). Let \(X\) be an arbitrary closed 4-manifold and \(F\) be a closed surface in \(X\) with self-intersection \(F\cdot F= 0\). Then the authors show that there is a broken, achiral Lefschetz fibration (BALF) from \(X\) to \(S^2\) with \(F\) as a fiber. To prove the result they take the concave BLF on \(F\times B^2\) and add enough round 1-handles so that the complement is built with \(0\)-, \(1\)- and \(2\)-handles. This induces an open book decomposition (OBD) on the boundary of this concave piece. Then they construct a convex BLF on the complement, inducing an OBD on its boundary which supports a contact structure homotopic to the contact structure supported by the OBD coming from the concave piece. They arrange that both contact structures are overtwisted and so isotopic. By Giroux’s work on open books the two OBDs have a common positive stabilization. The authors arrange concave boundaries and convex boundaries so that the open books match. They give a list of questions in the last section.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
57R65 Surgery and handlebodies
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] S Akbulut, R Kirby, An exotic involution of \(S^4\), Topology 18 (1979) 75 · Zbl 0465.57013
[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 · Zbl 0584.57009
[3] S Akbulut, B Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001) 319 · Zbl 1002.57062
[4] D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043 · Zbl 1077.53069
[5] R \DI Baykur, Near-symplectic broken Lefschetz fibrations and (smooth) invariants of \(4\)-manifolds
[6] R \DI Baykur, Kähler decomposition of \(4\)-manifolds, Algebr. Geom. Topol. 6 (2006) 1239 · Zbl 1133.57011
[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 · Zbl 0843.57020
[8] F Ding, H Geiges, A I Stipsicz, Surgery diagrams for contact \(3\)-manifolds, Turkish J. Math. 28 (2004) 41 · Zbl 1077.53071
[9] S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205 · Zbl 1040.53094
[10] S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743 · Zbl 1012.57040
[11] Y Eliashberg, Classification of overtwisted contact structures on \(3\)-manifolds, Invent. Math. 98 (1989) 623 · Zbl 0684.57012
[12] J B Etnyre, T Fuller, Realizing \(4\)-manifolds as achiral Lefschetz fibrations, Int. Math. Res. Not. (2006) 21 · Zbl 1118.57019
[13] J B Etnyre, B Ozbagci, Invariants of contact structures from open books · Zbl 1157.57015
[14] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357 · Zbl 0528.57011
[15] D T Gay, Open books and configurations of symplectic surfaces, Algebr. Geom. Topol. 3 (2003) 569 · Zbl 1035.57015
[16] D T Gay, R Kirby, Constructing symplectic forms on 4-manifolds which vanish on circles, Geom. Topol. 8 (2004) 743 · Zbl 1054.57027
[17] E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Higher Ed. Press (2002) 405 · Zbl 1015.53049
[18] R E Gompf, Killing the Akbulut-Kirby \(4\)-sphere, with relevance to the Andrews-Curtis and Schoenflies problems, Topology 30 (1991) 97 · Zbl 0715.57016
[19] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. \((2)\) 148 (1998) 619 · Zbl 0919.57012
[20] R E Gompf, A I Stipsicz, \(4\)-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999) · Zbl 0933.57020
[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 · Zbl 0504.57002
[23] C Hog-Angeloni, W Metzler, The Andrews-Curtis conjecture and its generalizations, London Math. Soc. Lecture Note Ser. 197, Cambridge Univ. Press (1993) 365 · Zbl 0814.57002
[24] R Kirby, Akbulut’s corks and \(h\)-cobordisms of smooth, simply connected \(4\)-manifolds, Turkish J. Math. 20 (1996) 85 · Zbl 0868.57031
[25] R Kirby, Problems in low-dimensional topology, 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 · Zbl 0242.57015
[27] W B R Lickorish, A representation of orientable combinatorial \(3\)-manifolds, Ann. of Math. \((2)\) 76 (1962) 531 · Zbl 0106.37102
[28] A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of \(B^4\), Invent. Math. 143 (2001) 325 · Zbl 0983.32027
[29] R Matveyev, A decomposition of smooth simply-connected \(h\)-cobordant \(4\)-manifolds, J. Differential Geom. 44 (1996) 571 · Zbl 0885.57016
[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 · Zbl 1143.53079
[32] T Perutz, Lagrangian matching invariants for fibred four-manifolds. II · Zbl 1144.53104
[33] C H Taubes, Pseudoholomorphic punctured spheres in \(\mathbbR\times(S^1\times S^2)\): moduli space parametrizations, Geom. Topol. 10 (2006) 1855 · Zbl 1161.53075
[34] C H Taubes, Pseudoholomorphic punctured spheres in \(\mathbb R\times (S^1\times S^2)\): properties and existence, Geom. Topol. 10 (2006) 785 · Zbl 1134.53045
[35] I Torisu, Convex contact structures and fibered links in \(3\)-manifolds, Internat. Math. Res. Notices (2000) 441 · Zbl 0978.53133
[36] M Usher, The Gromov invariant and the Donaldson-Smith standard surface count, Geom. Topol. 8 (2004) 565 · Zbl 1055.53064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.