Algebraic & Geometric Topology

Stein fillable contact $3$–manifolds and positive open books of genus one

Paolo Lisca

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A 2–dimensional open book (S,h) determines a closed, oriented 3–manifold Y(S,h) and a contact structure ξ(S,h) on Y(S,h). The contact structure ξ(S,h) is Stein fillable if h is positive, ie h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse holds, that is

ξ ( S , h )  Stein fillable h  positive .

On the other hand, results by Wand [Phd thesis (2010)] and by Baker, Etnyre and Van Horn–Morris [J. Differential Geom. 90 (2012) 1-80] imply the existence of counterexamples to the above implication with S of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the implication holds under the assumption that S is a one-holed torus and Y(S,h) is a Heegaard Floer L–space.

Article information

Algebr. Geom. Topol., Volume 14, Number 4 (2014), 2411-2430.

Received: 10 April 2013
Revised: 5 January 2014
Accepted: 7 January 2014
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

Stein fillings contact structures open books


Lisca, Paolo. Stein fillable contact $3$–manifolds and positive open books of genus one. Algebr. Geom. Topol. 14 (2014), no. 4, 2411--2430. doi:10.2140/agt.2014.14.2411.

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  • S Akbulut, B Ozbagci, Erratum: “Lefschetz fibrations on compact Stein surfaces” [Geom. Topol. 5 (2001), 319–334], Geom. Topol. 5 (2001) 939–945
  • S Akbulut, B Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001) 319–334
  • K L Baker, J B Etnyre, J Van Horn-Morris, Cabling, contact structures and mapping class monoids, J. Differential Geom. 90 (2012) 1–80
  • J A Baldwin, Heegaard Floer homology and genus one, one-boundary component open books, J. Topol. 1 (2008) 963–992
  • S K Donaldson, The orientation of Yang–Mills moduli spaces and $4$–manifold topology, J. Differential Geom. 26 (1987) 397–428
  • J B Etnyre, B Ozbagci, Invariants of contact structures from open books, Trans. Amer. Math. Soc. 360 (2008) 3133–3151
  • H Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics 109, Cambridge Univ. Press (2008)
  • E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: “Proceedings of the International Congress of Mathematicians, Vol. II”, (T Li, editor), Higher Ed. Press, Beijing (2002) 405–414
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)
  • K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007) 427–449
  • K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, II, Geom. Topol. 12 (2008) 2057–2094
  • K Honda, W H Kazez, G Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. 83 (2009) 289–311
  • R Kirby, P Melvin, Dedekind sums, $\mu$–invariants and the signature cocycle, Math. Ann. 299 (1994) 231–267
  • A G Lecuona, P Lisca, Stein fillable Seifert fibered $3$–manifolds, Algebr. Geom. Topol. 11 (2011) 625–642
  • A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of $B\sp 4$, Invent. Math. 143 (2001) 325–348
  • K Murasugi, On closed $3$–braids, Memoirs of the Amer. Math. Soc. 151, Amer. Math. Soc. (1974)
  • S Y Orevkov, Quasipositivity problem for $3$–braids, Turkish J. Math. 28 (2004) 89–93
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39–61
  • L Paris, D Rolfsen, Geometric subgroups of mapping class groups, J. Reine Angew. Math. 521 (2000) 47–83
  • O Plamenevskaya, Contact structures with distinct Heegaard Floer invariants, Math. Res. Lett. 11 (2004) 547–561
  • A Wand, Factorizations of diffeomorphisms of compact surfaces with boundary, PhD thesis, University of California, Berkeley (2010) Available at \setbox0\makeatletter\@url {\unhbox0
  • C Wendl, Strongly fillable contact manifolds and $J$–holomorphic foliations, Duke Math. J. 151 (2010) 337–384