Algebraic & Geometric Topology

Overtwisted open books from sobering arcs

Noah Goodman

Full-text: Open access

Abstract

We study open books on three manifolds which are compatible with an overtwisted contact structure. We show that the existence of certain arcs, called sobering arcs, is a sufficient condition for an open book to be overtwisted, and is necessary up to stabilization by positive Hopf-bands. Using these techniques we prove that some open books arising as the boundary of symplectic configurations are overtwisted, answering a question of Gay.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 3 (2005), 1173-1195.

Dates
Received: 24 July 2004
Revised: 15 June 2005
Accepted: 26 July 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796448

Digital Object Identifier
doi:10.2140/agt.2005.5.1173

Mathematical Reviews number (MathSciNet)
MR2171807

Zentralblatt MATH identifier
1090.57020

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57M99: None of the above, but in this section

Keywords
open book contact structure overtwisted sobering arc symplectic configuration graph

Citation

Goodman, Noah. Overtwisted open books from sobering arcs. Algebr. Geom. Topol. 5 (2005), no. 3, 1173--1195. doi:10.2140/agt.2005.5.1173. https://projecteuclid.org/euclid.agt/1513796448


Export citation

References

  • V Colin, Chirurgies d'indice un et isotopies de sphères dans les variétés de contact tendues, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 659–663
  • R J Daverman, R B Sher (editors), Handbook of geometric topology, North-Holland, Amsterdam (2002)
  • Y Eliashberg, Contact $3$-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165–192
  • Y M Eliashberg, W P Thurston, Confoliations, University Lecture Series 13, American Mathematical Society, Providence, RI (1998)
  • J B Etnyre, Introductory lectures on contact geometry, from: “Topology and geometry of manifolds (Athens, GA, 2001)”, Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence, RI (2003) 81–107
  • J B Etnyre, K Honda, On the nonexistence of tight contact structures, Ann. of Math. (2) 153 (2001) 749–766
  • J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31–39
  • D Gabai, Detecting fibred links in $S\sp 3$, Comment. Math. Helv. 61 (1986) 519–555
  • D T Gay, Open books and configurations of symplectic surfaces, \agtref3200319569586
  • E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637–677
  • 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, Beijing (2002) 405–414
  • E Giroux, N Goodman, On the Stable Equivalence of Open Books in Three-Manifolds, in preparation
  • N Goodman, Contact Structures and Open Books, PhD thesis, University of Texas at Austin (2003)
  • K Honda, On the classification of tight contact structures. I, \gtref4200011309368
  • A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of $B\sp 4$, Invent. Math. 143 (2001) 325–348
  • W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345–347
  • I Torisu, Convex contact structures and fibered links in 3-manifolds, Internat. Math. Res. Notices (2000) 441–454