Geodesic
Overtwisted open books from sobering arcs
Goodman, Noah1
1Massachusetts Institute of Technology, Cambridge MA 02139, USA
Algebraic and Geometric Topology, Tome 5 (2005) no. 3, pp. 1173-1195
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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.

DOI : 10.2140/agt.2005.5.1173
Keywords: open book, contact structure, overtwisted, sobering arc, symplectic configuration graph

Goodman, Noah  1

1 Massachusetts Institute of Technology, Cambridge MA 02139, USA 
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Goodman, Noah. Overtwisted open books from sobering arcs. Algebraic and Geometric Topology, Tome 5 (2005) no. 3, pp. 1173-1195. doi: 10.2140/agt.2005.5.1173

[1] 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

[2] , Handbook of geometric topology, North-Holland (2002)

[3] Y Eliashberg, Contact 3–manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165

[4] Y M Eliashberg, W P Thurston, Confoliations, University Lecture Series 13, American Mathematical Society (1998)

[5] 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. (2003) 81

[6] J B Etnyre, K Honda, On the nonexistence of tight contact structures, Ann. of Math. $(2)$ 153 (2001) 749

[7] J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31

[8] D Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv. 61 (1986) 519

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

[10] E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637

[11] 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

[12] E Giroux, N Goodman, On the stable equivalence of open books in three-manifolds, Geom. Topol. 10 (2006) 97

[13] N Goodman, Contact structures and open books, PhD thesis, University of Texas at Austin (2003)

[14] K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309

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

[16] W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345

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

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