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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symplectic $2$-handles and transverse links


Author: David T. Gay
Journal: Trans. Amer. Math. Soc. 354 (2002), 1027-1047
MSC (2000): Primary 57R17, 57R65; Secondary 57M99
Published electronically: October 11, 2001
MathSciNet review: 1867371
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Abstract: A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching $2$-handles to convex boundaries of symplectic $4$-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over $S^1$; in this case the handles can be attached with any framing larger than a lower bound determined by the fibration. The other method results in a weaker convexity condition on the new boundary (sufficient to imply tightness of the new contact structure), and in this case the handles can be attached with any framing less than a certain upper bound. These methods supplement methods developed by Weinstein and Eliashberg for attaching symplectic $2$-handles along Legendrian knots.


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Additional Information

David T. Gay
Affiliation: Department of Mathematics, University of Arizona, 617 North Santa Rita, Post Office Box 210089, Tucson, Arizona 85721
Email: dtgay@math.arizona.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02890-2
PII: S 0002-9947(01)02890-2
Keywords: Symplectic handle, symplectic handlebody, contact surgery, fibered link, fibered knot, symplectic filling, convexity, concavity, transverse link, transverse knot, symplectic germ
Received by editor(s): January 24, 2000
Received by editor(s) in revised form: June 4, 2001
Published electronically: October 11, 2001
Article copyright: © Copyright 2001 American Mathematical Society