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Forum Mathematicum

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Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 25, Issue 3


Knotted surfaces in 4-manifolds

Thomas E. Mark
Published Online: 2013-05-02 | DOI: https://doi.org/10.1515/form.2011.130


Fintushel and Stern have proved that if is a symplectic surface in a symplectic 4-manifold such that has simply-connected complement and nonnegative self-intersection, then there are infinitely many topologically equivalent but smoothly distinct embedded surfaces homologous to . Here we extend this result to include symplectic surfaces whose self-intersection is bounded below by , where g is the genus of .

We make use of tools from Heegaard Floer theory, and include several results that may be of independent interest. Specifically we give an analogue for Ozsváth–Szabó invariants of the Fintushel–Stern knot surgery formula for Seiberg–Witten invariants, both for closed 4-manifolds and manifolds with boundary. This is based on a formula for the Ozsváth–Szabó invariants of the result of a logarithmic transformation, analogous to one obtained by Morgan–Mrowka–Szabó for Seiberg–Witten invariants, and the results on Ozsváth–Szabó invariants of fiber sums due to the author and Jabuka. In addition, we give a calculation of the twisted Heegaard Floer homology of circle bundles of “large” degree over Riemann surfaces.

Keywords: Knotted surfaces; Heegaard Floer homology; gauge theory

About the article

Received: 2010-05-19

Revised: 2011-01-21

Published Online: 2013-05-02

Published in Print: 2013-05-01

Citation Information: Forum Mathematicum, Volume 25, Issue 3, Pages 597–637, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/form.2011.130.

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© 2013 by Walter de Gruyter Berlin Boston.Get Permission

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