Rounding corners of polygons and the embedded contact homology of T3

Hutchings, Michael; Sullivan, Michael G

doi:10.2140/gt.2006.10.169

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Rounding corners of polygons and the embedded contact homology of $T^3$

Michael Hutchings and Michael G Sullivan

Geometry & Topology 10 (2006) 169–266

DOI: 10.2140/gt.2006.10.169

arXiv: math.SG/0410061

Abstract

The embedded contact homology (ECH) of a 3–manifold with a contact form is a variant of Eliashberg–Givental–Hofer’s symplectic field theory, which counts certain embedded $J$ –holomorphic curves in the symplectization. We show that the ECH of $T^{3}$ is computed by a combinatorial chain complex which is generated by labeled convex polygons in the plane with vertices at lattice points, and whose differential involves “rounding corners”. We compute the homology of this combinatorial chain complex. The answer agrees with the Ozsváth–Szabó Floer homology $H F^{+} (T^{3})$ .

Keywords

embedded contact homology, Floer homology

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Publication

Received: 5 October 2004

Accepted: 25 January 2006

Published: 26 March 2006

Proposed: Robion Kirby

Seconded: Peter Ozsváth, Tomasz Mrowka

Authors

Michael Hutchings
Department of Mathematics University of California Berkeley CA 94720 USA
http://math.berkeley.edu/~hutching/
Michael G Sullivan
Department of Mathematics and Statistics University of Massachusetts Amhurst MA 01003-9305 USA
http://www.math.umass.edu/~sullivan/

Volume 10, issue 1 (2006)