Volume 10, issue 1 (2006)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Rounding corners of polygons and the embedded contact homology of $T^3$

Michael Hutchings and Michael G Sullivan

Geometry & Topology 10 (2006) 169–266

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 T3 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 HF+(T3).

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/