Rounding corners of polygons and the embedded contact homology of T3

Hutchings, Michael¹ ; Sullivan, Michael G²

¹ Department of Mathematics, University of California, Berkeley CA 94720, USA
² Department of Mathematics and Statistics, University of Massachusetts, Amhurst MA 01003-9305, USA

Geometry & topology, Tome 10 (2006) no. 1, pp. 169-266.

Résumé

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 $H{F}^{+}\left(T^3\right)$.

Détail

DOI : 10.2140/gt.2006.10.169

Keywords: embedded contact homology, Floer homology

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