Rounding corners of polygons and the embedded contact homology of<i>T</i><sup>3</sup>
Сглаживание углов многоугольников и встроенная контактная гомология T³
2006-03-26
SCID: 54.1/rbjuaew4
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3-toruscombinatorial chain complexembedded contact homologylattice polygonssymplectic field theory
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Abstract (AI)
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 Ozsvth-Szab Floer homology HF C .T 3 /.
Key Findings
1
The chain complex differential is defined through a geometric operation called rounding corners of polygons.
2
The embedded contact homology of T^3 is modeled by a combinatorial chain complex generated by labeled convex lattice polygons.
3
The homology of this combinatorial complex is computed explicitly.
4
The resulting homology agrees with the Ozsváth–Szabó Floer homology of T^3.
5
This provides a combinatorial computation of ECH, which ordinarily counts embedded holomorphic curves in a symplectization.
Research Object
embedded contact homology of the 3-torus T^3
Research Subject
the combinatorial chain-complex computation of ECH using labeled convex lattice polygons and a corner-rounding differential, and its resulting homology
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2006-03-26
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