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CONTACT HOMOLOGY OF GOOD TORIC CONTACT MANIFOLDS MIGUEL ABREU AND LEONARDO MACARINI
 

Summary: CONTACT HOMOLOGY OF GOOD TORIC CONTACT MANIFOLDS
MIGUEL ABREU AND LEONARDO MACARINI
Abstract. In this paper we show that any good toric contact manifold has well defined
cylindrical contact homology and describe how it can be combinatorially computed from the
associated moment cone. As an application we compute the cylindrical contact homology of
a particularly nice family of examples that appear in the work of Gauntlett-Martelli-Sparks-
Waldram on Sasaki-Einstein metrics. We show in particular that these give rise to a new
infinite family of non-equivalent contact structures on S2
× S3
in the unique homotopy class
of almost contact structures with vanishing first Chern class.
1. Introduction
Contact homology is a powerful invariant of contact structures, introduced by Eliashberg,
Givental and Hofer [14] in the bigger framework of Symplectic Field Theory. Its simplest
version is called cylindrical contact homology and can be briefly described in the following
way. Let (N, ) be a closed (i.e. compact without boundary) co-oriented contact manifold,
1(N) a contact form ( = ker ) and R X(N) the corresponding Reeb vector
field ((R)d 0 and (R) 1). Assume that is nondegenerate, with the meaning
that all contractible closed orbits of R are nondegenerate. Consider the graded Q-vector
space C(N, ) freely generated by the contractible closed orbits of R, where the grading

  

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

 

Collections: Mathematics