The fundamental group of the space of contact structures on the $3$-torus

Geiges, Hansjoerg; Klukas, Mirko

doi:10.4310/mrl.2014.v21.n6.a3

Cited by 6 publications

(4 citation statements)

References 9 publications

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“…Bourgeois reproved this using contact homology in [Bou06, Proposition 2]. Then Geiges and Klukas in [GK14] proved the theorem when and .…”

Section: Introductionmentioning

confidence: 99%

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On the contact mapping class group of Legendrian circle bundles

Giroux¹,

Massot²

2017

Compositio Math.

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In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in [Gir01b]. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.

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“…Bourgeois reproved this using contact homology in [Bou06, Proposition 2]. Then Geiges and Klukas in [GK14] proved the theorem when and .…”

Section: Introductionmentioning

confidence: 99%

“…Our main result is the following theorem, in which V d is endowed with any principal circle bundle structure inherited from one on V = T * 1 S-the case g = 1 = d was previously treated by H. Geiges and M. Klukas in [GK14]:…”

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confidence: 99%

On the contact mapping class group of Legendrian circle bundles

Giroux¹,

Massot²

2017

Compositio Math.

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“…In particular, one can extract information about G(X) through the study of K 0 (X). For instance, this has been exploited in [AM, Bu, Kr, Mc] for the symplectic case and studied in [CP,GK,GG1] in the contact case. The space K 0 (X) can be analyzed through the evaluation map at a point with the use of the space of linear almost complex structures.…”

Section: Homogeneous Indicesmentioning

confidence: 99%

Higher Maslov indices

Casals

Ginzburg

Presas

2017

Journal of Geometry and Physics

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Abstract. We define Maslov-type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of Z. These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these indices and their applications.

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“…For instance, one can find in the literature several examples of contact manifolds (V, ξ) for which ker (j * | π0 ) is non-trivial; the interested reader can consult [2,5,11,12,[17][18][19] and [6,23] for, respectively the tight and overtwisted 3-dimensional cases, and [2,20,22] and [15] for, respectively, the tight and overtwisted higher-dimensional cases. Notice that the examples in [2,20,22] are tight according to the definition of overtwistedness in higher dimensions in [1], which generalizes the 3-dimensional one given in [7].…”

Section: Introductionmentioning

confidence: 99%