A Smale Type Result and Applications to Contact Homology

Vittorio Martino


In this note we will show that the injection of a suitable subspace of the space of Legendrian loops into the full loop space is an S1-equivariant homotopy equivalence. Moreover, since the smaller space is the space of variations of a given action functional, we will compute the relative Contact Homology of a family of tight contact forms on the three-dimensional torus.


Contact Morse Homology; Legendrian loops

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A. Bahri, Compactness, Adv. Nonlinear Stud. 8 (2008), no. 3, 465-568

A.Bahri, Homology computation, Adv. Nonlinear Stud. 8 (2008), no. 1, 1-7

A. Bahri, Pseudo-orbits of contact forms, Pitman Research Notes in Mathematics Series 173, Longman Scientific and Technical, Longman, London, 1988

A. Bahri, A Lagrangian Method for the Periodic Orbit Problem of Reeb Vector-Fields, Geometric Methods in PDE's, 1-19, Lect. Notes Semin. Interdiscip. Mat., 7, Semin. Interdiscip. Mat. (S.I.M.), Potenza, 2008

A. Bahri, Classical and Quantic Periodic Motions of Multiply Polarized Spin-particles, Pitman Research Notes in Mathematics Series, 378. Longman, Harlow, 1998

A. Bahri, Flow lines and algebraic invariants in contact form geometry, Progress in Nonlinear Differential Equations and their Applications, 53. Birkhuser Boston, Inc., Boston, MA, 2003

A. Bahri, Classical and quantic periodic motions of multiply polarized spin-particles, Pitman Research Notes in Mathematics Series, 378. Longman, Harlow, 1998

A. Bahri; Y. Xu, Recent progress in conformal geometry, ICP Advanced Texts in Mathematics, 1. Imperial College Press, London, 2007

F. Bourgeois, A Morse-Bott approach to Contact Homology, in "Symplectic and Contact Topology: Interactions and Perspectives", Fields Institute Communications 35 (2003), 55-77

F. Bourgeois, V. Colin, Homologie de contact des variétés toroïdales, Geometry and Topology 9 (2005), 299-313

Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), no. 3, 623-637

E. Giroux, Une infinit de structures de contact tendues sur une infinit de varits, Invent. Math. 135 (1999), no. 3, 789-802

H. Geiges; J. Gonzalo, Contact geometry and complex surfaces, Invent. Math. 121 (1995), no. 1, 147-209

J. Gonzalo, A dynamical characterization of contact circles, Geom. Dedicata 132 (2008), 105-119

J. Gonzalo, F.Varela, Modèles globaux des variétés de contact, Third Schnepfenried geometry conference, Vol.1 (Schnepfenried, 1982), Astérisque, no.107-108, pp. 163-168, Soc. Math.France, Paris, 1983

Y. Kanda, The classification of tight contact structures on the 3-torus, Comm. Anal. Geom. 5 (1997), 413-438

E.B. Lebow, Embedded contact homology of 2-torus bundles over the circle. Thesis (Ph.D.), University of California, Berkeley. 2007. 165 pp

A. Maalaoui, V. Martino. The topology of a subspace of the Legendrian curves on a closed contact 3-manifold, Advanced Nonlinear Studies, 14 (2014), 393-426

A. Maalaoui, V. Martino. Homology computation for a class of contact structures on T3, Calculus of Variations and Partial Differential Equations, 50 (2014), 599-614

V. Martino, A Legendre transform on an exotic S3, Advanced Nonlinear Studies, 11 (2011), 145-156

S. Smale, Regular curves on Riemannian manifolds. Trans. Amer. Math. Soc. 87 (1958), 492-512

DOI: 10.6092/issn.2240-2829/4718


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