Publications

2012

• Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization
  
  • Monzner Alexandra
  • Vichery Nicolas
  • Zapolsky Frol
  Journal of modern dynamics, American Institute of Mathematical Sciences, 2012, pp.205-249. For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compact support on T^*N. These satisfy properties analogous to those of partial quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.
• On the topology of fillings of contact manifolds and applications.
  
  • Oancea Alexandru
  • Viterbo Claude
  Commentarii Mathematici Helvetici, European Mathematical Society, 2012, pp.41-69. The aim of this paper is to address the following question: given a contact manifold $(\Sigma, \xi)$, what can be said about the aspherical symplectic manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$ ? We first extend a theorem of Eliashberg, Floer and McDuff to prove that under suitable assumptions the map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We then apply this method in the case of contact manifolds having a contact embedding in $ {\mathbb R}^{2n}$ or in a subcritical Stein manifold. We prove in many cases that the homology of the fillings is uniquely determined. Finally we use more recent methods of symplectic topology to prove that, if a contact hypersurface has a Stein subcritical filling, then all its weakly subcritical fillings have the same homology. A number of applications are given, from obstructions to the existence of Lagrangian or contact embeddings, to the exotic nature of some contact structures.