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Register a new userLoops of Legendrians in contact 3-manifolds
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Javier Mart\\'inez-Aguinaga
Publication date
2018
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English
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We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.
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We prove that Besse contact forms on closed connected 3-manifolds, that is, contact forms with a periodic Reeb flow, are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.
We prove that that the homotopy type of the path connected component of the identity in the contactomorphism group is characterized by the homotopy type of the diffeomorphism group plus some data provided by the topology of the formal contactomorphism space. As a consequence, we show that every connected component of the space of Legendrian long knots in $R^3$ has the homotopy type of the corresponding smooth long knot space. This implies that any connected component of the space of Legendrian embeddings in $NS^3$ is homotopy equivalent to the space $K(G,1)timesU(2)$, with $G$ computed by A. Hatcher and R. Budney. Similar statements are proven for Legendrian embeddings in $R^3$ and for transverse embeddings in $NS^3$. Finally, we compute the homotopy type of the contactomorphisms of several tight $3$-folds: $NS^1 times NS^2$, Legendrian fibrations over compact orientable surfaces and finite quotients of the standard $3$-sphere. In fact, the computations show that the method works whenever we have knowledge of the topology of the diffeomorphism group. We prove several statements on the way that have interest by themselves: the computation of the homotopy groups of the space of non-parametrized Legendrians, a multiparametric convex surface theory, a characterization of formal Legendrian simplicity in terms of the space of tight contact structures on the complement of a Legendrian, the existence of common trivializations for multi-parametric families of tight $R^3$, etc.
We provide examples of contact manifolds of any odd dimension $geq 5$ which are not diffeomorphic but have exact symplectomorphic symplectizations.
We prove that closed connected contact manifolds of dimension $geq 5$ related by an h-cobordism with a flexible Weinstein structure become contactomorphic after some kind of stabilization. We also provide examples of non-conjugate contact structures on a closed manifold with exact symplectomorphic symplectizations.
Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in $S^{2n-1}_{text{std}}$. Loose Legendrians are a class of Legendrian spaces which satisfy an $h$--principle, meaning that their geometric classification is in bijective correspondence with their topological types. For the particular case of the linear arboreal singularities, we show that constructable sheaves suffice to detect whether any closed set of an arboreal link is loose.
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