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Fundamentals Of Differential Geometry

Consider a manifold, of dimension greater than 3, equipped with a bracket-generating distribution. In this article we prove complete h-principles for embedded regular horizontal curves and for embedded transverse curves. These results contrast with the 3-dimensional contact case, where the full h-principle for transverse/legendrian knots is known not to hold. We also prove analogous statements for immersions, with no assumptions on the ambient dimension.

Classification of tangent and transverse knots in bracket-generating distributions

It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was also claimed there without a proof that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on $S^3$, fixed at a point, is contractible. We prove this result in the current paper.

The space of tight contact structures on ${\mathbb R}^3$ is contractible

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 $K(G,1)\times\U(2)$, 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.

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I

. In every Engel manifold we construct an inﬁnite family of pairwise non-isotopic transverse tori that are all smoothly isotopic. To distinguish the transverse tori in the family we introduce a homological invariant of transverse tori that is similar to the self-linking number for transverse knots in contact 3-manifolds. Analogous results are presented for Legendrian tori in even contact 4-manifolds.

/pdf/non-isotopic-transverse-tori-in-engel-manifolds-23ejx03n.pdf

Non-isotopic transverse tori in Engel manifolds

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at [Pi]1 –level.

/pdf/fundamental-groups-of-formal-legendrian-and-horizontal-z1mleaexwy.pdf

Fundamental groups of formal Legendrian and horizontal embedding spaces

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of non trivial loops of Legendrian embeddings are already non trivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that this computes the fundamental group of the space of horizontal embeddings as well.

/pdf/fundamental-groups-of-formal-legendrian-and-horizontal-yolo0078oy.pdf

We study homotopically non-trivial spheres of Legendrians in the standard contact ${\mathbb {R}}^3$ and ${\mathbb {S}}^3$. We prove that there is a homotopy injection of the contactomorphism group of ${\mathbb {S}}^3$ 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.

Loops of Legendrians in contact 3-manifolds

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Papers

Fundamental groups of formal Legendrian and horizontal embedding spaces

Fundamental groups of formal Legendrian and horizontal embedding spaces

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I

Loops of Legendrians in contact 3-manifolds