The homotopy type of the contactomorphism group of a contact 3-manifold

Abstract

We show that the homotopy type of any connected component of the contactomorphism groupof a tight contact 3-manifold is characterized by the homotopy type of the di↵eomorphismgroup plus some data provided by the topology of the formal contactomorphism space. As aconsequence, we show that every connected component of the space of Legendrian long knotsin R3 has the homotopy type of the underlying smooth long knot space. This implies that anyconnected component of the space of Legendrian embeddings in S3 is homotopy equivalentto the space K(G, 1) ⇥ U(2), with G computed by A. Hatcher and R. Budney. Similarstatements are proven for Legendrian embeddings in R3 and for transverse embeddings inS3. We compute the homotopy type of the contactomorphism group of several tight 3-folds:S1 ⇥ S2, Legendrian fibrations over compact orientable surfaces and finite quotients of thestandard 3-sphere. In fact, the computations show that the method works whenever we haveknowledge of the topology of the di↵eomorphism group. We prove several statements on theway that have interest by themselves: the computation of the homotopy groups of the space ofnon-parametrized Legendrians, a multiparametric convex surface theory, a characterizationof formal Legendrian simplicity in terms of the space of tight contact structures on thecomplement of a Legendrian and the existence of common trivializations for multi-parametricfamilies of tight R3. We also compare the situation with the overtwisted case. Finally, weapply contact topological methods to study the topology of the space of parametrized smoothknots in R4...