We compute the homotopy type of any connected component of the space of tight contact structures on a 3-fold. In fact, we actually prove a partial h-principle for the inclusion of the contactomorphism group into the diffeomorphism group. The basic building block is the homotopy equivalence induced by the inclusion of the contactomorphism group of the sphere relative to a point and the diffeomorphism group relative to a point result recently proven by Elisahberg and Mishachev.
Then, we wonder how these sets of techniques work for overtwisted manifolds? i.e. we just try to prove the same theorem than in the tight case, assuming that the triangulation is very small, we easily obtain that all the cells are tight and then, everything looks like working, so however, there must be something wrong because we find several contradictions: the overtwisted mirage. Once the mistake is understood, we proceed to compute the homotopy type of the space of contact structures/contactomorphisms by using just Mishachev-Eliiashberg result, i.e. we reprove the 3-dimensional overtiwsted h-principle as a corollary. We will compute the space of embeddings of overtwisted disks in some particular manifolds. Finally we end by explaining the conjecture tight overtwisted.
This is joint work with Dahyana Farias, Eduardo Fernández and Xabi Martínez
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
