GT Seminar with Álvaro del Pino Gomez (Utrecht university)

Abstract: A distribution is maximally non-integrable if its sheaf of tangent vector fields has "as many non-trivial Lie brackets as possible". This condition defines a partial differential relation on the space of distributions that turns out to be rather non-trivial to tackle.

In this talk I will focus on distributions of rank 2. In this case, the maximal non-integrability condition is beyond the applicability of the classic h-principle approaches due to Gromov. However, the 3-dimensional situation (i.e. for 3-dimensional contact structures) was addressed successfully by Eliashberg in 1989 using a so-called removal of singularities approach. He proved that there is a class of 3-dimensional contact structures, called overtwisted, that can be completely classified in homotopical terms.

In 2018, Vogel and myself proved an analogous statement in dimension 4 (i.e. for so-called Engel structures). The proof we gave recovers Eliashberg's statement as well, suggesting that Engel and contact overtwistedness are concrete incarnations of a more general phenomenon.

In this talk I will discuss the following conjecture: maximally non-integrable distributions of rank-2 admit an overtwisted class that satisfies the complete h-principle. My goal will be to outline a strategy of proof generalising the 4-dimensional approach. The crucial technical step is a flexibility statement for submanifolds transverse to a bracket-generating distribution.

This is on-going work with F. ter Haar and F.J Martínez-Aguinaga.

See upcoming seminars in our seminar series on Geometry and Topology (GT).