The h-principles fails for preglendrian in corank 2 fat distributions

A central question in the topology of distributions is whether there exist maximally non-integrable distributions beyond contact structures for which the h-principle fails, i.e. whether there are genuinely new invariants not dictated by algebraic topology. Most known classes instead satisfy strong flexibility results and are classified by their obvious homotopical data. In this talk I will briefly survey this landscape and then present a recent joint result with Álvaro del Pino and Eduardo Fernández giving a first example of rigidity beyond the contact world: a class of fat (corank- 2) distributions that naturally generalise contact structures, together with their adapted submanifolds, the prelegendrians. Using a canonical contactisation, we construct families of prelegendrians detected by contact-topological invariants, thereby showing that the ℎ h-principle fails in this setting.