In the 1950s, Smale and Hirsch proved that the space of immersions of an m-dimensional manifold into an n-dimensional manifold is homotopy equivalent, as long as m < n, to the space of monomorphisms between the tangent spaces. Any statement of this form (i.e. a comparison theorem between a space of geometric structures and an associated space that is purely algebraic topological in nature), is known as a homotopy principle, or h-principle.

Later on, in the late 60s and early 70s, Gromov developed (or generalised) various techniques capable of proving h-principles. Since then, these ideas have been impactful in the study of many geometric structures (including immersions, submersions, foliations, symplectic structures, contact structures, embeddings, and Riemannian metrics).

The goal of the talk will be to sketch some of these techniques and state some consequences in the homotopical study of tangent distributions (i.e., subbundles of the tangent bundle).