The Gromov weak homotopy equivalence principle

The h-principle is a general homotopic way to solve partial differential equations and, more generally, partial differential relations. The theory was started by Y. Eliashberg, M. Gromov and A. V. Philips and it allows one to reduce a differential topological problem to an algebraic topological problem. A way to prove the h-principle is by Convex Integration Theory. Developed originally by Gromov, it is applied to solve relations in jet spaces, including certain classes of undetermined non-linear systems of partial differential equations. The h-principle occurs for instance in immersion problems, isometric immersion problems and other areas. A counter-intuitive result which can be proved by applying the h-principle is the sphere eversion without creasing or tearing. This thesis consists of three parts. In the first chapter we introduce the concept of fiber bundle, which is a space that is locally a product space but globally may have a different topological structure, and the concept of jet bundle, a construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. In the second chapter we develop the Gromov Convex Integration Theory that provides the main general topological method for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDEs theory and to optimal control theory. Finally, in the last chapter, we examine some relations for which the h-principle holds.