The notion of homotopy principle or $h$-principle is one of the key concepts in an elegant language developed by Gromov to deal with a host of questions in geometry and topology. Roughly speaking, for a certain differential geometric problem to satisfy the $h$-principle is equivalent to saying that a solution to the problem exists whenever certain obvious topological obstructions vanish.

The foundational examples for applications of Gromov's ideas include

• (i) Hirsch-Smale immersion theory,
• (ii) Nash-Kuiper $C^1$–isometric immersion theory,
• (iii) existence of symplectic and contact structures on open manifolds.

Gromov has developed several powerful methods that allow one to prove $h$-principles. These notes, based on lectures given in the Graduiertenkolleg of Leipzig University, present two such methods which are strong enough to deal with applications (i) and (iii).

Readership

Graduate students and research mathematicians interested in geometry and topology.