Homotopy Principle - Rough Idea

Rough Idea

Assume we want to find a function ƒ on Rm which satisfies a partial differential equation of degree k, in co-ordinates . One can rewrite it as

where stands for all partial derivatives of ƒ up to order k. Let us exchange every variable in for new independent variables Then our original equation can be thought as a system of

and some number of equations of the following type

A solution of

is called a non-holonomic solution, and a solution of the system (which is a solution of our original PDE) is called a holonomic solution.

In order to check whether a solution exists, first check if there is a non-holonomic solution (usually it is quite easy and if not then our original equation did not have any solutions).

A PDE satisfies the h-principle if any non-holonomic solution can be deformed into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a holonomic solution. The topological problem of finding a non-holonomic solution is much easier to handle and can be addressed with the obstruction theory for topological bundles.

Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded Lagrangians in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from pseudo-holomorphic curves.