Homotopy principle

Remarks

Every genuine solution gives a formal solution, so the h-principle says that the only obstruction to solving $\mathcal{R}$ is the homotopy-theoretic obstruction to finding a formal solution. Stronger versions ask that the inclusion of holonomic solutions into formal solutions be a weak homotopy equivalence; the parametric h-principle is of this kind. Example: let $\dim M < \dim N$ and let $\mathcal{R} \subseteq J^1(M, N)$ be the set of $1$-jets whose linear part is injective. Genuine solutions are immersions, formal solutions are fibrewise injective bundle maps $TM \to TN$, and the Smale–Hirsch theorem shows that $\mathcal{R}$ satisfies the h-principle. Non-example: for $M = S^1$, $N = \R$ and the analogous relation of injective $1$-jets, the section $F(x) = (0, \text{the linear map } v \mapsto v)$, using the standard trivialisation of $TS^1$, is a formal solution. But there is no genuine solution, because a smooth function on the compact circle has a maximum, where its derivative is zero; so the h-principle fails.