# nLab h-principle

Gromovs homotopy principle or h-principle

# Gromov’s homotopy principle or h-principle

## Definition

Let $p \colon X\to B$ be a smooth fiber bundle and write $X^{(r)}$ for the $r$-jets of sections of $p$; there is an induced projection $X^{(r)} \to B$ which makes it into a smooth bundle.

One considers a fixed partial differential relation (see differential equation) which is a subset $R\subset X^{(r)}$. One is interested in finding formal sections $s$ of $X^{(r)}\to B$ whose image is in $R$, and which are, in addition, holonomic sections, i.e. equal to the $r$-jet $J_f^r$ of a $C^r$-section $f$ of $p:X\to B$.

One says that $R$ satisfies the h-principle if every formal section of $R$ is homotopic to a holonomic section within the appropriate topological space of formal sections.

## Applications and examples

For example, in the case of holomorphic fiber bundles $R$ may be the Cauchy-Riemann relation, thus the question is if there is a deformation of a continuous section into a holomorphic one. In other words, one wants to reduce the problem of existence of maps of certain type to the analogous topological problem. Applications include results on the spaces of immersions, submersions, k-mersions, holomorphic maps, symplectic and isometric embeddings, contact structures and so on.

Mikhail Gromov demonstrated the result that if $B$ is an open $n$-manifold, and if $R$ is open as a subspace of $X^{(r)}$ and is $Diff(B)$-invariant, then $R$ satisfies the $h$-principle (EliasMisha 01, sec 2.1, Haefliger 71, p. 128)

## General methods

Gromov introduced many techniques of proving the h-principle including the method of microflexible/continuous sheaves, the methods of convex integration and the removal of singularities.

## References

Review:

• Y. Eliashberg, N. Mishachev, Introduction to the h-principle, Graduate Studies in Mathematics 48. Amer. Math. Soc. 2002 gBooks

• Andrés Angel, Johannes Ebert, Gromov’s h-principle and its applications research seminar, 2010, outline pdf (with useful bibliography)

Wikipedia: homotopy principle

Original articles:

• Mikhail Gromov, Partial differential relations, Ergebn. Math. Grenzgeb. (3), 9, Springer (1986) (doi:10.1007/978-3-662-02267-2)

• Mikhail Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Proc. Int. Congress Math. Nice 1970, vol. 2, 221-225 (djvu w OCR, pdf)

• Mikhail Gromov, Papers on h-principle: geometric methods for solving partial differential equations/inequalities and the homotopy structure in the spaces of their solutions, list (with some links)

• André Haefliger, Lectures on the Theorem of Gromov, Proceedings of Liverpool Singularities Symposium, II (1969/1970), p. 128-141. Lecture Notes in Math. 209, Springer, Berlin, 1971 (pdf).

• John Francis, The h-principle in topology (overview pdf)