---
tag: Y2S9GWG1J1
kind: concept
name: Homotopy principle
msc: [58A20, 57R42]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_principle
same_as:
  wikidata: Q5891835
  wikipedia: Homotopy principle
---

## Summary

The homotopy principle, or h-principle, is a property of a partial differential relation.
It says that every formal solution can be deformed through formal solutions into a genuine solution.
A formal solution assigns to each point a value and candidate derivatives that satisfy the relation pointwise.
It does not require the candidates to be the true derivatives of the values.
When the h-principle holds, the existence and classification of solutions reduce to a problem in homotopy theory.
It was formulated by Mikhail Gromov, building on results of Smale, Hirsch and Nash.
It covers immersions, many symplectic and contact problems, and isometric $C^1$ embeddings.

## Explanations

### L1 human

Many geometric problems ask for a map whose derivative obeys some condition, for example a map whose derivative is never zero.
The easy part is to choose, at each point, a value and a matrix that obeys the condition, with no link between the two.
The hard part is to find a map whose real derivative obeys it.
The h-principle says that whenever the easy problem can be solved, so can the hard one, and up to deformation in the same number of ways.
A famous consequence is that a sphere can be turned inside out through immersions.