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Questions tagged [h-principle]

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0answers
140 views

h-principle for pairs

Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \...
7
votes
1answer
135 views

Short embeddings for open manifolds and dimension reduction of sets

The question is maybe a bit technical, but I find the related construction very beautiful. In the very famous work - "$C^1$-isometric imbeddings" by J.Nash (1954) the author presented the ...
7
votes
1answer
121 views

Immersions of manifolds with boundary (regular homotopy classes, h-principle)

Have regular homotopy classes of immersions of manifolds with boundary been studied? i.e. immersions $(M, \partial M) \looparrowright (N, \partial N)$. The references I've looked at from this question ...
2
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0answers
89 views

Wrinkling smooth functions

I am interested in applying a result from the work by Eliashberg and Mishachev on wrinkling. Namely, in their first paper on wrinkling, they prove Theorem 1.6 B (Theorem 1.6 A is a non-parameterized ...
4
votes
0answers
202 views

H-principle for smoothing

I'm trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically. It's not hard (e.g. using the methods in Hartshorne-...
4
votes
1answer
171 views

Almost complex structures on a 4-ball that are not tamed

Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...
5
votes
0answers
135 views

h-principle on Hilbert manifolds

Gromov's h-principle is a powerful tool in studying various geometric structure on open, finite-dimensional manifolds. Is there any generalization of h-principle to (necessarily open) infinite-...
2
votes
1answer
369 views

Question about the h-principle

So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions ...
3
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0answers
187 views

The relative h-principle and extension problems

As a beginner for h-principles, I want to know why the relative h-principle cannot imply a positive solution to the problems for extending symplectic structures. The following is a relative h-...
24
votes
0answers
624 views

Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
27
votes
5answers
4k views

H-principle and PDE's

According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations ...
1
vote
1answer
316 views

Holomorphic h-principle for compact manifolds

The Oka principle for Stein manifolds says (roughly) that the only obstructions for "things" are topological obstructions (for instance every smooth complex vector bundle admits a holomorphic ...