The H-principle for Ample Relations

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

The H -principle for Ample Relations

Vincent Borrelli

Université Lyon 1

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Ample Relations

Definition.– A subsetA⊂Rⁿisampleif for everya∈Athe interior of the convex hull of the connected component to whichabelongs isRⁿi. e.:IntConv(A,a) =Rⁿ(in particular A=∅is ample).

Ais not ample Ais ample Ais not ample.

Example.– The complement of a linear subspaceF ⊂Rⁿ is ample if and only if CodimF ≥2.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Ample Relations

Definition.– A subsetA⊂Rⁿisampleif for everya∈Athe interior of the convex hull of the connected component to whichabelongs isRⁿi. e.:IntConv(A,a) =Rⁿ(in particular A=∅is ample).

Ais not ample Ais ample Ais not ample.

Example.– The complement of a linear subspaceF ⊂Rⁿ is ample if and only if CodimF ≥2.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Ample Relations

Definition.– LetE =P×Rⁿ−→^π P be a fiber bundle, a subsetR ⊂E is said to beampleif, for everyp∈P, R_p :=π⁻¹(p)∩ Ris ample inRⁿ.

Remark.– IfR ⊂E is ample andz :P −→E is a section, then, for everyp∈P,the conditionz(p)∈Conv(R_p, σ(p)) necessarily holds.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

One Jet Space

•The1-jet space of maps

J¹(M,N) ={(x,y,L)|x ∈M,y ∈N,L∈ L(T_xM,TyN)}

is a natural fiber bundle overM×N

L(T_xM,T_yN)−→J¹(M,N)−→^p M×N and overM

L(R^m,Rⁿ)×N−→J¹(M,N)−→^π M

•Adifferential relationof order 1 is a subset R ⊂J¹(M,N).

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

One Jet Space

•The1-jet space of maps

J¹(M,N) ={(x,y,L)|x ∈M,y ∈N,L∈ L(T_xM,TyN)}

is a natural fiber bundle overM×N

L(T_xM,T_yN)−→J¹(M,N)−→^p M×N and overM

L(R^m,Rⁿ)×N−→J¹(M,N)−→^π M

•Adifferential relationof order 1 is a subset R ⊂J¹(M,N).

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Ample Relations in J

(M, N)

•Locally, we identifyJ¹(M,N)with

J¹(U,V) = U × V × L(R^m,Rⁿ) =U × V ×Qm i=1Rⁿ.

= {(x,y,v₁, ...,v_m)}

whereU andV are charts ofMandN.

•We set :

J¹(U,V)^⊥:={(x,y,v₁, ...,v_m−1)}.

•We have

R_U,V −→ J¹(U,V)

↓p^⊥ J¹(U,V)^⊥.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Ample Relations in J

(M, N)

•Letz∈J¹(U,V)^⊥,we set

R^⊥_z = (p^⊥)⁻¹(z)∩ R_U,V.

• R^⊥is a differential relation of the bundle

J¹(U,V) ^p

⊥

−→J¹(U,V)^⊥.

Definition. – A differential relationR ⊂J¹(M,N)isampleif for every local identificationJ¹(U,V)and for every

z∈J¹(U,V)^⊥,the spaceR^⊥_z is ample in(p^⊥)⁻¹(z)'Rⁿ.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Ample Relations in J

(M, N)

Example. –The differential relationI of immersions from M^m to Nⁿis ample if n>m.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Formal solutions

•Aformal solutionof a differential relationR ⊂J¹(M,N) is any sectionσ∈Γ(R).

•AsolutionofRis a mapf :M−→Nsuch thatj¹f ∈Γ(R). We denote bySol(R)thespace of solutionsofR.

•The natural inclusion

J : C¹(M,N) −→ J¹(M,N) f 7−→ j¹f. induces a map

J :Sol(R)−→Γ(R).

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Formal solutions

•Aformal solutionof a differential relationR ⊂J¹(M,N) is any sectionσ∈Γ(R).

•AsolutionofRis a mapf :M−→Nsuch thatj¹f ∈Γ(R).

We denote bySol(R)thespace of solutionsofR.

•The natural inclusion

J : C¹(M,N) −→ J¹(M,N) f 7−→ j¹f. induces a map

J :Sol(R)−→Γ(R).

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Formal solutions

•Aformal solutionof a differential relationR ⊂J¹(M,N) is any sectionσ∈Γ(R).

•AsolutionofRis a mapf :M−→Nsuch thatj¹f ∈Γ(R).

We denote bySol(R)thespace of solutionsofR.

•The natural inclusion

J : C¹(M,N) −→ J¹(M,N) f 7−→ j¹f. induces a map

J :Sol(R)−→Γ(R).

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

H-principle

Definition.– A differential relationRsatisfies the h-principleif every formal solutionσ:M −→ Ris homotopic inΓ(R)to the 1-jet of a solution ofR.

This is asking the mapJ to induce an onto mapping between the components ofSol(R)andΓ(R)

π₀J :π₀Sol(R)π₀Γ(R). Definition.– A differential relationRsatisfiesthe parametrich-principleif the map

J :Sol(R)−→Γ(R) is a weak homotopy equivalence.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

H-principle

Definition.– A differential relationRsatisfies the h-principleif every formal solutionσ:M −→ Ris homotopic inΓ(R)to the 1-jet of a solution ofR.

This is asking the mapJ to induce an onto mapping between the components ofSol(R)andΓ(R)

π₀J :π₀Sol(R)π₀Γ(R).

Definition.– A differential relationRsatisfiesthe parametrich-principleif the map

J :Sol(R)−→Γ(R) is a weak homotopy equivalence.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

H-principle

Definition.– A differential relationRsatisfies the h-principleif every formal solutionσ:M −→ Ris homotopic inΓ(R)to the 1-jet of a solution ofR.

This is asking the mapJ to induce an onto mapping between the components ofSol(R)andΓ(R)

π₀J :π₀Sol(R)π₀Γ(R).

Definition.– A differential relationRsatisfiesthe parametrich-principleif the map

J :Sol(R)−→Γ(R) is a weak homotopy equivalence.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

H -principle for Ample Relations

Theorem (Gromov 69-73). –LetR ⊂J¹(M,N)be an open and ample differential relation. ThenRsatisfies the

parametric h-principle i. e.

J :Sol(R)−→Γ(R) is a weak homotopy equivalence.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Immersions

Smale Paradox (1958).– The parametrich-principle holds for the differential relation of immersions ofM^m intoNⁿwith n>m.A homotopic computation shows that ifM^m=S²and Nⁿ=R³then

π₀(I(S²,R³)) =π₂(Gl₊(3,R)) =0.

Thus there is only one class of immersions of the sphere inside the three dimensional space and in particular, the sphere can be everted among immersions.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Eversion of the Sphere

Thurston eversion, 1994

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Thurston Corrugations

•Thurston eversion introduces oscillations to avoid the creation of singular points.

•These oscillations, called by Thurstoncorrugations, soon proved to be a powerful tool to generate regular homotopies between two given immersions.

•Thurston corrugations are quite similar to the oscillations created by a convex integration. In fact, it now appears that the Theory of Corrugations can be seen as a simplified version of the Convex Integration Theory (all quantitative aspects are ignored).

•It is likely that, at that time, Thurston was unaware of the Convex Integration Theory.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Thurston Corrugations

•Thurston eversion introduces oscillations to avoid the creation of singular points.

•These oscillations, called by Thurstoncorrugations, soon proved to be a powerful tool to generate regular homotopies between two given immersions.

•Thurston corrugations are quite similar to the oscillations created by a convex integration. In fact, it now appears that the Theory of Corrugations can be seen as a simplified version of the Convex Integration Theory (all quantitative aspects are ignored).

•It is likely that, at that time, Thurston was unaware of the Convex Integration Theory.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Thurston Corrugations

•Thurston eversion introduces oscillations to avoid the creation of singular points.

•These oscillations, called by Thurstoncorrugations, soon proved to be a powerful tool to generate regular homotopies between two given immersions.

•Thurston corrugations are quite similar to the oscillations created by a convex integration. In fact, it now appears that the Theory of Corrugations can be seen as a simplified version of the Convex Integration Theory (all quantitative aspects are ignored).

•It is likely that, at that time, Thurston was unaware of the Convex Integration Theory.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Thurston Corrugations

•Thurston eversion introduces oscillations to avoid the creation of singular points.

•These oscillations, called by Thurstoncorrugations, soon proved to be a powerful tool to generate regular homotopies between two given immersions.

•Thurston corrugations are quite similar to the oscillations created by a convex integration. In fact, it now appears that the Theory of Corrugations can be seen as a simplified version of the Convex Integration Theory (all quantitative aspects are ignored).

•It is likely that, at that time, Thurston was unaware of the Convex Integration Theory.

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

Proof of Gromov Theorem

The pull-back bundle E

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

C

-density

AC⁰-denseh-principleholds forR ⊂J¹(M,N)if the (usual)h-principle holds and if for every formal solution σ:M −→ Rand every arbitrarily small neighborhoodU ⊂N of the image of the underlying mapf₀=bsσ :M−→N, the homotopyσt :M−→ Rjoiningf₀to a solutionf :=bsσ1

can be chosen such thatbs σt(M)⊂U,for allt ∈[0,1].

Similar definition for theC⁰-dense parametrich-principle.

Theorem (Gromov). –LetR ⊂J¹(M,N)be open and ample, thenRsatisfies to the C⁰-dense parametric h-principle .

The H-principle for

Ample Relations V.Borrelli

Ample Differential Relations H-principle for ample relations

William Thurston