Introduction to the h -principle

2 Holonomic approximation

The goal of this chapter is to prove Eliashberg and Mishachev’s holonomic approximation theorem:

Theorem 2.1
Let π \! :X → V be a locally trivial fibration, and A a polyhedron with positive codimension in V . Let \mathcal{F}\! :P × V → X^{(r)} be a smooth family of sections of π^ r parametrized by a compact manifold P , and such that \mathcal{F}_ p is holonomic for p in \operatorname{Op}(∂P) . For every positive functions ε and δ on V , there exists a family of isotopies φ^ p_ t\! :V → V and a family of holonomic sections F such that:

  • φ^ p_ t = \operatorname{Id} when t = 0 or p ∈ \operatorname{Op}(P)

  • each φ^ p_ t is δ -close to \operatorname{Id} in C^0 topology

  • each F_ p is defined on \operatorname{Op}(φ^ p_1(A))

  • d\left(F_ p(x), \mathcal{F}_ p(x)\right) ≤ ε(x) for all x in \operatorname{Op}(φ^ p_1(A)) .

Notations

Ambient source space is ℝ^ n , target space is ℝ^ p . For every point x = (x_1, … , x_ n) in ℝ^ n and every l ≤ n we set x^ l = (x_1, …, x_ l) . We set I = [-1, 1] and, for each positive s , I_ s = [-1/s, 1/s] . We always use the \max distance on products: d(x, x’) = \max d(x_ i, x’_ i) . The ε -neighborhood of a subset A ⊂ ℝ^ n is \mathcal{N}_{\! \! ε}(A) := \{ x ∈ ℝ^ n \; ;\; d(x, A) ≤ ε \} .

We also fix an integer k with 0 ≤ k < n . Sections will be defined near the cell C= I^ k × \{ 0_{n-k}\} ⊂ ℝ^ n , where 0_{n-k} is the origin of ℝ^{n-k} .

For any l ≤ k and y in I^{k - l} , we set C_ y = \{ y\} × I^ l × \{ 0_{n-k}\} ⊂ C .

Let F = (F_ y) be a smooth family of holonomic sections of J^ r(ℝ^ n, ℝ^ p) defined on \operatorname{Op}(C_ y) , where y is in I^{k-l} . We say that F is consistent near ∂C if there is some holonomic section 𝔽 defined near ∂C such that, for all x in \operatorname{Op}(∂C) , F_ y(x) = 𝔽(x) whenever both are defined (in particular when y is in ∂I^{k-l} or x is in ∂C_ y ). The size S(F) of such a family F is the smallest integer such that:

We fix once and for all a cut-off function χ \! :ℝ^+ → [0, 1] such that χ(d) = 0 when d ≤ 1/4 and χ(d) = 1 when d ≥ 3/4 .

Still keeping n , k and l fixed, for every pair of positive integers N and S , we set:

φ(x) = \left( x^{n-1}, x_ n + χ\big (Sd(x, ∂C)\big )\frac{3}{4S}\cos (2πNSx_{k-l})\right).

Note that this map commutes with the projection x ↦ x^ j , for any j < n , and is the identity on \mathcal{N}_{\! \! 1/4S}(∂C) .

Proposition 2.2
Let F be a family of holonomic sections as above. For every positive integer N and every integer S ≥ S(F) , there is a family \bar F = (\bar F_ z) where z is in I^{k-l-1} , which is consistent near ∂C (with the same 𝔽 ), each \bar F_ z is defined on \operatorname{Op}(φ(C_ z)) and, for all x in \operatorname{Op}(C) ,
d\left(F_{x^{k-l}}(φ(x)), \bar F_{x^{k-l-1}}(φ(x))\right) ≤ \frac{B(S)}N
for some function B (recall that x^ j is the projection of x onto ℝ^ j ).

  1. Explain why the inequality makes sense.

Preuve

We fix a cutoff function ρ \! :ℝ → [0, 1] which equals 0 before -1/2 and 1 after 1/2 . Let f be a family of sections representing F , where each f_ y , y ∈ I^{k-l} , is defined on a neighborhood of \mathcal{N}_{\! \! 1/S}(y) . For each integer i in [0, 2NS] , we set t_ i = -1 + i/(NS) ∈ I . We first note that the mean value theorem gives

\begin{multline} \label{eq:meanvalue} ∀ z ∈ I^{k-l-1}, ∀ t, t’ ∈ I, ∀ x ∈ \mathcal{N}_{\! \! 1/S}(z, t) ∩ \mathcal{N}_{\! \! 1/S}(z, t’),\\ d(F_{z, t}(x), F_{z, t'}(x)) ≤ S(F)|t - t’|. \end{multline}

Then for each i < 2NS and each z in I^{k-l-1} , we set

f^ i_ z(x) = [1 - ρ(Sx_ n)]f_{z, t_ i}(x) + ρ(Sx_ n)f_{z, t_{i+1}}(x)

which is defined on a neighborhood of (z + I_ S^{k-l-1}) × [t_ i, t_{i+1}]× I^ l × I_ S^{n-k} .

  1. Prove that

    \begin{equation*} \begin{aligned} j^ rf^ i_ z(x) & = F_{z, t_ i}(x) + j^ r\left[x ↦ ρ(Sx_ n)\left(f_{z, t_{i+1}}(x)- f_{z, t_ i}(x)\right) \right](x)\\ & = F_{z, t_ i}(x) + O\left(S^ r\frac{S(F)}{NS}\right). \end{aligned}\end{equation*}

Also note that f^ i_ z(x) = f_{z, t_ i}(x) when x_ n ≤ -1/(2S) and f^ i_ z(x) = f_{z, t_{i+1}}(x) when x_ n ≥ 1/(2S) . In particular f^ i_ z(φ(x)) = f_{z, t_ i}(φ(x)) when x is close to (z, t_ i, 0) while f^ i_ z(φ(x)) = f_{z, t_{i+1}}(φ(x)) when x is close to (z, (t_ i + t_{i+1})/2, 0) .

So we define a family of sections near φ(C_ z) by:

\bar f_ z(φ(x)) = \begin{cases} f^ i_ z(φ(x)) & \text {if $x_{k-l} ∈ [t_ i, (t_ i + t_{i+1})/2]$} \\ f_{z, t_{i+1}}(φ(x)) & \text {if $x_{k-l} ∈ [(t_ i + t_{i+1})/2, t_{i+1}]$} \end{cases}

and then define \bar F_ z = j^ r\bar f_ z .

  1. Prove that \bar F has the announced properties (don’t forget to discuss what happens near ∂C ).

Proposition 2.3
Let \mathcal{F} be a smooth family of sections of J^ r(ℝ^ n, ℝ^ p) defined on \operatorname{Op}(C) . Let ε be a positive real number. For every sequence of positive integers N_ l , 1 ≤ l ≤ k , there exists a sequence of families F^ l of holonomic sections defines near sub-cells of dimension l , and diffeomorphisms φ_ l preserving x^ k such that:

  1. φ_0 = \operatorname{Id} and, for all x in \operatorname{Op}(C) ,

    7

    \begin{equation} d \left( F^0_{x^ k}(x), \mathcal{F}(x) \right) ≤ \frac{ε}{k+1}. \end{equation}

  2. Each pair (F^ l, φ_ l) for l ≥ 1 depends on previous ones and N_ l .

  3. for all x in \operatorname{Op}(C_{x^{k-k-1}}) ,

    d\left((φ_{l+1})_*F^{l+1}_{x^{k-l-1}}(φ_{l+1}(x)),\; F^ l_{x^{k-l}}(φ_{l+1}(x))\right) ≤ \frac{B(S(F^ l))}{N_{l+1}}.

  1. Prove the above proposition.

We now prove the holonomic approximation lemma over a cube. For each sequences F^ l , φ_ l with parameters N_ l from the proposition, we set φ = φ_1 ∘ ⋯ ∘ φ_ k and F := φ_* F^ k . We also set, for 0 ≤ l ≤ k , φ^ l = φ_{l+1} ∘ ⋯ ∘ φ_ k , so that φ^0 = φ , φ^ l = φ_{l+1} ∘ φ^{l+1} and φ^ k = \operatorname{Id} .

  1. Prove that, for all x in \operatorname{Op}(C) and 0 ≤ l < k :

    \begin{multline*} d \left( φ^ l_*F^ k(φ^ l(x)), F^ l_{x^{k-l}}(φ^ l(x)) \right) ≤\\ K(φ_{l+1}) d\left( φ^{l+1}_*F^ k(φ^{l+1}(x)), F^{l+1}_{x^{k-l-1}}(φ^{l+1}(x)) \right) + \frac{B(S(F^ l))}{N_{l+1}} \end{multline*}

    for some nonnegative function K .

  2. Prove that, for all x in \operatorname{Op}(C) :

    d(F(φ(x)), \mathcal{F}(φ(x))) ≤ \frac{ε}{k+1} + \sum _{l=0}^{k-1} \prod _{j=1}^ l K(φ_ j)\frac{B(S(F^ l))}{N_{l+1}}.

  3. Conclude the proof of the proposition.

  1. Explain how to add a parameter p ∈ I^ d in the holonomic approximation theorem over a cube by reduction to the unparametric case in ℝ^{n + d} .

  2. Prove the theorem by induction over the cell decomposition of A and some cell decomposition of P .