Dependencies

Let \(U_0\) and \(U_1\) be open sets in \(E\). Let \(K_0 ⊆ U_0\) and \(K_1 ⊆ U_1\) be compact subsets. For any \(γ_0 ∈ \operatorname{\mathcal{L}}(U_0, g, β, Ω)\) and \(γ_1 ∈ \operatorname{\mathcal{L}}(U_1, g, β, Ω)\), there exists \(U ∈ 𝓝(K_0 ∪ K_1)\) and there exists \(γ ∈ \operatorname{\mathcal{L}}(U, g, β, Ω)\) which coincides with \(γ_0\) near \(K_0\cup U_1^c\).

A relation \(\mathcal{R}\) is ample if, for every \(σ = (x, y, φ)\) in \(\mathcal{R}\) and every \((λ, v)\), the slice \(\mathcal{R}_{σ, λ, v}\) is ample in \(T_yY\).

A first order differential relation \(\mathcal{R}\) is ample if all its slices are ample.

A subset \(Ω\) of a real vector space \(E\) is ample if the convex hull of each connected component of \(Ω\) is the whole \(E\).

The average of a loop \(γ\) is \(\barγ := \int _{𝕊^1} γ(s)\, ds\).

The map obtained by corrugation of \(f\) in direction \((π, v)\) using \(γ\) with oscillation number \(N\) is

A dual pair on a vector space \(E\) is a pair \((π, v)\) where \(π\) is a linear form on \(E\) and \(v\) a vector in \(E\) such that \(π(v) = 1\).

A continuous family of loops \(γ \! :E × [0, 1] × 𝕊^1 → F, (x, t, s) ↦ γ^t_x(s)\) surrounds a map \(g \! :E → F\) with base \(β \! :E → F\) on \(U ⊆ E\) in \(Ω ⊆ E × F\) if, for every \(x\) in \(U\), every \(t ∈ [0, 1]\) and every \(s ∈ 𝕊^1\),

• \(γ^t_x\) is based at \(β(x)\)

• \(γ^0_x(s) = β(x)\)

• \(γ^1_x\) surrounds \(g(x)\)

• \((x,γ^t_x(s)) ∈ Ω\).

The space of such families will be denoted by \(\operatorname{\mathcal{L}}(g, β, U, Ω)\).

A formal solution of a differential relation \(\mathcal{R}⊆ J^1(M, N)\) is a section of \(J^1(M, N) → M\) taking values in \(\mathcal{R}\). A solution of \(\mathcal{R}\) is a map from \(M\) to \(N\) whose \(1\)–jet extension is a formal solution.

A formal solution of a differential relation \(\mathcal{R}\) over \(U ⊂ E\) is a map \(\mathcal{F}= (f, φ) \! :E → F × \operatorname{Hom}(E, F)\) such that, for every \(x\) in \(U\), \((x, f(x), φ(x))\) is in \(\mathcal{R}\).

A first order differential relation \(\mathcal{R}⊆ J^1(M, N)\) satisfies the \(h\)-principle if every formal solution of \(\mathcal{R}\) is homotopic to a holonomic one. It satisfies the parametric \(h\)-principle if, for every manifold with boundary \(P\), every family \(\mathcal{F}: P × M → J^1(M, N)\) of formal solutions which are holonomic for \(p\) in \(𝓝(∂P)\) is homotopic to a family of holonomic ones relative to \(𝓝(∂P)\). It satisfies the parametric \(h\)-principle if, for every manifold with boundary \(P\), every family \(\mathcal{F}: P × M → J^1(M, N)\) of formal solutions is homotopic to a family of holonomic ones.

Let \(E'\) be a linear subspace of \(E\). A map \(\mathcal{F}= (f, φ) : E → F × \operatorname{Hom}(E, F)\) is \(E'\)–holonomic if, for every \(v\) in \(E'\) and every \(x\), \(Df(x)v = φ(x)v\).

A section \(\mathcal{F}\) of \(J^1(M, N) → M\) is called holonomic if it is the \(1\)–jet of its base map. Equivalently, \(\mathcal{F}\) is holonomic if there exists \(f \! :M → N\) such that \(\mathcal{F}= j^1f\), since such a map is necessarily \(\operatorname{bs}\mathcal{F}\).

Let \(E → B\) and \(F → B\) be two vector bundles over some smooth manifold \(B\). The bundle \(\operatorname{Hom}(E, F) → B\) is the set of linear maps from \(E_b\) to \(F_b\) for some \(b\) in \(B\), with the obvious project map.

A homotopy of formal solutions of \(\mathcal{R}\) is a family of sections \(\mathcal{F}: ℝ × M → J^1(M, N)\) which is smooth over \([0, 1] × M\) and such that each \(m ↦ \mathcal{F}(t, m)\) is a formal solution when \(t\) is in \([0, 1]\).

A homotopy of formal solutions over \(U\) is a map \(\mathcal{F}: ℝ × E → F × \operatorname{Hom}(E, F)\) which is smooth over \([0, 1] × U\) and such that each \(x ↦ \mathcal{F}(t, x)\) is a formal solution over \(U\) when \(t\) is in \([0, 1]\).

The support of a family \(γ\) of loops in \(F\) parametrized by \(E\) is the closure of the set of \(x\) in \(E\) such that \(γ_x\) is a constant loop.

The \(1\)-jet of a smooth map \(f \! :M → N\) is the map from \(m\) to \(J^1(M, N)\) defined by \(j^1f(m) = (m, f(m), T_mf)\).

Let \(M\) and \(N\) be smooth manifolds. Denote by \(p_1\) and \(p_2\) the projections of \(M × N\) to \(M\) and \(N\) respectively.

The space \(J^1(M, N)\) of \(1\)-jets of maps from \(M\) to \(N\) is \(Hom(p_1^*TM, p_2^*TN)\)

For every bundle \(p : E → B\) and every map \(f \! :B' → B\), the pull-back bundle \(f^*E → B'\) is defined by \(f^*E = \{ (b', e) ∈ B' × E \; |\; p(e) = f(b')\} \) with the obvious projection to \(B'\).

A first order differential relation for maps from \(M\) to \(N\) is a subset \(\mathcal{R}\) of \(J^1(M, N)\).

A first order differential relation for maps from \(E\) to \(F\) is a subset \(\mathcal{R}\) of \(E × F × \operatorname{Hom}(E, F)\).

For every \(σ = (x, y, φ)\), the slice of \(\mathcal{R}\) at \(σ\) with respect to \((π, v)\) is:

A formal solution \(\mathcal{F}\) of \(\mathcal{R}\) over \(U\) is \((π, v)\)–short if, for every \(x\) in \(U\), \(Df(x)v\) belonds to the interior of the convex hull of \(\operatorname{Conn}_{φ(v)}\mathcal{R}((x, f(x), φ(x)), π, v)\).

We say a loop \(γ\) surrounds a vector \(v\) if \(v\) is surrounded by a collection of points belonging to the image of \(γ\). Also, we fix a base point \(0\) in \(𝕊^1\) and say a loop is based at some point \(b\) if \(0\) is sent to \(b\).

A point \(x\) in \(E\) is surrounded by points \(p_0\), …, \(p_d\) if those points are affinely independent and there exist weights \(w_i ∈ (0, 1)\) such that \(x = \sum _i w_i p_i\).

The complement of a linear subspace of codimension at least 2 is ample.

If a relation is ample then it is ample if the sense of Definition 2.14 when seen in local charts.

If \(\mathcal{R}\) is ample then, for any parameter space \(P\), \(\mathcal{R}^P\) is also ample.

If a point \(x\) of \(E\) lies in the convex hull of a set \(P\), then \(x\) belongs to the convex hull of a finite set of affinely independent points of \(P\).

Let \(\mathcal{F}\) be a formal solution of \(\mathcal{R}\) over an open set \(U\). Let \(K_1 ⊂ U\) be a compact subset, and let \(K_0\) be a compact subset of the interior of \(K_1\). Assume \(\mathcal{F}\) is holonomic near a subset \(C\) of \(U\). Let \(ε\) be a positive real number.

If \(\mathcal{R}\) is open and ample over \(U\) then there is a homotopy \(\mathcal{F}_t\) such that:

1. \(\mathcal{F}_0 = \mathcal{F}\)

2. \(\mathcal{F}_t\) is a formal solution of \(\mathcal{R}\) over \(U\) for all \(t\) ;

3. \(\mathcal{F}_t(x) = \mathcal{F}(x)\) for all \(t\) when \(x\) is near \(C\) or outside \(K_1\).

4. \(d(\operatorname{bs}\mathcal{F}_t(x), \operatorname{bs}\mathcal{F}(x)) ≤ ε\) for all \(t\) and all \(x\) ;

5. \(\mathcal{F}_1\) is holonomic near \(K_0\).

If a point \(x\) of \(E\) lies in the convex hull of an open set \(P\), then it is surrounded by some collection of points belonging to \(P\).

Given a point \(c\) of \(E\) and a real number \(t\), let:

be the homothety which dilates about \(c\) by a scale of \(t\).

Suppose \(c\) belongs to the interior of a convex subset \(C\) of \(E\) and \(t {\gt} 1\), then

Let \(\mathcal{F}\) be a formal solution of \(\mathcal{R}\) over an open set \(U\). Let \(K_1 ⊂ U\) be a compact subset, and let \(K_0\) be a compact subset of the interior of \(K_1\). Let \(C\) be a subset of \(U\). Let \(E'\) be a linear subspace of \(E\) contained in \(\ker π\). Let \(ε\) be a positive real number.

Assume \(\mathcal{R}\) is open over \(U\). Assume that \(\mathcal{F}\) is \(E'\)–holonomic near \(K_0\), \((π, v)\)–short over \(U\), and holonomic near \(C\). Then there is a homotopy \(\mathcal{F}_t\) such that:

1. \(\mathcal{F}_0 = \mathcal{F}\) ;

2. \(\mathcal{F}_t\) is a formal solution of \(\mathcal{R}\) over \(U\) for all \(t\) ;

3. \(\mathcal{F}_t(x) = \mathcal{F}(x)\) for all \(t\) when \(x\) is near \(C\) or outside \(K_1\) ;

4. \(d(\operatorname{bs}\mathcal{F}_t(x), \operatorname{bs}\mathcal{F}(x)) ≤ ε\) for all \(t\) and all \(x\) ;

5. \(\mathcal{F}_1\) is \(E' ⊕ ℝv\)–holonomic near \(K_0\).

Given an affine basis \(b\) of \(E\), the interior of the convex hull of \(b\) is the set of points with strictly positive barycentric coordinates.

Assume \(Ω\) is open and connected over some neighborhood of \(x_0\). If \(g(x)\) is in the convex hull of \(Ω_x\) for \(x\) near \(x_0\) then there is a continuous family of loops defined near \(x_0\), based at \(β\), taking value in \(Ω\) and surrounding \(g\).

If a vector \(v\) is in the convex hull of a connected open subset \(O\) then, for every base point \(b ∈ O\), there is a continuous family of loops \(γ \! :[0, 1] × 𝕊^1 → E, (t, s) ↦ γ^t(s)\) such that, for all \(t\) and \(s\):

• \(γ^t\) is based at \(b\)

• \(γ^0(s) = b\)

• \(γ^t(s) ∈ O\)

• \(γ^1\) surrounds \(v\)

For every smooth map \(f \! :M → N\),

1. \(j^1f\) is smooth

2. \(j^1f\) is a section of \(J^1(M, N) → M\)

3. \(j^1f\) composed with \(J^1(M, N) → N\) is \(f\).

The relation of immersions of \(M\) into \(N\) in positive codimension is open and ample.

The relation of immersions in positive codimension is open and ample.

Let \(\mathcal{R}\) be a first order differential relation for maps from \(M\) to \(N\). If, for every manifold with boundary \(P\), \(\mathcal{R}^P\) satisfies the \(h\)-principle then \(\mathcal{R}\) satisfies the parametric \(h\)-principle. Likewise, the \(C^0\)-dense and relative \(h\)-principle for all \(\mathcal{R}^P\) imply the parametric \(C^0\)-dense and relative \(h\)-principle for \(\mathcal{R}\).

In the above setup, we have:

• \(\bar F\) is holonomic at \((x, p)\) if and only if \(F_p\) is holonomic at \(x\).

• \(F\) is a family of formal solutions of some \(\mathcal{R}⊂ J^1(X, Y)\) if and only if \(\bar F\) is a formal solution of \(\mathcal{R}^P := Ψ^{-1}(\mathcal{R})\).

Let \(γ \! :E × 𝕊^1 → F\) be a smooth family of loops surrounding a map \(g\) with base \(β\) over some \(U ⊆ E\). There is a family of circle diffeomorphisms \(φ \! :U × 𝕊^1 → 𝕊^1\) such that each \(γ_x ∘ φ_x\) has average \(g(x)\) and \(φ_x(0) = 0\).

Each \(\operatorname{\mathcal{L}}(g, β, U, Ω)\) is path connected: for every \(γ_0\) and \(γ_1\) in \(\operatorname{\mathcal{L}}(g, β, U, Ω)\), there is a continuous map \(δ \! :[0, 1] × E × [0, 1] × 𝕊^1 → F, (τ, x, t, s) ↦ δ^t_{τ, x}(s)\) which interpolates between \(γ_0\) and \(γ_1\) in \(\operatorname{\mathcal{L}}(g, β, U, Ω)\).

For every \(x\) in \(E\) and every collection of points \(p ∈ E^{d+1}\) surrounding \(x\), there is a neighborhood \(U\) of \(\{ (x, p)\} \) and a function \(w \! :E × E^{d+1} → ℝ^{d+1}\) such that, for every \((y, q)\) in \(U\),

• \(w\) is smooth at \((y, q)\)

• \(w(y, q) {\gt} 0\)

• \(\sum _{i=0}^d w_i(y, q) = 1\)

• \(y = \sum _{i=0}^d w_i(y, q)q_i\)

The above definitions translate to the definitions of the previous chapter in local charts. (We’ll need more precise statements...)

The linear map \(φ + (w - φ(v)) ⊗ π)\) coincides with \(φ\) on \(\ker π\) and sends \(v\) to \(w\). If \(\sigma \) belongs to \(\mathcal{R}\) then \(φ(v)\) belongs to \(\{ w ∈ F, (x, y, φ + (w - φ(v)) ⊗ π) ∈ \mathcal{R}\} \).

In the setup of Proposition 1.3, assume we have a continuous family \(γ\) of loops defined near \(K\) which is based at \(β\), surrounds \(g\) and such that each \(γ_x^t\) takes values in \(Ω_x\). Then there such a family which is defined on all of \(U\) and agrees with \(γ\) near \(K\).

Let \(U\) be an open set in \(E\) and \(K ⊆ U\) a compact subset and \(U'\) an open neighborhood of \(\overline U\). Let \(Ω\) be a set in \(E × F\) such that, for each \(x\) in \(U'\), \(Ω_x := Ω ∩ (\{ x\} × F)\) is open and connected in \(\{ x\} × F\).

Let \(β\) and \(g\) be maps from \(E\) to \(F\) that are smooth on \(U'\). Assume that \(β(x) ∈ Ω_x\) for all \(x\) in \(U'\), and \(g(x) = β(x)\) near \(K\).

If, for every \(x\) in \(U'\), \(g(x)\) is in the convex hull of \(Ω_x\), then there exists a smooth family of loops

such that, for all \(x\) in \(U\), and all \((t, s) ∈ [0, 1] × 𝕊^1\)

• \(γ^t_x(s) ∈ Ω_x\)

• \(γ^0_x(s) = β(x)\)

• \(\barγ^1_x = g(x)\)

• \(γ^t_x(s) = β(x)\) if \(x\) is near \(K\).

Let \(f\) be a \(\mathcal{C}^1\) function from \(E\) to \(F\). Let \((π, v)\) be a dual pair on \(E\). Let \(γ \! :E × 𝕊^1 → F\) be a compactly supported \(\mathcal{C}^1\) family of loops. Assume that \(\overline{γ_x} = Df(x)v\) for all \(x\) in some set \(U\).

For any positive \(ε\), the function \(f'\) obtained by corrugation of \(f\) in direction \((π, v)\) using \(γ\) with large enough oscillation number \(N\) satisfies:

1. \(∀ x, ‖f'(x) - f(x)‖ ≤ ε\)

2. \(∀ x, ‖(Df'(x) - Df(x))_{|\ker π}‖ ≤ ε\).

3. \(∀ x ∈ U, ‖Df'(x)v - γ(x, Nπ(x))‖ ≤ ε\)

In addition, all the differences estimated above vanish if \(x\) is outside the support of \(γ\) (in addition to being in \(U\) in the last estimate).

If \(\mathcal{R}\) is open and ample then it satisfies the relative and parametric \(C^0\)-dense \(h\)-principle.

There is a homotopy of immersions of \(𝕊^2\) into \(ℝ^3\) from the inclusion map to the antipodal map \(a : q ↦ -q\).