# 3 Global theory of open and ample relations

# 3.1 Preliminaries

## 3.1.1 Vector bundles operations

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 $B’$.

The case of vector bundles.

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.

Set-theoretically, one can define $\operatorname{Hom}(E, F)$ as the set of subsets $S$ of $E × F$ such that there exists $b$ such that $S ⊂ E_b × F_b$ and $S$ is the graph of a linear map. But the type theory formalization will use other tricks here. The facts that really matter are listed in lemma 3.5.

## 3.1.2 Jets spaces

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 $Hom(p_1^*TM, p_2^*TN)$

We will use notations like $(m, n, φ)$ to denote an element of $J^1(M, N)$, but one should keep in mind that $J^1(M, N)$ is not a product, since $φ$ lives in $\operatorname{Hom}(T_mM, T_nN)$ which depends on $m$ and $n$.

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

The composition of a section $\mathcal{F}\! :M → J^1(M, N)$ with the projection onto $N$ will sometimes be denoted by $\operatorname{bs}\mathcal{F}\! :M → N$ and called the base map of $\mathcal{F}$.

For every smooth map $f \! :M → N$,

$j^1f$ is smooth

$j^1f$ is a section of $J^1(M, N) → M$

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

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 $\operatorname{bs}\mathcal{F}$.

# 3.2 First order differential relations

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

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 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 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.

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

## Parametricity for free

In many cases, relative parametric $h$-principles can be deduced from relative non-parametric ones with a larger source manifold. Let $X$, $P$ and $Y$ be manifolds, with $P$ seen a parameter space. Denote by $Ψ$ the map from $J^1(X × P, Y)$ to $J^1(X, Y)$ sending $(x, p, y, ψ)$ to $(x, y, ψ ∘ ι_{x, p})$ where $ι_{x, p} : T_xX → T_xX × T_pP$ sends $v$ to $(v, 0)$.

To any family of sections $F_p : x ↦ (f_p(x), φ_{p, x})$ of $J^1(X, Y)$, we associate the section $\bar F$ of $J^1(X × P, Y)$ sending $(x, p)$ to $\bar F(x, p) := (f_p(x), φ_{p, x} ⊕ ∂f/∂p(x, p))$.

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 $\mathcal{R}$ 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 $\mathcal{R}$.

# 3.3 The $h$-principle for open and ample differential relations

In this chapter, $X$ and $Y$ are smooth manifolds and $\mathcal{R}$ is a first order differential relation on maps from $X$ to $Y$: $\mathcal{R}⊂ J^1(X, Y)$. For any $σ = (x, y, φ)$ in $\mathcal{R}$ and any dual pair $(λ, v) ∈ T^*_xV × T_xV$, we set:

where $\operatorname{Conn}_a A$ is the connected component of $A$ containing $a$. In order to decipher this definition, it suffices to notice that $φ + (w - φ(v))⊗λ$ is the unique linear map from $T_xX$ to $T_yY$ which coincides with $φ$ on $\ker λ$ and sends $v$ to $w$. In particular, $w = φ(v)$ gives back $φ$.

Of course we will want to deal with more that one point, so we will consider a vector field $V$ and a $1$–form $λ$ such that $λ(V) = 1$ on some subset $U$ of $X$, a formal solution $F$ (defined at least on $U$), and get the corresponding $\mathcal{R}_{F, λ, v}$ over $U$.

One easily checks that $\mathcal{R}_{σ, κ^{-1}λ, κv} = κ\mathcal{R}_{σ, λ, v}$ hence the above definition only depends on $\ker λ$ and the direction $ℝV$.

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 $T_yY$.

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

The relation of immersions of $M$ into $N$ in positive codimension is open and ample.

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

We first explain how to get rid of parameters, using the relation $\mathcal{R}^P$ for families of solutions parametrized by $P$.

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

We fix $σ = (x, y, ψ)$ in $\mathcal{R}^P$. For any $λ = (λ_X, λ_P) ∈ T^*_xX × T^*_pP$ and $v = (v_X, v_P) ∈ T_xX × T_pP$ such that $λ(v) = 1$, we need to prove that $\operatorname{Conv}\mathcal{R}^P_{σ, λ, v} = T_yY$. Unfolding the definitions gives:

A degenerate but easy case is when $λ_X = 0$. Then the condition on $w$ becomes $ψ ∘ ι_{x, p} ∈ \mathcal{R}$, which is true by definition of $\mathcal{R}^P$, so $\mathcal{R}^P_{σ, λ, v} = T_yY$.

We now assume $λ_X$ is not zero and choose $u ∈ T_xX$ such that $λ_X(u) = 1$. We then have $\mathcal{R}^P_{σ, λ, v} = \mathcal{R}_{Ψσ, λ_X, u} + ψ(v) - ψ∘ι_{x, p}(u)$. Because $\mathcal{R}$ is ample and taking convex hull commutes with translation, we get that $\operatorname{Conv}\mathcal{R}^P_{σ, λ, v} = T_yY$.

Lemmas 3.13 and 3.18 prove we can assume there are no parameters. So we start with a single formal solution $F$ of $\mathcal{R}$, which is holonomic near some closed subset $A ⊂ X$.

We first assume $X$ is closed, and will then explain the proof adjustments needed in the non-compact case. By compactness, there are finite many compact subsets $(K_i)_{1 ≤ i ≤ N}$ contained in coordinate charts $U_i$ and such that $\bigcup K_i = X$.

We prove by induction on $i$ from $0$ to $N$ that there are formal solutions $F_i$, starting with $F_0 = F$, that are homotopic to $F$ relative to $A$, holonomic on $K_{≤ i} := \bigcup _{j ≤ i} K_j$ and whose base maps are $(1-2^{-i})ε$-close to that of $F$ (this contrived bound will be convenient for the non-compact case).

Assume $F_i$ has been constructed for some $i < N$. We now want to construct it on $U_{i+1}$. Lemma 3.15 ensures the pull-back of $\mathcal{R}$ in the chart corresponding to $U_{i+1}$ is ample in the sense of the preceding chapter. Hence Lemma 2.16 can be applied to construct $F_{i+1}$, using the image of $K_{i+1}$ as $K_0$ and the image of $A ∩ K_{i+1} ∩ K_{≤ i}$ as $C$. It is holonomic on $K_{≤ i+1}$ because it agrees with $F_i$ near $K_{≤ i}$ and is holonomic near $K_{i+1}$.

Once all $F_i$ are constructed, we define $F^t$ to be the concatenation of all homotopies relating $F_i$ to $F_{i+1}$.

If $X$ is not compact, then one can use a countable family of subsets $(U_i, K_i)$ which is locally finite (ie. every point $x$ has a neighborhood intersecting only finitely many $U_i$). The way we have chosen $C^0$-bounds ensures that each $\operatorname{bs}F_i$ is still at distance at most $ε$ from $\operatorname{bs}F$. We now have countably many homotopies to concatenate, so we need to reparametrize the $i$-th homotopy by an interval of length $2^{-i}$. This give a family of sections of $J^1(X, Y)$ parametrized by $t ∈ [0, 1)$. But our local finiteness assumption implies that, for each each $x$, there is some $t_0 < 1$ and some neighborhood $U$ of $x$ such that our family is $t$–independent on $U$ for $t ≥ t_0$. So we can extend to $t = 1$. The resulting family is smooth since smoothness is a local condition in both $x$ and $t$.

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

We denote by $ι$ the inclusion of $𝕊^2$ into $ℝ^3$. We set $j_t = (1-t)ι + ta$. This is a homotopy from $ι$ to $a$ (but not animmersion for $t=1/2$). Using the canonical trivialization of the tangent bundle of $ℝ^3$, we can set, for $(q, v) ∈ T𝕊^2$, $G_t(q, v) = \mathrm{Rot}_{Oq}^{πt}(v)$, the rotation around axis $Oq$ with angle $πt$. The family $σ : t ↦ (j_t, G_t)$ is a homotopy of formal immersions relating $j^1ι$ to $j^1a$. It is homotopic by reparametrization to a homotopy of formal immersions relating $j^1ι$ to $j^1a$ which are holonomic for $t$ near the $0$ and $1$.

The above theorem ensures this family is homotopic, relative to $t = 0$ and $t = 1$, to a family of holonomic formal immersions, ie a family $t ↦ j^1f_t$ with $f_0 = ι$, $f_1 = a$, and each $f_t$ is an immersion.