# 3.1 Preliminaries

## 3.1.1 Vector bundles operations

Definition 3.1

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.

Definition 3.2

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

Definition 3.3
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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 $n$.

Definition 3.4
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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 $\mathcal{F}$.

Lemma 3.5
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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 $f$.

Proof

This is obvious by construction…

Definition 3.6
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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

Definition 3.7
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A first order differential relation for maps from $M$ to $N$ is a subset $\mathcal{R}$ of $J^1(M, N)$.

Definition 3.8
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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.

Definition 3.9
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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 $[0, 1]$.

Definition 3.10
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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.

Lemma 3.11
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The above definitions translate to the definitions of the previous chapter in local charts. (We’ll need more precise statements...)

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 $\bar F(x, p) := (f_p(x), φ_{p, x} ⊕ ∂f/∂p(x, p))$.

Lemma 3.12
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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 $\mathcal{R}^P := Ψ^{-1}(\mathcal{R})$.

Proof

TODO…

Lemma 3.13
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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 $\mathcal{R}$.

Proof

This obviously follows from lemma 3.12.

# 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:

$\mathcal{R}_{σ, λ, v} = \operatorname{Conn}_{φ(v)}\left\{ w ∈ T_yY \; ;\; \big (x,\; y,\; φ + (w - φ(v))⊗λ\big ) ∈ \mathcal{R}\right\}$

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 $ℝV$.

Definition 3.14
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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 $T_yY$.

Lemma 3.15
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If a relation is ample then it is ample if the sense of Definition 2.14 when seen in local charts.

Proof

This follows from the fundamental properties of the tangent bundle.

Lemma 3.16

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

Proof

This obviously follows from lemma 2.15.

Theorem 3.17 Gromov
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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 $P$.

Lemma 3.18
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If $\mathcal{R}$ is ample then, for any parameter space $P$, $\mathcal{R}^P$ is also ample.

Proof

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:

$\mathcal{R}^P_{σ, λ, v} = \operatorname{Conn}_{φ(v)}\left\{ w ∈ T_yY \; ;\; \big (x,\; y,\; ψ ∘ ι_{x, p} + (w - ψ(v))⊗λ_X\big ) ∈ \mathcal{R}\right\} .$

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 $\operatorname{Conv}\mathcal{R}^P_{σ, λ, v} = T_yY$.

Proof of Theorem 3.17

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 $i {\lt} N$. 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 $t_0 {\lt} 1$ 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 $t$.

Theorem 3.19 Smale 1958

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

Proof

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.