a neighborhood $U \subset M$ of $m_0$ such that $f|_U : U \to \mathbb{R}^D$ is an embedding and that each $z \in B(z_0, \delta)$ admits a unique projection $p(z)$ on $f(U)$ with $p(z) \in f(U)$. The lemma also gives an expression for the differential $dp_{z_0} : \mathbb{R}^D \to \text{Im}(df_{m_0})$ at $z_0$ of $p : B(z_0, \delta) \to f(U)$, and this expression also only depends on the first two differentials of $f$ at $m_0$. Hence, given $r \ge 2$, a jet $(m, x, [f]) \in J^r(M, \mathbb{R}^D)$ such that $df_m$ has full rank and a point $z \in \mathbb{R}^D$ such that $\text{Im}(df_m) \perp z - x$, and assuming that $v \mapsto \|d(f \circ \phi)_0(v)\|^2 - \langle z - x, d^2(f \circ \phi)_0(v, v) \rangle$ is positive definite (for some chart $\phi : (V, 0) \to (M, m_i)$) and $\text{Im}((df_i)_{m_i}) \perp (z - x_i)$.

We define the quadratic form

$$B_{a,z}(h) := \sum_{i,j=1}^{s} \det(A_{a,z,i}(h)^{\top} A_{a,z,j}(h))$$

on $\mathbb{R}^D$, where $A_{a,z,i}(h)$ is the $D \times s$ matrix whose $j$-th column is equal to $x_j - z$ if $j \ne i$ and whose $i$-th column is $P_{[f_i],x_i,z}(h) - h$.

We will typically consider the case where $z \in \text{Conv}(x_1, \dots, x_s)$ and $h \in \text{Vec}(x_1 - z, \dots, x_s - z)^{\perp}$, but this is not needed for the definition to make sense.

We can now finally formulate (the negation of) the Big Simplex Property (BSP) in a way that is compatible with jets and thence define $W_4^{(s)}$.

**Definition 8.22.** For any $s, r \in \mathbb{N}$ with $s \ge 2$ and $2 \le r \le k$, we let $(W_4^{\circ})^{(s)}$ be the set of multijets $(m_1, \dots, m_s, x_1, \dots, x_s, [f_1], \dots, [f_s]) \in J_s^r(M, \mathbb{R}^D)$ that belong to $W_2^{(s)}$ and such that

* for all $i \in [s]$, $\text{rank}((df_i)_{m_i}) = m$;

* for all $i \in [s]$ and for any chart $\phi : (U, 0) \to (M, m_i)$, the form $v \mapsto \|d(f_i \circ \phi)_0(v)\|^2 - \langle z - x_i, d^2(f_i \circ \phi)_0(v, v) \rangle$ is positive definite;

* and the quadratic form $B_{a,z}$ is degenerate on $\text{Vec}(x_1 - z, \dots, x_s - z)^{\perp}$.

Furthermore, we let $W_4^{(s)}$ be the closure of $(W_4^{\circ})^{(s)}$ in $J_s^r(M, \mathbb{R}^D)$.

**Lemma 8.23.** For any $s, r \in \mathbb{N}$ with $2 \le s \le D + 1$ and $2 \le r \le k - 1$, the set $W_4^{(s)}$ is a finite union of closed Whitney stratifiable subsets $W_{4,\nu}^{(s)}$ of $J_s^r(M, \mathbb{R}^D)$ of codimension at least $sm + 1$.

*Proof.* The proof is yet again similar to that of Lemma 8.10; as for Lemmas 8.14 and 8.18, we omit details and focus on the differences of import. As in the previous proofs, we fix $s \in \{2, \dots, D + 1\}$ and write $W_4 = W_4^{(s)}$.
The definition of $W_4^\circ$ is similar to that of $W_2$, but three conditions have been added: the first is that for $(m_1, \dots, m_s, x_1, \dots, x_s, [f_1], \dots, [f_s]) \in J_s^r(M, \mathbb{R}^D)$ the rank of $(df_i)_{m_i}$ must be $m$ for $i \in [s]$. This is an algebraic constraint in any adapted chart, and an open condition, and does not add any difficulty to the proof. The second is that for any $i \in [s]$ and any chart $\phi : (U, 0) \to (M, m_i)$, the form $v \mapsto \|d(f_i \circ \phi)_0(v)\|^2 - \langle z - x_i, d^2(f_i \circ \phi)_0(v, v) \rangle$ must be positive definite (where $z$, if it exists, is the only point in $\text{Conv}(x_1, \dots, x_s)$ such that $d(x_1, z) = \dots = d(x_s, z)$). We know from the proof of Lemma 8.14 that this condition is algebraic in adapted charts, and it is also open. The third and last addition is that we ask that the form $B_{a,z}$ degenerate on $\text{Vec}(x_1 - z, \dots, x_s - z)^\perp$. If we prove that this condition is semialgebraic in adapted charts and that it does increase the codimension by one, the same arguments as in Lemmas 8.10, 8.14 and 8.18 show that $W_4^\circ$ is locally semialgebraic of codimension $sm + 1$, hence that so is its closure $W_4$, which can then be decomposed as a union of finitely many closed Whitney stratified sets $\{W_{4,\nu}\}_{\nu \in N}$ of codimension at least $sm + 1$ (for some finite set of indices $N$), as claimed.

Let $(m, x, [f]) \in J^r(M, \mathbb{R}^D)$ and $z \in \mathbb{R}^D$ be such that the conditions of Lemma 4.3 are satisfied by $x, f, z$. Let $\phi : (V, 0) \to (M, m)$ be a chart of $M$ around $m$, and let us place ourselves in the adapted chart of $J^r(M, \mathbb{R}^D)$ to which $\phi$ gives rise. In Lemma 4.3, we defined the linear endomorphism $\Lambda : \text{Im}(df_m) \to \text{Im}(df_m)$ associated to $x, z, f$ by stating that $\Lambda(v)$ must be the only vector in $\text{Im}(df_m)$ such that

$$ \langle z - x, d^2(f \circ \phi)_0(d(f \circ \phi)_0^{-1}(v), d(f \circ \phi)_0^{-1}(w)) \rangle = \langle \Lambda(v), w \rangle $$

for any $w \in \text{Im}(df_{m_0})$. Let $A$ be the $D \times m$ matrix that represents $d(f \circ \phi)_0 : \mathbb{R}^m \to \mathbb{R}^D$, and let $B$ be the $m \times m$ matrix that represents the bilinear form $\mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R}, (v, w) \mapsto \langle z - x, d^2(f \circ \phi)_0(v, w) \rangle$. Then $\tilde{\Lambda} := d(f \circ \phi)_0^{-1} \circ \Lambda \circ d(f \circ \phi)_0 : \mathbb{R}^m \to \mathbb{R}^m$ is such that

$$ \langle z - x, d^2(f \circ \phi)_0(v, w) \rangle = \langle d(f \circ \phi)_0(\tilde{\Lambda}(v)), d(f \circ \phi)_0(w) \rangle $$

for any $v, w \in \mathbb{R}^m$, which in terms of matrices means that

$$ w^\top B v = w^\top A^\top A \tilde{\Lambda}(v) $$

for all $v, w \in \mathbb{R}^m$, or equivalently

$$ (A^\top A)^{-1} B v = \tilde{\Lambda}(v) $$

(where we use the same notation for the map $\tilde{\Lambda}$ and the matrix that represents it in the chosen system of coordinates). We also see that the map $d(f \circ \phi)_0^{-1} \circ \pi_{\text{Im}(df_m)} : \mathbb{R}^D \to \mathbb{R}^m$ is represented by the pseudo-inverse matrix

$$ (A^\top A)^{-1} A^\top. $$

This expression is well-defined because $A$ is of full rank, and the coefficients of the resulting matrix are polynomial fractions in the coefficients of $A$.

Finally, remember that we defined $P_{[f],x,z}$ as the differential at $z$ on the image of $f$, which was shown in Lemma 4.3 to be equal to

$$ P_{[f],x,z} = (\text{id} - \Lambda)^{-1} \circ \pi_{\text{Im}(df_m)} : \mathbb{R}^D \to \text{Im}(df_m). $$

This can be rewritten as

$$ \begin{aligned} P_{[f],x,z} &= (\text{id} - \Lambda)^{-1} \circ \pi_{\text{Im}(df_m)} = d(f \circ \phi)_0 d(f \circ \phi)_0^{-1} (\text{id} - \Lambda)^{-1} d(f \circ \phi)_0 d(f \circ \phi)_0^{-1} \pi_{\text{Im}(df_m)} \\ &= d(f \circ \phi)_0 (\text{id} - d(f \circ \phi)_0^{-1} \Lambda d(f \circ \phi)_0)^{-1} d(f \circ \phi)_0^{-1} \pi_{\text{Im}(df_m)} \\ &= d(f \circ \phi)_0 (\text{id} - \tilde{\Lambda})^{-1} d(f \circ \phi)_0^{-1} \pi_{\text{Im}(df_m)} \end{aligned} $$
which is thus represented by the $D \times D$ matrix

$$A(I_m - (A^\top A)^{-1}B)^{-1}(A^\top A)^{-1}A^\top$$

whose coefficients are polynomial fractions in the coordinates of the adapted chart.

For $a = (m_1, \dots, m_s, x_1, \dots, x_s, [f_1], \dots, [f_s]) \in J_s^r(M, \mathbb{R}^D)$ and $z \in \mathbb{R}^D$, we defined $B_{a,z}$ as

$$B_{a,z}(h) := \sum_{i,j=1}^{s} \det(A_{a,z,i}(h)^\top A_{a,z,j}(h))$$

for $h \in \mathbb{R}^D$, where $A_{a,z,i}(h)$ is the $D \times s$ matrix whose $j$-th column is equal to $x_j - z$ if $j \neq i$ and whose $i$-th column is $P_{[f_i],x_i,z}(h) - h$. Sums, scalar products and determinants are all polynomial operations, and we have shown that the coefficients of the matrices representing the maps $P_{[f_i],x_i,z}$ are polynomial fractions in the coordinates, hence so are the coefficients of the $D \times D$ matrix $O$ representing $B_{a,z}$. One can locally define a linear map $\iota: \mathbb{R}^{D-m} \to \mathbb{R}^D$ such that the image of $\iota$ is $E^\perp = \text{Vec}(x_2 - x_1, \dots, x_s - x_2)^\perp$ and such that it is represented by a matrix $J$ whose coefficients are polynomial fractions in the coordinates of $x_1, \dots, x_s$; then $B_{a,z}$ is degenerate when restricted to $E^\perp$ if and only if the determinant of $J^\top O J$ is zero, and this determinant is a polynomial fraction in the coordinates of the adapted chart (possibly restricted to a smaller domain). This proves that asking that $B_{a,z}$ be degenerate is a semialgebraic constraint in the adapted charts.

We now turn to the codimension of $W_4^\circ$. In what follows, we restrict ourselves to an adapted chart of sufficiently small domain $U$ of $J_s^r(M, \mathbb{R}^D)$ that there exists a polynomial $P$ in the coordinates of the chart and in those of $z \in \mathbb{R}^D$ such that wherever $B_{a,z}$ is well-defined, it is degenerate on $E^\perp$ if and only if $P(a, z) = 0$ (to simplify notations, we write $P(a, z)$ for the value of $P$ evaluated in the coordinates corresponding to $a$ and $z$). Such a polynomial exists because we have shown that the determinant of $B_{a,z}$ can be locally expressed as a polynomial fraction, which is 0 if and only if its numerator is 0.

We only need to show that the intersection of $W_4^\circ$ with the domain of such a chart is of codimension at least $sm + 1$: then the same arguments as in the proofs of Lemmas 8.10, 8.14 and 8.18 are enough to conclude. In what follows, we do not always specify "in the coordinates of the chart" or "restricted to the domain of the chart" and write $f_i$ for the precomposition of $f_i$ with the coordinates of the chart.

We face two challenges: the first is geometric in nature. Indeed, it is not trivial that asking that the form $B_{a,z}$ be degenerate on $E^\perp$ truly increases the codimension by one; after all, the form $B_{a,z}$ is degenerate on $\mathbb{R}^D$. Nonetheless, we will show that it is true. The second difficulty is that as in the proof of Lemma 8.10, the polynomial constraint $P(a, z) = 0$ is not a priori everywhere transverse to the other constraints, as $P$ might have singularities. A few tricks are enough to circumvent this mostly technical problem.

Write $a \in U$ as $a = (a_{\le 1}, a_{\ge 2})$, where

$$a_{\le 1} := (m_1, \dots, m_s, x_1, \dots, x_s, (df_1)_{m_1}, \dots, (df_s)_{m_s})$$

$$a_{\ge 2} = ((d^2 f_1)_{m_1}, \dots, (d^2 f_s)_{m_s}, (d^{\ge 3} f_1)_{m_1}, \dots, (d^{\ge 3} f_s)_{m_s}),$$

(up to some reordering of the coordinates). Let $a$ be such that for some (unique) $z \in \mathbb{R}^D$, we have $\dim \text{Aff}(x_1, \dots, x_s) = s - 1$, $\|x_1 - z\|^2 = \dots = \|x_s - z\|^2$, $\text{Im}((df_i)_{m_i}) \perp (z - x_i)$ and $\text{rank}((df_i)_{m_i}) = m$ for $i = 1, \dots, s$. Let us consider all $a' \in U$ such that the sources $m'_i$, the
targets $x'_i$ and the first differential $(df'_i)_{m'_i}$ of $a'$ are equal to those of $a$, i.e. such that $a'_{\le 1} = a_{\le 1}$ (we let the $a'_{\ge 2}$ vary freely). All the conditions listed above are true for all such $a'$.

The key observation is the following: we can make the eigenvalues associated to the form $v \mapsto \langle z - x_i, d^2(f'_i)_{m_i}(v, v) \rangle$ arbitrarily smaller than 0 (and doing so will ensure, as a side effect, that $v \mapsto \|d(f'_i)_{m_i}(v)\|^2 - \langle z - x_i, d^2(f'_i)_{m_i}(v, v) \rangle$ be positive definite). To summarize what follows, this makes the differential $P_{[f'_i], x_i, z}$ at $z$ of the projection on the image of $f'_i$ tend to 0, which in turn makes $B_{a',z}(h)$ approximately equal to the square of the $s$-volume of the simplex $\text{Conv}(x_1, \dots, x_s, z + h)$ for $h \in E^{\perp}$, which is non-zero. This shows that the polynomial that associates the differentials of degree 2 of $a'$ to $P(a', z)$ is non-trivial, thus that this polynomial is non-zero for most $a'$ and defines a codimension 1 constraint. Now let us formalize this argument.

Remember that we defined the linear endomorphism $\Lambda : \text{Im}(d(f'_i)_{m_i}) \to \text{Im}(d(f'_i)_{m_i})$ associated to $x_i, z, f'_i$ by stating that $\Lambda(v)$ must be the only vector in $\text{Im}(d(f'_i)_{m_i})$ such that (in the coordinates)

$$ \langle z - x, d^2(f'_i)_{m_i}((d(f'_i)_{m_i})^{-1}(v), (d(f'_i)_{m_i})^{-1}(w)) \rangle = \langle \Lambda(v), w \rangle $$

for any $w \in \text{Im}(d(f'_i)_{m_i})$. As the eigenvalues associated to the form $v \mapsto \langle z - x_i, d^2(f'_i)_{m_i}(v, v) \rangle$ go to minus infinity, so must the eigenvalues of $\Lambda$, hence the linear map

$$ P_{[f'_i], x_i, z} = (\text{id} - \Lambda)^{-1} \circ \pi_{\text{Im}(d(f'_i)_{m_i})} : \mathbb{R}^D \to \text{Im}(d(f'_i)_{m_i}) $$

tends to 0. Then $B_{a',z}$ tends to

$$ h \mapsto \sum_{i,j=1}^{s} \det(\tilde{A}_{a,z,i}(h)^{\top} \tilde{A}_{a,z,j}(h)), $$

where $\tilde{A}_{a,z,i}(h)$ is the $D \times s$ matrix whose $j$-th column is equal to $x_j - z$ if $j \ne i$ and whose $i$-th column is $-h$. Now the same arguments as in the proof of Lemma 6.7 show that

$$ \sum_{i,j=1}^{s} \det(\tilde{A}_{a,z,i}(h)^{\top} \tilde{A}_{a,z,j}(h)) = (s!)^2 \text{Vol}_s(\text{Conv}(x_1, \dots, x_s, z+h))^2 + o(\|h\|^2). $$

But as dim $\text{Aff}(x_1, \dots, x_s) = s-1$, if $h \in E^{\perp} = \text{Vec}(x_1-z, \dots, x_s-z)^{\perp}$ then $\text{Vol}_s(\text{Conv}(x_1, \dots, x_s, z+h)) \ge C\|h\|$ for some constant $C > 0$ that depends on the positions of $x_1, \dots, x_s \in \mathbb{R}^D$. This shows that $B_{a',z}$ is non-degenerate if the eigenvalues associated to $v \mapsto \langle z - x_i, d^2(f'_i)_{m_i}(v, v) \rangle$ are negative enough.

For the same fixed $a = (a_{\le 1}, a_{\ge 2}) \in U$ and associated $z \in \mathbb{R}^D$, the map

$$ P_{a_{\le 1}} : a'_{\ge 2} \mapsto P((a_{\le 1}, a'_{\ge 2}), z) $$

is polynomial in the coordinates. As $B_{a',z}$ is non-degenerate for an appropriate choice of $a'_{\ge 2}$, the polynomial $P_{a_{\le 1}}$ is non-trivially 0, hence

$$ S_{a_{\le 1}} := \{a'_{\ge 2} : B_{(a_{\le 1}, a'_{\ge 2}), z} \text{ is well-defined and } P_{a_{\le 1}}(a'_{\ge 2}) = 0\} $$

is an algebraic hypersurface (possibly with singularities) in the open set

$$ \{a'_{\ge 2} : B_{(a_{\le 1}, a'_{\ge 2}), z} \text{ is well-defined}\}, $$

and in particular a semialgebraic set of codimension (at least) 1.
Consider the projection

$$ \pi_{\le 1} : (a_{\le 1}, a_{\ge 2}) \mapsto a_{\le 1} $$

defined on the domain $U$ of the adapted chart. The image of the restriction of $\pi_{\le 1}$ to $W_4^\circ \cap U$ is included in the set $W_2 = W_2^{(s)}$ of Definition 8.8 for $r = 1$. We know from Lemma 8.10 that $W_2$ is a finite union of Whitney stratified sets of codimension $sm$ in $J_s^1(M, \mathbb{R}^D)$, and we can in fact assume that the image of $\pi_{\le 1}|_{W_4^\circ \cap U}$ is included in one of those Whitney stratified sets by making $U$ small enough. An easy consequence of Theorem 2.7.1 from [BR90] is that if $f: A \to B$ is a semialgebraic map between $A$ and $B$ two semialgebraic sets, the dimension of $A$ is less than or equal to the dimension of $B$ plus the maximum of the dimensions of the fibers $f^{-1}(b)$ (for $b \in B$). Now $\pi_{\le 1}: W_4^\circ \cap U \to W_2$ is a semialgebraic map, and we have just shown that the fibers $(\pi_{\le 1}|_{W_4^\circ \cap U})^{-1}(a_{\le 1})$ for $a_{\le 1} \in \text{Im}(\pi_{\le 1}|_{W_4^\circ \cap U})$ have codimension at least 1. Hence $W_4^\circ \cap U$ has codimension at least $sm + 1$, which is enough to conclude. $\square$

As for Lemmas 8.10, 8.14 and 8.18, we choose for a given $M$ and for each $s, r \in \mathbb{N}$ with $2 \le s \le D+1$ and $2 \le r \le k-1$ a decomposition $W_4^{(s)} = \cup_{\nu \in N} W_{4,\nu}^{(s)}$ of $W_4^{(s)}$ into a finite union of closed Whitney stratifiable sets of codimension at least $sm+1$ whose existence is guaranteed by Lemma 8.23, and we simply refer to it as “the decomposition from Lemma 8.23” or “the sets $\{W_{4,\nu}\}_{\nu \in N}$ from Lemma 8.23” in what follows.

**Lemma 8.24.** For any $2 \le r \le k-1$, if $i \in \text{Emb}^k(M, \mathbb{R}^D)$ is such that $M = i(M)$ satisfies condition (P1), (P2) and (P3) and is such that for all $s = 2, \dots, D+1$, $j_s^r(i): M^{(s)} \to J_s^r(M, \mathbb{R}^D)$ is transverse to the sets $\{W_{4,\nu}^{(s)}\}_{\nu \in N}$ from Lemma 8.23, then $M$ satisfies condition

(P4) There exist constants $C > 0$ and $\mu_0 \in (0,1)$ such that for every $\mu \in [0, \mu_0)$, the set $Z_\mu(M)$ is included in a tubular neighborhood of size $C\mu$ of $Z(M)$, that is every point of $Z_\mu(M)$ is at distance less than $C\mu$ from $Z(M)$.

*Proof.* Fix $2 \le r \le k1$ and $i \in \text{Emb}^k(M, \mathbb{R}^D)$ as in the statement. Our goal is to show that each critical point of $i(M)$ satisfies the criterion in Corollary 6.8; then the corollary will guarantee that $i(M)$ satisfies Condition (P4).

Let $z_0 \in Z(M)$ with $\pi_M(z_0) = \{x_1, \dots, x_s\}$ for some $s \in \{2, \dots, D+1\}$, and $E = \text{Vec}(x_j - x_l : j, l \in [s]\}$. Let $\delta > 0$ be small enough that there exist maps $p_j : B(z_0, \delta) \to B(x_j, \delta') \cap M$ that satisfy the conclusions of Lemma 4.8 for $j = 1, \dots, s$ (and for some $\delta' > 0$). We have to prove that there exist constants $\delta > \tilde{\delta} > 0$ and $L > 0$ such that for any $h \in E^\perp$ with $\|h\| < \tilde{\delta}$, the $s$-volume of the $s$-simplex $\Delta(h) := \text{Conv}(\{z_0 + h\} \cup \{p_j(z_0 + h) : j \in [s]\})$ satisfies

$$ \text{Vol}_s(\Delta(h)) \ge L \|h\|. \qquad (32) $$

As in the proof of Lemma 8.15, the point $a = (m_1, \dots, m_s, x_1, \dots, x_s, [i], \dots, [i]) \in J_s^r(M, \mathbb{R}^D)$ (where $i(m_j) = x_j$) is such that $d(x_1, z_0) = \dots = d(x_s, z_0)$, $z_0 \in \text{Conv}(x_1, \dots, x_s)$, the rank of $d m_j$ is $m$, and for any chart $\phi : (U, 0) \to (M, m_j)$ the quadratic form $\mathbb{R}^m \to \mathbb{R}$, $v \mapsto \|d(i \circ \phi)_0(v)\|^2 - \langle z_0 - x_j, d^2(i \circ \phi)_0(v, v) \rangle$ is positive definite (for $j = 1, \dots, s$). The last point is true because we have assumed that $i$ satisfies Condition (P3), which is equivalent to this property (as was shown in Lemma 3.1). If the form $B_{a,z_0}$ from Definition 8.21 was degenerate on $E^\perp$, the point $a$ would satisfy all the conditions of the definition of $(W_4^\circ)^{(s)}$ and hence would belong to $W_4^{(s)} \cap \text{Im}(j_s^r(i)) \subset J_s^r(M, \mathbb{R}^D)$. But by hypothesis $j_s^r(i) : M^{(s)} \to J_s^r(M, \mathbb{R}^D)$ is transverse to the Whitney stratifiable sets $\{W_{4,\nu}^{(s)}\}_{\nu \in N}$ and those are of codimension at least
$sm + 1$; hence the intersection of $\mathrm{Im}(j_s^r(i))$ with $W_{4,\nu}^{(s)}$ is trivial for any $\nu \in N$, and so is its intersection with $W_4^{(s)}$. Consequently, the form $B_{a,z_0}$ is non-degenerate on $E^{\perp}$.

Remember that by definition, the form $B_{a,z_0}$ is defined as

$$B_{a,z_0}(h) := \sum_{j_1,j_2=1}^{s} \det(A_{a,z_0,j_1}(h)^{\top} A_{a,z_0,j_2}(h))$$

for $h \in \mathbb{R}^D$, where $A_{a,z_0,j}(h)$ is the $D \times s$ matrix whose $l$-th column is equal to $x_l - z_0$ if $l \neq j$ and whose $j$-th column is $P_{[i],x_j,z_0}(h) - h$, with $P_{[i],x_j,z_0}$ being the differential at $z_0$ of the projection on a small neighborhood of $x_j$ in $\mathbb{M}$ (as in the statement of Lemma 4.3). This projection coincides with $p_j : B(z_0, \delta) \to B(x_j, \delta') \cap \mathbb{M}$, hence by definition of the form $B$ from Equation 25 in Lemma 6.7 we have $B(h) = B_{a,z_0}(h)$ for all $h \in E^{\perp}$ with $\|h\| < \bar{\delta}$ for some $0 < \bar{\delta} < \delta$. By applying Corollary 6.8 and using the same reasoning for each critical point of $\mathbb{M}$, this is enough to conclude that $\mathbb{M}$ satisfies Condition (P4). $\square$

## 8.4 Density of the transversality conditions

In the previous subsection, we defined five sets in the space of multijets that can be written as finite unions of closed Whitney stratified sets, and shown that the transversality to those sets of the map $j_s^r(i) : M^{(s)} \to J_s^r(M, \mathbb{R}^D)$ (respectively the map $(j_s^r)_{M\rho_i} : M^{(s)} \times \mathbb{R}^D \to J_s^r(M, \mathbb{R}))$ induced by an embedding $i : M \to \mathbb{R}^D$ implies that $i(M)$ satisfies the conditions (P1)-(P4) that are of interest to us.<fnref n="9" /> It remains to show that the set of embeddings for which these properties of transversality are satisfied is dense in the Whitney topology; with this, we will have proved the second half of the Genericity Theorem 1.1, as well as Remark 1.3. To that end, the transversality theorems presented in Subsection 8.2 are the key ingredients.

**Theorem 8.25 (The Density Theorem).** Let $M$ be a compact $C^k$ manifold for some $k \in \mathbb{N} \cup \{\infty\}$ with $k \ge 1$. Then, the set of $C^k$ embeddings $i : M \to \mathbb{R}^D$ such that $M = i(M)$ satisfies conditions (P1), (P2), (P3) and (P4) is dense in $Emb^k(M, \mathbb{R}^D)$ for the Whitney $C^k$-topology.

*Proof.* The only difficulty here is that the transversality theorems 8.1 and 8.3 only state results regarding the density of $C^{r+1}$ functions whose multijets are transverse to sets in $J_s^r(M, \mathbb{R})$ and $J_s^r(M, \mathbb{R}^D)$, as opposed to $C^r$ functions, and relatedly, we had to assume throughout Subsection 8.3 that $M$ was a $C^k$ manifold for $k \ge 3$. Hence we have to be somewhat cautious to account for the cases $k = 1, 2$, as proceeding naively would force us to ask that $M$ be at least $C^3$. The key ingredient here will be that $C^{r_1}$ maps are dense in $C^{r_2}$ maps for $r_1 \ge r_2$ and for the Whitney $C^{r_2}$-topology.

Throughout the proof, the sets $W_0^{(s)}, W_1^{(s)}, W_2^{(s)}, W_3^{(s)}$ and $W_4^{(s)}$ are as defined in Subsection 8.3. Let us start by assuming that $k \ge 3$. For $r = 2$ and a given $s = 2, \dots, D+2$, the set $W_0^{(s)} \subset J_s^2(M)$ is a closed submanifold (hence in particular a closed Whitney stratifiable set). As $k \ge r+1 = 3$, we can apply the transversality theorem adapted to the case of distance functions (Theorem 8.3) to $W_0^{(s)}$ (simply with $A = M^{(s)} \times \mathbb{R}^D$, in the notations of the theorem) to get that the set

$$F_{W_0^{(s)}} := \{f \in C^k(M, \mathbb{R}^D) : (j_s^2)_{M\rho_f} \text{ is transverse to } W_0^{(s)} \text{ in } J_s^2(M, \mathbb{R})\}$$

<footnote n="9">⁹The property of transversality must be verified for either $s = 2, \dots, D+2$, $s = 2, \dots, D+1$ or $s = 3, \dots, D+1$ depending on the set considered.</footnote>
is residual in $C^k(M, \mathbb{R}^D)$ (note that we have not yet restricted ourselves to embeddings). Similarly, we have shown in Lemmas 8.10, 8.14, 8.18 and 8.23 that $W_1^{(s)}$, $W_2^{(s)}$, $W_3^{(s)}$, $W_4^{(s)} \subset J_s^2(M, \mathbb{R}^D)$ are finite unions of closed Whitney stratifiable subsets $\{W_{2,\alpha}^{(s)}\}_{\alpha \in A}$, $\{W_{3,\beta}^{(s)}\}_{\beta \in B}$, $\{W_{1,\gamma}^{(s)}\}_{\gamma \in C}$ and $\{W_{4,\nu}^{(s)}\}_{\nu \in N}$ respectively. The transversality theorem 8.1 (with $A = M^{(s)}$) then yields that for any $s = 2, \dots, D+1$ and $\alpha \in A$, the set

$$F_{W_{2,\alpha}^{(s)}} := \{f \in C^k(M, \mathbb{R}^D) : j_s^2(f) \text{ is transverse to } W_{2,\alpha}^{(s)} \text{ in } J_s^2(M, \mathbb{R})\}$$

is residual in $C^k(M, \mathbb{R}^D)$, and similarly for the sets

$$F_{W_{3,\beta}^{(s)}} := \{f \in C^k(M, \mathbb{R}^D) : j_s^2(f) \text{ is transverse to } W_{3,\beta}^{(s)} \text{ in } J_s^2(M, \mathbb{R})\}$$

and

$$F_{W_{4,\nu}^{(s)}} := \{f \in C^k(M, \mathbb{R}^D) : j_s^2(f) \text{ is transverse to } W_{4,\nu}^{(s)} \text{ in } J_s^2(M_k, \mathbb{R})\}$$

for $\beta \in B$, $\nu \in N$ and $s = 2, \dots, D+1$ and the sets

$$F_{W_{1,\gamma}^{(s)}} := \{f \in C^k(M, \mathbb{R}^D) : j_s^2(f) \text{ is transverse to } W_{1,\gamma}^{(s)} \text{ in } J_s^2(M, \mathbb{R})\}$$

for $\gamma \in C$ and $s = 3, \dots, D+1$. As $C^k(M, \mathbb{R}^D)$ is a Baire space (see e.g. [Hir94, Thm 4.4]), the (finite) intersection

$$F := \bigcap_{s=2}^{D+2} F_{W_0^{(s)}} \cap \bigcap_{s=3}^{D+1} \left( \bigcap_{\gamma \in C} F_{W_{1,\gamma}^{(s)}} \right) \cap \bigcap_{s=2}^{D+1} \left( \bigcap_{\alpha \in A} F_{W_{2,\alpha}^{(s)}} \cap \bigcap_{\beta \in B} F_{W_{3,\beta}^{(s)}} \cap \bigcap_{\nu \in N} F_{W_{4,\nu}^{(s)}} \right)$$

of residual sets is yet again residual, and in particular dense in $C^k(M, \mathbb{R}^D)$. As the set of embeddings $Emb^k(M, \mathbb{R}^D)$ is open in $C^k(M, \mathbb{R}^D)$ (see [Mic80, Prop 5.3]), the set $E := F \cap Emb^k(M, \mathbb{R}^D)$ is dense in $Emb^k(M, \mathbb{R}^D)$.

Let $i \in E$; then $(j_s^2)_{M\rho_i}$ is transverse to $W_0^{(s)}$ for $s = 2, \dots, D+2$, hence we know from Yomdin's Lemma 8.19 that $i(M)$ satisfies Condition (P0). From Lemma 8.15 (which we can apply, as we have assumed that $i$ is $C^k$ with $k \ge 3 = r+1$), and as $j_s^2(i)$ is transverse to the sets $\{W_{2,\alpha}^{(s)}\}_{\alpha \in A}$ and $\{W_{3,\beta}^{(s)}\}_{\beta \in B}$ for $s = 2, \dots, D+1$, we get that $i(M)$ satisfies conditions (P2) and (P3). This, in addition to the transversality of $j_s^2(i)$ to $\{W_{1,\gamma}^{(s)}\}_{\gamma \in C}$ for $s = 3, \dots, D+1$ and to $\{W_{4,\nu}^{(s)}\}_{\nu \in N}$ for $s = 2, \dots, D+1$, allows us to apply Lemmas 8.19 and 8.24 (again, $i$ is at least $C^3$) to get that $i(M)$ also fulfills Conditions (P1) and (P4). Hence we have shown that the $C^k$ embeddings that satisfy conditions (P1)-(P4) are dense in $Emb^k(M, \mathbb{R}^D)$ for the Whitney $C^k$-topology, which completes the proof when $k \ge 3$.

Assume now that $k \in \{1, 2\}$. Using Theorem 2.9 from [Hir94, Chapter 2], we can equip $M$ with a $C^3$ differential structure that is compatible with its given $C^k$ differential structure. To make a clear distinction, we write $M_3$ to refer to $M$ equipped with this structure. Let $U \subset Emb^k(M, \mathbb{R}^D) = Emb^k(M_3, \mathbb{R}^D)$ be any non-empty open set of the Whitney $C^k$-topology. As the $C^3$ embeddings $Emb^3(M_3, \mathbb{R}^D)$ seen as a subset of $Emb^k(M_3, \mathbb{R}^D)$ are dense in $Emb^k(M_3, \mathbb{R}^D)$ for the Whitney $C^k$-topology (see e.g. [Hir94, Section 2, Thm 2.7]), there is a $C^3$ embedding $i \in U \cap Emb^3(M_3, \mathbb{R}^D)$. As we have shown above that the $C^3$ embeddings $i'$ such that $i'(M)$ satisfies Conditions (P1-4) are dense in $Emb^3(M_3, \mathbb{R}^D)$ for the Whitney $C^3$-topology, we can
find such embeddings arbitrarily close to $i$ for the $C^3$-norm, hence also for the coarser $C^k$-norm.
Such an embedding $i'$ is in particular a $C^k$ embedding for the initial structure on $M$, and if
$\|i - i'\|_{C^k}$ is small enough, then $i'$ must belong to $U$. Hence we can find $i' \in U$ such that $i'(M)$
satisfies Conditions (P1-4), which is enough to conclude. $\square$

**Remark 8.26.** The observant reader might wonder why we did not use the “openness” part of the transversality theorems 8.1 and 8.3 to directly get that the set of $C^k$ embeddings $i: M \to \mathbb{R}^D$ such that $i(M)$ satisfies conditions (P1), (P2), (P3) and (P4) is open in $Emb^k(M, \mathbb{R}^D)$ for the Whitney $C^k$-topology. The even more observant reader remembers that Theorem 8.1 guarantees that if $k \ge r+1$, the set of $C^k$ embeddings $i$ such that $j_s^r(i)$ is transverse on $A \subset M^{(s)}$ to some closed Whitney stratified set $W \subset J_s^r(M, \mathbb{R}^D)$ is open in the Whitney $C^{r+1}$-topology under the condition that $A$ be compact (and similarly for Theorem 8.3). Here we face two problems: the first is that we need transversality on $A = M^{(s)}$, which is not compact. This hurdle can in fact be avoided by controlling the distance between the projections of critical points, though some work is required. The second problem is that for our sets $W \subset J_s^r(M, \mathbb{R}^D)$ to be defined, we need $r \ge 2$, hence $k \ge 3$, and we could not find a way around this limitation; hence, though we could have shown that the set of suitable embeddings is open in the Whitney $C^3$-topology using the two transversality theorems, we needed the more concrete and geometric proof of the $C^2$ Stability Theorem 1.4 to show that this set is also open in the Whitney $C^2$-topology.

As a corollary of the Density Theorem and the $C^2$ Stability Theorem 1.4, we finally get the Genericity Theorem 1.1.

*Proof of the Genericity Theorem 1.1.* The density part of the statement comes directly from the Density Theorem 8.25. From the $C^2$ Stability Theorem 1.4, we get that the set of embeddings $i \in Emb^2(M, \mathbb{R}^D)$ such that $i(M)$ satisfies (P1), (P2), (P3) and (P4) is open in $Emb^2(M, \mathbb{R}^D)$ for the Whitney $C^2$-topology. As the Whitney $C^2$-topology on $Emb^k(M, \mathbb{R}^D) \subset Emb^2(M, \mathbb{R}^D)$ is the subspace topology inherited from the Whitney $C^2$-topology on $Emb^2(M, \mathbb{R}^D)$, the set of $C^k$ embeddings such that $i(M)$ satisfies (P1), (P2), (P3) and (P4) is also open in $Emb^k(M, \mathbb{R}^D)$. $\square$

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