# Isolated singularities of hypersurfaces

Rustam Sadykov  and  Stanislav Trunov

###### Abstract.

Introduced by Seifert and Threlfall, cylindrical neighborhoods is an essential
tool in the Lusternik-Schnirelmann theory. We conjecture that every isolated
critical point of a smooth function admits a cylindrical ball neighborhood. We
show that the conjecture is true for cone-like critical points, Cornea
reasonable critical points, and critical points that satisfy the Rothe $H$
hypothesis. In particular, the conjecture holds true at least for those
critical points that are not infinitely degenerate.

###### 1991 Mathematics Subject Classification:

58C05, 58K05, 55M30

##  1\. Introduction

Given a smooth function $f\colon M\to\mathbb{R}$ on a manifold $M$ without
boundary, a point $x$ in $M$ is said to be _critical_ if the differential
$d_{x}f$ of $f$ at $x$ is trivial. A point that is not critical is said to be
_regular_. We say that a value $c$ of $f$ is _critical_ if the fiber
$f^{-1}(c)$ contains a critical point. Critical points of functions could be
extremely complicated. For example, the fiber $f^{-1}(c)$ over a regular value
$c$ is a hypersurface of $M$ with no singularities. On the other hand, every
closed subset $V\subset M$ is the hypersurface level $f^{-1}(c)$ of an
appropriately chosen smooth function $f$ on $M$. In other words, critical
points of smooth functions are at least as complicated as closed subsets of a
manifold.

Let $x_{0}$ be a critical point of a smooth function $f$ with critical value
$f(x_{0})=c$. We say that an isolating neighborhood $U$ of $x_{0}$ is
_cylindrical_ if it consists of trajectories in
$f^{-1}[c-\varepsilon,c+\varepsilon]$ of the gradient vector field of $f$. We
will always assume that $U$ is a smooth manifold of dimension $\dim M$ with
corners such that $U\cap f^{-1}(c-\varepsilon)$ and $U\cap
f^{-1}(c+\varepsilon)$ are smooth manifolds with (smooth) boundary.
Cylindrical neighborhoods were introduced by Seifert and Threlfall [16] in
1938, and were used in [13, 19, 4] and [3]. The existence of a cylindrical
ball neighborhood, i.e., a cylindrical neighborhood homeomorphic to a ball, is
an important property as it allows one to apply, for example, the Lusternik-
Schnirelman type argument and deduce Lusternik-Schnirelman inequalities [19,
3, 15].

In this paper we study cylindrical ball neighborhoods and give evidence
supporting Conjecture 1. We note that in dimensions $\neq 4,5$ all smooth
manifolds homeomorphic to a ball are diffeomorphic to a ball.

######  Conjecture 1.

Every isolated critical point of a smooth function admits a cylindrical ball
neighborhood.

If there is an open subset $U\subset\mathbb{R}^{m}$, and a coordinate
neighborhood $\tau\colon U\to M$ of a point $x\in M$ such that $f\circ\tau$ is
the restriction of a polynomial, we say that the critical point $x$ of $f$ is
_algebraic_. In this case, the singular hypersurface $V=f^{-1}(c)$ is locally
an algebraic set. It is known (e.g. see [10]) that for every algebraic
critical point $x_{0}$ of $f$ with $f(x_{0})=c$, there is a closed disc
neighborhood $B_{\varepsilon}$ of $x_{0}$ in $M$ such that $\partial
B_{\varepsilon}$ is transverse to $V$, and the pair $(B_{\varepsilon},V\cap
B_{\varepsilon})$ is homeomorphic to the cone over the pair $(\partial
B_{\varepsilon},V\cap\partial B_{\varepsilon})$. We say that
$(B_{\varepsilon},V\cap B_{\varepsilon})$ is a _cone neighborhood pair_ for
$x_{0}$. This essential property of algebraic singular points is a starting
point for the theory of hypersurface singularities, see [10]. It follows (see
[6, 7] and Proposition 17) that a critical point $x_{0}$ admits a cone
neighborhood pair if and only if it is _cone-like_ , i.e., if it admits a cone
neighborhood in $f^{-1}(f(x_{0}))$.

######  Theorem 2.

Every cone-like critical point of a smooth function $f\colon M\to\mathbb{R}$
admits a cylindrical ball neighborhood.

A version of Theorem 2 appears in [6, p. 396] for manifolds $M$ of dimension
$m\neq 4$.

In [3], Cornea defined a _reasonable critical point_ as an isolated critical
point $x_{0}$ of $f$ such that $f^{-1}(f(x_{0}))$ admits a Whitney
stratification into the stratum $\\{x_{0}\\}$ and its complement. By [3, Lemma
2] every reasonable critical point admits a cone neighborhood pair, and,
therefore, it is cone-like.

######  Corollary 3.

Every reasonable critical point admits a cylindrical ball neighborhood.

Another large class of critical points was studied by E. Rothe in [13] and
[14]. We say that an isolated critical point $x_{0}$ satisfies the hypothesis
$H$ if there is an isolating coordinate neighborhood $U$ of $x_{0}$ such that
for all $x\neq x_{0}$ in $U\cap f^{-1}(x_{0})$ the vectors $x-x_{0}$ and
$\mathop{\mathrm{missing}}{grad}f(x)$ are linearly independent.

######  Corollary 4.

Every critical point satisfying the hypothesis $H$ admits a cylindrical ball
neighborhood.

We will show that the set of jets of non cone-like critical points is of
infinite codimension in the space of all jets, see Corollary 20. In other
words, non cone-like critical points are extremely rare and infinitely
degenerate. We also deduce that the critical point of any infinitely
determined map germ is cone-like, see Proposition 23.

Finally, we show that the class of critical points that admit a cylindrical
ball neighborhood diffeomorphic to a ball is larger than the class of cone-
like critical points.

######  Theorem 5.

For every $m\geq 4$ there is a smooth function
$f\colon\mathbb{R}^{m}\to\mathbb{R}$ with an isolated critical point $x_{0}$
such that $x_{0}$ is not cone-like and such that $x_{0}$ admits a cylindrical
neighborhood diffeomorphic to a ball.

The functions $f$ that appear in Theorem 5 are constructed by Takens in [18].
Theorem 5 implies that critical points of Takens functions are removable, see
Corollary 27. This is closely related to the Funar theorem [5, Proposition
2.1] asserting that cone-like critical points of smooth maps $f\colon M\to N$
of a manifold of dimension $m\leq 2k-1$ to a manifold of dimension $k\geq 2$
are topologically removable except possibly when
$(m,k)\in\\{(2,2),(4,3),(8,5),(16,9)\\}$, and under an additional condition
that the critical point admits an adapted neighborhood diffeomorphic to a ball
when $(m,k)=(5,3)$.

Theorem 2 and its corollaries are related to the Lusternik-Schnirelmann
category [3, 17, 19]. In fact, in [15] we will show that a closed manifold of
dimension at least $6$ admits a Singhof-Takens filling by $n$ smooth balls if
and only if it admits a function with $n$ critical points each of which admits
a cylindrical ball neighborhood.

In section 2 we give a detailed proof of Theorem 2. Next, in section 3 we
prove Corollary 4, and show that every cone-like critical point admits a cone
neighborhood pair. In section 4 we review relevant theorems from singularity
theory and deduce that the set of jets of map germs with non cone-like
critical points is a subset of a proalgebraic set of infinite codimension in
the space of all jets. We also show that critical points of infinitely
determined map germs are cone-like. Finally, in section 5, we list properties
of Takens critical points, namely: the Takens function germs are flat
(Proposition 24), the critical points of Takens functions admit cylindrical
neighborhoods (Proposition 25) diffeomorphic to a ball, and the critical
points of Takens functions are removable (Corollary 27). The proof of Theorem
5 can also be found in section 5.

We are grateful to Louis Funar for valuable comments, and references.

##  2\. Proof of Theorem 2

Let $f$ be a smooth function on a manifold $M$ without boundary. We choose a
Riemannian metric on $M$. It defines a gradient flow $\gamma_{t}$ on $M$ such
that the trajectory $t\mapsto\gamma_{t}(y)$ of any point $y$ is a curve in $M$
parametrized by an open interval $(t_{1},t_{2})$ where $-\infty\leq
t_{1}<t_{2}\leq\infty$. Every trajectory $t\mapsto\gamma_{t}(y)$ is either
closed in $M$ or the limit of $\gamma_{t}(y)$ as $t\to\infty$ or $t\to-\infty$
is a critical point of $f$. For example, for every critical point $x\in M$,
its trajectory is a constant curve $t\mapsto x$ parametrized by
$t\in(-\infty,\infty)$. The union $D$ of all non-closed trajectories of the
gradient flow as well as all critical points of $f$ is a closed subset of $M$.
The closed set $D$ is a union of closed subsets $D(x)$ of points $y$ such that
for the trajectory $t\mapsto\gamma_{t}(y)$ the limit of $\gamma_{t}(y)$ as
$t\to\infty$ or $t\to-\infty$ is the critical point $x$.

For any value $a$ of the function $f$, let $M_{a}$ denote the level
$f^{-1}(a)$. The intersection of $D$ with $M_{a}$ will be denoted by $D_{a}$.
Similarly, $D_{a}(x)$ denotes the intersection of $D(x)$ and $M_{a}$.

The gradient flow $\gamma_{t}$ defines a diffeomorphism

$h_{a,b}\colon M_{a}\setminus D_{a}\to M_{b}\setminus D_{b}$

for every pair $(a,b)$ of real numbers such that $a<b$ by associating to a
point $x$ a unique point in $M_{b}\setminus D_{b}$ on the trajectory
$t\mapsto\gamma_{t}(x)$ of $x$. We will also write $h_{b,a}$ for the inverse
of $h_{a,b}$.

Suppose now that $x_{0}$ is a cone-like singular point of a smooth function
$f$. Since $x_{0}$ is isolated, we may assume that $f(x_{0})=0$, and $x_{0}$
is the only critical point on the singular hypersurface $M_{0}$. Similarly, we
may assume that there are regular values $c$ and $-c$ of $f$ such that $0$ is
a unique critical value in the interval $(-c-\epsilon,c+\epsilon)$ for some
$\epsilon>0$.

Let $(B_{\varepsilon},M_{0}\cap B_{\varepsilon})$ be an isolating cone
neighborhood pair for $x_{0}$. We recall that $\partial B_{\varepsilon}$
intersects $M_{0}$ transversally. We will write $V$ for $M_{0}\cap
B_{\varepsilon}$. Let $H$ denote the subset of $M_{[-c,c]}$ of points $x$ such
that either the gradient curve through $x$ intersects $V$ or one of the limits
of the gradient curve through $x$ is $x_{0}$, where $M_{[-c,c]}$ is the
submanifold of $M$ of points $x$ such that $f(x)\in[-c,c]$. We write $H_{a}$
for the intersection of $H$ and $M_{a}$. By choosing $V$ and $c$ sufficiently
small, we may assume that $H\subset B_{\varepsilon}$.

######  Lemma 6.

The subsets $H_{c}\subset M_{c}$ and $H_{-c}\subset M_{-c}$ are smooth
submanifolds with (smooth) boundary. The subset $H\subset M$ is a smooth
manifold of codimension $0$ with corners $\partial H_{-c}\sqcup\partial
H_{c}$. In other words, $H$ is a smooth cylindrical neighborhood of $x_{0}$.

###### Proof.

Let $x$ be a point in $H\setminus D$. Suppose that $f(x)\in(-c,c)$. We will
omit the cases $f(x)=c$ and $f(x)=-c$ as the argument in these cases is
similar. Without loss of generality we may assume that $f(x)\leq 0$; the case
$f(x)\geq 0$ is similar.

Let $\delta$ denote the gradient flow of $f$. Then for some $t\geq 0$, the
point $\delta_{t}(x)$ is in $V$. Suppose $x\in H$ is a point with
$\delta_{t}(x)\in\partial V$. Then for a small neighborhood $U$ of $x$ in $M$,
there is a smooth map $\varphi\colon U\to M_{0}$ that takes $y\in U$ to a
unique point $\varphi(y)\in M_{0}$ on the trajectory of $y$. In view of the
diffeomorphism $h_{f(x),0}$, the map $\varphi$ is a submersion onto a small
neighborhood of $\delta_{t}(x)$. Therefore $x$ admits a half disc neighborhood
in $H$. The case where $\delta_{t}(x)$ is in the interior of
$V\setminus\\{x_{0}\\}$ is similar.

Suppose now that $x\in D\cap H$. We still assume that $f(x)\leq 0$. We need to
show that for every point $y$ sufficiently close to $x$, either $y\in D$ or
the trajectory of the gradient flow of $f$ through $y$ either intersects $V$.
Assume that $y$ is not in $D$, and the trajectory through $y$ does not
intersect $V$. Then there is a sequence of points $y_{i}$ approaching $x$ such
that each trajectory $t\to\delta_{t}(y_{i})$ passes through a point $z_{i}$ in
$M_{0}\setminus V$ at some moment $t_{i}$. We may assume $f(y_{i})=f(x)$.
Since $M_{0}$ is compact, there is an accumulating point $z$ of the points
$z_{i}$ in $M_{0}$. In fact, by dropping some of the points $y_{i}$ (and
$z_{i}$), we may assume that $z$ is a limit point of $z_{i}$. We claim that
the trajectory of the gradient flow of $f$ through the point $z$ passes
through the point $x$, which contradicts the assumption that the trajectory of
the gradient flow through $x$ has a limit point at $x_{0}\in V$. To prove the
claim, let $y$ denote the point on the trajectory $t\mapsto\delta_{-t}(z)$
such that $f(y)=f(x)$. Then the trajectories through the points $z_{i}$
sufficiently close to $z$ intersect the level set $f^{-1}(f(y))$ at points
close to $y$, i.e., the points $y_{i}$ approach the point $y$. Thus, $y$
coincides with $x$.

Finally, the maps $h_{-c,0}$ and $h_{0,c}$ define diffeomorphisms $\partial
H_{-c}\to\partial V$ and $V\to\partial H_{c}$, and therefore, the subsets
$H_{c}$ and $H_{-c}$ are smooth submanifolds with smooth boundary. ∎

######  Proposition 7.

The manifold $H$ is a cone over $\partial H$ with vertex at $x_{0}$.

###### Proof.

Since $V$ is a cone over its boundary with vertex at $x_{0}$, it is a union of
$x_{0}$ and $\partial V\times[0,1]$. Let $v^{\prime}$ denote a vector field
along the second component in $V\times[0,1]$. Let $\lambda$ denote a monotonic
function on $[0,1]$ that is $0$ only at $0$ and $1$ at $1$. Let $\pi$ denote
the projection $V\times[0,1]\to[0,1]$. Then $(\lambda\circ\pi)v^{\prime}$
extends to a vector field $v$ on $V$.

The vector field $v$ restricts to a tangent radial vector field over the
manifold $V\setminus\\{x_{0}\\}$, it is trivial only at $0$, and any point in
$V\setminus\\{x_{0}\\}$ escapes $V$ along $v$ in finite time.

We will now use the gradient vector field $\mathop{\mathrm{missing}}{grad}(f)$
to extend the vector field $v$ over $H$, see Lemma 8. For any regular point
$x\in H$ of $f$, the tangent space $T_{x}M$ is the direct sum of the
_horizontal_ component $T_{x}M_{f(x)}$ and the _vertical_ component generated
by $\mathop{\mathrm{missing}}{grad}(f)$. We say that a vector field $u$ over
$H\setminus\\{x_{0}\\}$ is _horizontal_ if the vector $u(x)$ is in
$T_{x}M_{f(x)}$ for all $x\in H\setminus\\{x_{0}\\}$.

######  Lemma 8.

The vector field $v$ extends to a continuous vector field over $H$ such that
$v|H\setminus\\{x_{0}\\}$ is a nowhere zero horizontal vector field.

###### Proof.

For any point $x\in H\setminus D(x_{0})$ with $f(x)<0$, there is a unique
value $t>0$ such that $\delta_{t}(x)$ is a point $y$ in $V$, where
$\delta_{t}$ is the gradient flow of $f$. We define $v(x)$ by

$v(x)=\pi\circ(d\delta_{t})^{-1}(v(y)),$

where the map $\pi$ projects the tangent space $T_{y}M$ at $y\in
M_{[-c,c]}\setminus x_{0}$ to the tangent space $T_{y}M_{f(y)}$ of the level
manifold.

We claim that $v(x)$ is non-zero. Since $\delta_{t}$ takes gradient curves of
$f$ to gradient curves of $f$, it takes $\nabla f(y)$ to $\nabla f(x)$, and
therefore it takes $v(y)$ to a vector linearly independent from $\nabla f(x)$,
i.e., to a vector that is not in the kernel of $\pi$.

Similarly, for $x$ in $H\setminus D(x_{0})$ with $f(x)>0$ we define

$v(x)=(\pi\circ d\delta_{t})(v(y)),$

where $x=\delta_{t}(y)$ for some $t>0$ and $y\in V$. Finally, we set $v(x)=0$
for all $x\in D(x_{0})$. Then $v$ is a desired continuous vector field over
$H$. ∎

Let $w$ denote $f\cdot\nabla f$. Then $w(x)$ vanishes precisely over the set
$M_{0}$ as well as over the critical points of $f$.

######  Lemma 9.

The vector field $v+w$ is a continuous vector field on $H$ which is nowhere
zero on $H\setminus\\{x_{0}\\}$.

###### Proof.

Since $w$ and $v$ are continuous vector fields, we deduce that $w+v$ is a
continuous vector field over $H$. It is easily verified that $w+v$ is trivial
only at $x_{0}$. ∎

######  Lemma 10.

The corners of $H$ can be smoothened so that $v+w$ is still defined on the
modified manifold $H$ and $v+w$ is nowhere tangent to $\partial H$.

###### Proof.

Recall that the vector field $w(x)$ is a positively scaled gradient vector
field over $M_{(0,c]}$ and it is a negatively scaled gradient vector field
over $M_{[-c,0)}$. For all non-corner points $x$ of $\partial H$ on the level
$\pm c$, the tangent space of $\partial H$ is tangent to the level $M_{\pm
c}$, the vector $v(x)$ is also tangent to the level $M_{\pm c}$ while $w(x)$
is normal to $\partial H$ directed outward $H$. Therefore the vector field
$v+w$ over $\partial H\cap M_{\pm c}$ is outward normal.

For all non-corner points $x$ in $\partial H$ that are not on the levels $\pm
c$, the vector $v(x)$ is outward normal, while $w(x)$ is tangent to $\partial
H$. Again, the vector field $v+w$ is outward normal over $\partial H\setminus
M_{\pm c}$.

Finally, at any corner point $x$, locally $H$ is a quarter subspace of
$\mathbb{R}^{m}$ bounded by the horizontal space $f^{-1}(c_{+})$ or
$f^{-1}(c_{-})$, and a vertical space composed of gradient flow curves. It
follows that the corner of $H$ can be smoothened, and the vector field $v+w$
can be modified near the smoothened corner so that it is outward normal
everywhere over $\partial H$, see [12]. ∎

To summarize, we smoothened the corners of $H$ so that $H$ is a manifold with
smooth boundary $\partial H$. We also constructed a vector field $v+w$ with a
unique critical point at $x_{0}$. The vector field $v+w$ is outward normal
everywhere over $\partial H$.

######  Lemma 11.

The manifold $H\setminus\\{x_{0}\\}$ is diffeomorphic to $\partial
H\times[0,\infty)$.

###### Proof.

Let $\delta^{-v-w}_{t}$ denote the flow along the vector field $-v-w$. Suppose
that the trajectory $t\mapsto\delta^{-v-w}_{t}(z)$ is well-defined for
$t\in[0,a]$. Since $-v-w$ is inward normal over $\partial H$, the trajectory
of any point $z$ does not escape $H$, and therefore $\delta^{-v-w}_{a}(z)$ is
in $H$. Consequently, the curve $t\mapsto\delta^{-v-w}_{t}(z)$ is well-defined
for $t\in[0,a+\varepsilon)$ for some $\varepsilon$. Suppose now that
$t\mapsto\delta^{-v-w}_{t}(z)$ is well-defined for $t\in[0,a)$ and it is not
well-defined for $t\in[0,a]$. Since $-v-w$ is nowhere zero, we deduce that the
limit of $\delta^{-v-w}_{t}(z)$ as $t\to a$ is $x_{0}$.

Next, we claim that for every point $x\in H\setminus\\{x_{0}\\}$, its flow
curve along $-v-w$ approaches $x_{0}$. Indeed, along the vector field $-w$
every point of $H$ flows towards $V$, while along the vector field $-v$ the
points of $H$ stay on the same level of $f$. Consequently, any point in
$H\setminus\\{x_{0}\\}$ appears arbitrarily close to $V$ after flowing along
$-v-w$ for sufficiently long time. On the other hand, by the definition of $v$
any point on $V$ flows towards $x_{0}$ along $-v$.

Let $\lambda>0$ denote a function on $H\setminus\\{x_{0}\\}$ such that
$\lambda(x)$ and all its derivatives tend to $0$ as $x\to x_{0}$. We may
choose the function $\lambda$ so that $\lambda\equiv 1$ near $\partial H$, and
every trajectory $\delta^{-\lambda(v+w)}_{t}(z)$ of $-\lambda(v+w)$ starting
from any point $z\in\partial V$ is well-defined for $t\in[0,\infty)$. Then the
vector field $-\lambda(v+w)$ defines a diffeomorphism $\varphi\colon
H\setminus\\{x_{0}\\}\to\partial H\times[0,\infty)$ by $\varphi(x)=(y,t)$ for
a unique point $y\in\partial H$ and a unique value $t\geq 0$ such that
$\delta_{t}^{-\lambda(v+w)}(y)=x$. ∎

Thus, for every point $x\in H\setminus\\{x_{0}\\}$, the flow curve through $x$
passes through a unique point of $\partial H$, and has $x_{0}$ as its limit
point. This implies that $H$ is a cone over $\partial H$. ∎

######  Corollary 12.

The manifold $H$ is a smooth manifold homeomorphic to a ball.

###### Proof.

Since $H$ is a cone over a manifold, and $H$ is itself a manifold, it follows
that $H$ is a disc, see [15, Lemma 12]. ∎

Since the vector field $-v-w$ is tangent to $V\setminus\\{x_{0}\\}$ over
$V\setminus\\{x_{0}\\}$, it follows that $(H,V)$ is a cone neighborhood pair
for the critical points $x_{0}$. This completes the proof of Theorem 2.

######  Remark 13.

In the theory of maps with isolated cone-like critical points to manifolds of
arbitrary dimension adapted neighborhoods [5] play an important role. In
general the definition involves a list of assumptions, but in the case of a
map to $\mathbb{R}$ the definition is equivalent to the following one. An
_adapted_ neighborhood is a closed isolated connected neighborhood $U$ of a
critical point of a continuous function such that the level set $f^{-1}(a)$ is
transverse to $\partial U$ for each $a$ in the interior of $f(U)$. Clearly,
every cylindrical neighborhood is an adapted neighborhood.

##  3\. Proof of Corollary 4

Let $x_{0}$ be an isolated critical point of a function $f\colon
M\to\mathbb{R}$ on a smooth manifold of dimension $n$. By restricting the
function $f$ to a coordinate neighborhood about $x_{0}$, we may assume that
$M$ is $\mathbb{R}^{n}$. In [13] E. Rothe studied the class of critical points
satisfying the hypothesis $\mathcal{H}$.

Hypothesis $\mathcal{H}$. There exists an isolating neighborhood $U$ of
$x_{0}$ such that for all $x\neq 0$ in $U\cap f^{-1}(x_{0})$ the vectors
$x-x_{0}$ and $\nabla f(x)$ are linearly independent.

The hypothesis $\mathcal{H}$ is closely related to the hypothesis
$\mathcal{H}^{\prime}$.

Hypothesis $\mathcal{H}^{\prime}$. There exists an isolating neighborhood $U$
of $x_{0}$ such that over the complement to $x_{0}$ in $V=U\cap f^{-1}(x_{0})$
there is a smooth vector field $v$ which is nowhere zero and which is outward
normal over $\partial V$.

######  Lemma 14.

The hypothesis $\mathcal{H}$ implies the hypothesis $\mathcal{H}^{\prime}$.

###### Proof.

Let $f$ be a smooth function with an isolated critical point $x_{0}$ that
satisfies the hypothesis $\mathcal{H}$, i.e., there exists an isolating
neighborhood $U$ of $x_{0}$ with a certain property. Let $r\colon
U\to\mathbb{R}$ denote the function that associates with $x$ the distance
between $x$ and $x_{0}$ with respect to the Euclidean metric on
$\mathbb{R}^{n}$. We may choose a ball neighborhood $U_{\varepsilon}$ of
$x_{0}$ of radius $\varepsilon$ so that $U_{\varepsilon}\subset U$ and
$\varepsilon$ is a regular value of $r$. Put
$V_{\varepsilon}=U_{\varepsilon}\cap f^{-1}(x_{0})$.

Since the vectors $x-x_{0}$ and $\nabla f(x)$ are linearly independent, the
projection $v(x)$ of the vector $x-x_{0}$ to $T_{x}V$ is non-zero for all
$x\in V\setminus\\{x_{0}\\}$. Then the vector field $v$ satisfies the
hypothesis $\mathcal{H}^{\prime}$. ∎

The proof of the following Lemma 15 is similar to one of Lemma 11.

######  Lemma 15.

The manifold $V\setminus\\{x_{0}\\}$ is diffeomorphic to $\partial
V\times[0,\infty)$.

We also note that the limit of $\delta^{-\lambda v}_{t}(z)$ as $t\to\infty$ is
$x_{0}$ for all $z\in\partial V$. Therefore we deduce Corollary 16.

######  Corollary 16.

The set $V$ is homeomorphic to the cone over its boundary.

Then the proof of Proposition 7 shows that a critical point satisfying the
hypothesis $H$ admits a cylindrical ball neighborhood.

The above argument proves a characterizations of a cone neighborhood pair for
an isolated critical point.

######  Proposition 17.

Let $B_{\varepsilon}$ be an $\varepsilon$-disc neighborhood of $x_{0}$ such
that $\partial B_{\varepsilon}$ is transverse to $M_{0}$. Suppose that
$V=B_{\varepsilon}\cap M_{0}$ is a cone over its boundary with vertex at
$x_{0}$. Then there exists a cone neighborhood pair for $x_{0}$.

###### Proof.

Since $V$ is a cone over its boundary with vertex at $x_{0}$, the complement
$V\setminus\\{x_{0}\\}$ is diffeomorphic to $V\times(-\infty,0]$.
Consequently, there is a nowhere zero vector field $v$ on
$V\setminus\\{x_{0}\\}$ that is outward normal over $\partial V$. By an
argument as in the proof of Proposition 7, we conclude that the critical point
$x_{0}$ admits a cylindrical ball neighborhood $H$. In fact, the construction
in the proof of Proposition 7 also produces a nowhere zero vector field $w+v$
over $H\setminus\\{0\\}$ that restricts to $v$ over $V$. Consequently, the
pair $(H,H\cap V)$ is a cone neighborhood pair for $x_{0}$. ∎

##  4\. Algebraic singular points

In this section we make precise the statement that the set of Taylor series of
map germs with non cone-like critical points is of infinite codimension in the
space of all Taylor series. In subsection 4.1 we recall the Tourgeron theorem
on finitely determined map germs and deduce that map germs with non cone-like
critical points are extremely rare, see Corollary 20. In subsection 4.2 we
will discuss infinitely determined map germs, which is a larger class than the
class of finitely determined map germs, and show that if a singular map germ
is infinitely determined, then its critical point is cone-like, see
Proposition 23.

###  4.1. Finitely determined map germs

Let $\mathcal{E}(n,p)$ denote the vector space of map germs at $0$ of smooth
functions $\mathbb{R}^{n}\to\mathbb{R}^{p}$. The vector space
$\mathcal{E}(n)=\mathcal{E}(n,1)$ of function germs is a ring with
multiplication and addition induced by multiplication and addition of
functions. Let $\mathcal{B}(n)$ denote the subset of invertible map germs $f$
in $\mathcal{E}(n,n)$ with an additional condition $f(0)=0$. The set
$\mathcal{B}(n)$ is a group with operation given by the composition. We say
that map germs $f_{0},f_{1}\in\mathcal{E}(n,p)$ are _right equivalent_ if
there is a map germ $h\in\mathcal{B}(n)$ such that $f_{0}\circ h=f_{1}$. For a
non-negative integer $k$, the _$k$ -jet_ $\hat{f}$ of a map germ $f$ at $0$ is
the Taylor polynomial of $f$ at $0$ of order $k$. Similarly, for $k=\infty$,
the $k$-jet of $f$ is the Taylor series of $f$ at $0$. The vector space of all
$k$-jets at $0$ is denoted by $\hat{\mathcal{E}}_{k}(n,p)$. The vector space
of Taylor series of map germs $\mathbb{R}^{n}\to\mathbb{R}^{p}$ at $0$ is
called the (infinite) _jet space_. It is denoted by
$\hat{\mathcal{E}}(n,p)\colon=\hat{\mathcal{E}}_{\infty}(n,p)$. There is a
sequence of projections of Euclidean spaces

$\hat{\mathcal{E}}(n,p)\to\cdots\to\hat{\mathcal{E}}_{k+1}(n,p)\to\hat{\mathcal{E}}_{k}(n,p)\to\cdots$

that truncate the series/polynomials. The truncation
$\hat{\mathcal{E}}(n,p)\to\hat{\mathcal{E}}_{k}(n,p)$ is denoted by $\pi_{k}$.
We say that a subset $X$ in $\hat{\mathcal{E}}(n,p)$ is _proalgebraic_ if it
is of the form $\cap\pi^{-1}_{k}X_{k}$ for some algebraic sets
$X_{k}\subset\hat{\mathcal{E}}_{k}(n,p)$. By definition, the _codimension_ of
$X$ is the upper limit of codimensions of the algebraic subsets $X_{k}$.

A map germ $f$ is said to be _$k$ -definite_ if every map germ $g$ with the
same $k$-jet $\hat{g}=\hat{f}$ is right equivalent to $f$. For example, a
function germ $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is $1$-determined at $0$ if
and only if it is non-singular at $0$, i.e., $df(0)\neq 0$. A function germ
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ is $2$-determined at a critical point $0$
if and only if $0$ is a Morse critical point. We say that a map germ
$f\colon(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ is _finite_ if
$\mathcal{E}(n)/\langle f_{1},...,f_{p}\rangle$ is finite dimensional, where
$\langle f_{1},...,f_{p}\rangle$ is the ideal generated by the components
$f_{1},...,f_{p}$ of $f$. We note that the condition $\mathcal{E}(n)/\langle
f_{1},...,f_{p}\rangle=k$ implies

$\dim\mathcal{E}(n)/(\langle
f_{1},...,f_{p}\rangle+m(n)^{k+1})=\dim\hat{\mathcal{E}}_{k}(n)/\langle\hat{f}_{1},...,\hat{f}_{p}\rangle\leq
k,$

where $m(n)$ is the maximal ideal in $\mathcal{E}(n)$. The converse is also
true, i.e., if we have
$\dim\hat{\mathcal{E}}_{k}(n)/\langle\hat{f}_{1},...,\hat{f}_{p}\rangle\leq k$
for some $k$, then $f$ is finite. It is known that if
$df\colon\mathbb{R}^{n}\to\mathbb{R}$ is a finite map germ at $0$, then the
map germ $f\colon\mathbb{R}^{n}\to\mathbb{R}$ at $0$ is finitely determined,
e.g., see [1, §11.10].

On the other hand, let $Y$ denote the set of Taylor series of non-finite map
germs. Then $Y=\cap\pi_{k}^{-1}Y_{k}$ for the sets

$Y_{k}=\\{\hat{g}\in\hat{\mathcal{E}}_{k}(n,p)\ |\
\dim\hat{\mathcal{E}}_{k}(n)/\langle\hat{g}_{1},...,\hat{g}_{p}\rangle>k\\},$

where $\hat{g}_{1},...,\hat{g}_{p}$ are components of $\hat{g}$.

######  Theorem 18 (Tourgeron Theorem).

The sets $Y_{k}$ are algebraic. In fact, if $n\leq p$, then $Y$ is a
proalgebraic set of infinite codimension.

######  Corollary 19.

The set of jets of map germs $f\in\mathcal{E}(n)$ for which $df$ is not finite
is a subset of infinite codimension. Thus, the set of jets of non-finitely
determined map germs is a subset of a proalgebraic set of infinite
codimension.

For proofs of the Tourgeron Theorem and its corollary we refer to [1, Theorem
13.4 and Remark 13.6] respectively.

We note that a $k$-determined map germ $f$ is right equivalent to a polynomial
$g$ of degree $k$. Since the critical point of a polynomial $g$ is algebraic,
we deduce that the critical point of any finitely determined map germ $f$ is
cone-like.

######  Corollary 20.

The set of jets of map germs with non cone-like critical points is a subset of
a proalgebraic set of infinite codimension.

###  4.2. Infinitely determined map germs

Next, we turn to infinitely determined map germs
$f\colon\mathbb{R}^{n}\to\mathbb{R}^{p}$ at $0$. We say that a map germ $f$ at
$0$ is _infinitely determined_ if any map germ $g$ at $0$ that has the same
Taylor series as $f$ is right equivalent to $f$.

Let $J_{f}$ denote the ideal in $\mathcal{E}(n)$ generated by $p\times p$
minors of the differential $d_{0}f$ of $f$ at $0$, where $f$ is a map germ at
$0$ of a function $\mathbb{R}^{n}\to\mathbb{R}^{p}$. We say that a finitely
generated ideal $I$ in $\mathcal{E}(n)$ is _elliptic_ if it contains
$m_{n}^{\infty}$, where $m_{n}$ is the maximal ideal in $\mathcal{E}(n)$. Part
of Theorem 21 was announced in [8] and for the proof, we refer to [2], [11],
and [21], see also [20, Theorem 6.1 and Lemma 6.2].

######  Theorem 21.

The following conditions are equivalent:

  * •

A map germ $f$ is infinitely determined.

  * •

$J_{f}$ is elliptic.

  * •

$||d_{x}f||^{2}\geq C||x||^{\alpha}$ holds on some neighborhood of $0$ for
some constants $C$ and $\alpha>0$, where $||d_{x}f||^{2}$ is the sum of
squares of $p\times p$ minors of $d_{x}f$.

By Theorem 21, if $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is not an infinitely
determined function germ at $0$, then there is a sequence of points
$z_{1},z_{2},....$ converging to $0$ such that

$\left(\frac{\partial f}{\partial
x_{1}}(z_{i})\right)^{2}+\cdots+\left(\frac{\partial f}{\partial
x_{n}}(z_{i})\right)^{2}=o||z_{i}||^{N}$

for all $N$.

######  Example 22.

The map germ $f(x,y)=(x^{2}+y^{2})^{2}$ is not finitely determined [22, p.
237], but it is infinitely determined. The map germ $g(x,y)=(x,y^{4}+xy^{2})$
is not infinitely determined.

######  Proposition 23.

If a singular map germ is infinitely determined, then its critical point is
cone-like.

If $E$ is an equivalence relation on map germs, then we say that a map germ
$f$ is $k$-$E$-determined if every map germ $g$ with the same $k$-jet
$\hat{g}$ as the $k$-jet $\hat{f}$ of $f$ is $E$-equivalent to $f$. An
important example of an $E$-equivalence is a $\mathcal{K}$-equivalence. If $f$
and $g$ are $\mathcal{K}$-equivalent, then there is a diffeomorphism of source
spaces of $f$ and $g$ that takes $f^{-1}(0)$ to $g^{-1}(0)$.

###### Proof.

Suppose that $f$ is infinitely determined. Then it is infinitely
$\mathcal{K}$-determined. Then, by the Brodersen theorem [2], it is finitely
$\mathcal{K}$-determined. In particular, the map germ $f$ is
$\mathcal{K}$-equivalent to a polynomial. Thus, there is a diffeomorphism of
the source spaces of $f$ and $g$ that takes $f^{-1}(0)$ to an algebraic set
$g^{-1}(0)$. ∎

##  5\. The Takens functions

In [19], Takens constructed functions on manifolds of dimension $\geq 3$ with
non cone-like critical points. In this section we study Takens functions, and
show that the critical points of these functions still admit cylindrical
neighborhoods diffeomorphic to a ball.

Given a smooth manifold $X$ with corners and a smooth manifold $Y$ with smooth
boundary, a map $f\colon X\to Y$ as well as its inverse is said to be a
_special almost diffeomorphism_ if it is continuous and restricts to a
diffeomorphism $X\setminus\Sigma\to Y\setminus f(\Sigma)$ where $\Sigma$ is
the set of points of the boundary of $X$ at which $\partial X$ is not smooth.
An _almost diffeomorphism_ of manifolds with corners is a composition of
special almost diffeomorphisms.

The Takens construction is based on the existence for $n\geq 4$ of a set
$Q\subset\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$ such that

  * •

there exists a continuous map $\tau\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ with
$\tau(Q)=0$ such that the restriction $\tau|\mathbb{R}^{n}\setminus Q$ is a
diffeomorphism onto $\mathbb{R}^{n}\setminus\\{0\\}$, and

  * •

there is no closed neighborhood $U$ of $Q$ in $\mathbb{R}^{n-1}$ such that
$U\setminus Q$ is homeomorphic with $\partial U\times(0,1]$.

When $n=4$, we may choose $Q\subset\mathbb{R}^{3}\subset\mathbb{R}^{4}$ to be
the Fox-Artin arc, which is an embedded segment in $\mathbb{R}^{3}$ with non-
trivial fundamental group $\pi_{1}(\mathbb{R}^{3}\setminus Q)$. When $n\geq
5$, the space $Q\subset\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$ can be
constructed by means of the Newman-Mazur compact contractible manifold $M$ of
dimension $n-1$ with non-trivial $\pi_{1}(\partial M)$. The Newman-Mazur
compact manifold $M$ has the properties that

  * •

$M\times[0,1]$ is almost diffeomorphic to a round ball,

  * •

the double of $M$ is diffeomorphic to $S^{n}$, and

  * •

the disc $D^{n-1}$ can be decomposed into a union $A_{1}\cup A_{2}$ of two
manifolds with corners each of which is almost diffeomorphic to the Newman-
Mazur manifold $M$.

Namely, starting with a disc $B_{0}$ of dimension $n-1$, Takens defines a
sequence of compact $(n-1)$-manifolds $B_{i}$, where $B_{i+1}$ is obtained
from $B_{i}$ by removing from $B_{i}$ a collar neighborhood of $B_{i}$ as well
as a subset diffeomorphic to $A_{1}$ so that $B_{i+1}$ is diffeomorphic to the
boundary connected sum $B_{i}\\#_{b}A_{2}$, which is almost diffeomorphic to
$B_{i}\\#_{b}M$. Then $Q=\cap B_{i}$. It follows that in all these cases $Q$
is a compact subset of $\mathbb{R}^{n-1}$.

To construct a smooth function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ with an
isolated non cone-like critical point $0$, we first construct a smooth
function $F\colon\mathbb{R}^{n}\to\mathbb{R}$ with
$F^{-1}(0)=\mathbb{R}^{n-1}$ and $\partial F/\partial x_{n}\equiv 0$ precisely
over $Q$. Then $f$ is defined by $F=f\circ\tau$.

The function $F$ in [18] is of the form

$F(x_{1},...,x_{n})=\lambda(x_{1},...,x_{n-1})\cdot x_{n}+h(x_{n}),$

where $\lambda(x_{1},...,x_{n-1})$ and $h(x_{n})$ are certain smooth functions
such that $\lambda>0$ over the complement to $Q$, $h^{\prime}\geq 0$, and
$h(0)=0$. Namely, let $\\{Q_{i}\\}$ be a basis of compact neighborhoods of $Q$
in $\mathbb{R}^{n-1}$ such that $Q_{i}$ is in the interior of $Q_{i-1}$. Let
$\\{\lambda_{i}\\}$ denote the partition of unity of
$\mathbb{R}^{n-1}\setminus Q$ with support of $\lambda_{1}$ in the complement
of $Q_{2}$, and support of all other functions $\lambda_{i}$ in
$Q_{i-1}\setminus Q_{i+1}$. Then $\lambda=\sum a_{i}\lambda_{i}$, where the
strictly positive constants $a_{i}$ are chosen so that $a_{i}$ is less than
the minimum of $1$ and

$\frac{||x||^{2}}{2^{i}||(\lambda_{i}\cdot x_{n})\tau^{-1}||_{C^{i}}},$

where $||x||^{2}=x_{1}^{2}+\cdots x_{n}^{2}$, and $||g||_{C^{i}}$ is the
maximum of values of derivatives of $g$ of orders at most $i$ over the compact
support of the smooth function $g$ restricted to the disc $||x||\leq 1$.
Therefore, the series $\sum(\lambda_{i}\cdot x_{n})\circ\tau^{-1}$ converges
to a smooth function $(\lambda\cdot x_{n})\circ\tau^{-1}$ whose derivative of
any order is obtained by term by term differentiating the series.

Let $\\{h_{j}\\}$ denote a sequence of functions on $\mathbb{R}^{1}$
parametrized by $j=1,2,...$ such that $h_{j}(x)=x$ if $|x|\geq 1/i$,
$h_{i}(x)=0$ if $|x|<1/(i+1)$ and $h^{\prime}_{j}(x)\geq 0$. As above, we
define a smooth function $h(x_{n})=\sum b_{j}h_{j}\circ x_{n}$, where the
strictly positive constants $b_{j}$ are chosen so that $b_{j}$ is less than
the minimum of $1$ and

$\frac{||x||^{2}}{2^{j}||h_{j}\circ x_{n}\circ\tau^{-1}||_{C^{i}}}.$

We immediately deduce the following proposition.

######  Proposition 24.

The Takens function germs $f$ are flat, i.e., derivatives of all orders of $f$
at $0$ are trivial.

######  Proposition 25.

The critical points of Takens functions admit cylindrical neighborhoods
diffeomorphic to a ball.

###### Proof.

To begin with let us show that the critical points of Takens functions admit
cylindrical ball neighborhoods.

Since $F|Q\equiv 0$, and $\partial F/\partial x_{n}>0$ over the complement to
$Q$, the implicit function theorem implies that for each $\varepsilon\neq 0$
there is a function $g_{\varepsilon}(x_{1},...,x_{n-1})$ such that
$F(x_{1},...,x_{n-1},g_{\varepsilon}(x_{1},...,x_{n-1}))=\varepsilon$. In
other words, the level set $F(x_{1},...,x_{n})=\varepsilon$ is the graph of
the function $x_{n}=g_{\varepsilon}(x_{1},...,x_{n-1})$. Furthermore, let
$B_{r}$ denote a closed ball in $\mathbb{R}^{n-1}$ centered at $0$ of radius
$r$ such that $B_{r}$ contains the compact set $Q$ in its interior. Then the
absolute value of the restriction $g_{\varepsilon}|B_{r}$ is bounded by a
constant $c_{\varepsilon}$ such that $c_{\varepsilon}\to 0$ as $\varepsilon\to
0$.

For any subset $X$ of $\mathbb{R}^{n-1}$, let $C_{\varepsilon}(X)$ denote the
compact rectangular cylindrical subset of $\mathbb{R}^{n}$ over $X$ bounded by
the graphs of the functions $g_{-\varepsilon}|X$ and $g_{\varepsilon}|X$. In
other words, the set $C_{\varepsilon}(X)$ consists of points
$(x_{1},...,x_{n})$ such that $(x_{1},...,x_{n-1})$ is a point in $X$ and

$g_{-\varepsilon}(x_{1},...,x_{n-1})\leq x_{n}\leq
g_{\varepsilon}(x_{1},...,x_{n}).$

Then $C_{\varepsilon}(X)$ is diffeomorphic to the cylinder
$X\times[-\varepsilon,\varepsilon]$.

Let $H(r,\varepsilon)$ denote a neighborhood of $Q$ that consists of all
trajectories of the gradient vector field of $F$ in
$F^{-1}[-\varepsilon,\varepsilon]$ that either contain a point in $B_{r}$ or
has a limit point in $Q$.

######  Lemma 26.

There is a number $r_{3}>0$ such that the cylindrical neighborhood
$H(r_{3},\varepsilon)$ is diffeomorphic to the rectangular cylinder
$C_{\varepsilon}(B_{r_{3}})$.

###### Proof.

Let $r_{0}<r_{1}<r_{2}<r_{3}<r_{4}$ be numbers such that $B_{r_{0}}$ contains
$Q$ in its interior, and $H(r_{i},\varepsilon)\subset
C_{\varepsilon}(B_{r_{i+1}})$ for $i=0,...,3$. There is a smooth homotopy
$v_{t}$ parametrized by $t\in[0,1]$ of the gradient vector field $v_{0}$ of
$F$ to a vector field $v_{1}$ such that $v_{t}$ is constant over
$H(r_{0},\varepsilon)$, $v_{t}\cdot\partial/\partial x_{n}>0$ over
$C_{\varepsilon}(B_{r_{3}})$ for all $t$, and $v_{1}$ is vertical up over
$C_{\varepsilon}(B_{r_{4}}\setminus B_{r_{2}})$.

Given a point $x$ in $\mathbb{R}^{n-1}\setminus Q$, there is a unique
trajectory $\gamma_{x}$ of the vector filed $v_{t}$ passing through $x$. Let
$y\in\mathbb{R}^{n}$ denote the point in $\mathbb{R}^{n}$ on the trajectory
$\gamma_{x}$ such that $F(y)=s$. Then $j_{t,s}\colon\mathbb{R}^{n-1}\setminus
Q\to\mathbb{R}^{n}$ defined by $x\mapsto y$ is a smooth map that takes
$B_{r_{3}}\setminus Q$ to the upper horizontal part $\\{F\equiv s\\}\cap
H(r_{3},s)$ of the boundary of $H(r_{3},s)$.

Then there is a smooth isotopy $\varphi_{t}$ of a neighborhood
$H(r_{3},\varepsilon)$ of $Q$ to $C_{\varepsilon}(B_{r_{3}})$ defined by

$\varphi_{t}(x_{1},...,x_{n})=j_{t,F(x_{n})}(j_{0,F(x_{n})}^{-1}(x_{1},...,x_{n})).$

We claim that $\varphi_{1}$ is a diffeomorphism onto
$C_{\varepsilon}(B_{r_{3}})$. Indeed, let $(x_{1},...,x_{n})$ be a point in
$C_{\varepsilon}(B_{r_{3}})$ that is not on the trajectories of $v_{1}$ with
limit points on $Q$. Then the point $(x_{1},...,x_{n})$ is the image of the
point

$j_{0,F(x_{n})}\circ j_{1,F(x_{n})}^{-1}(x_{1},...,x_{n}).$

On the other hand, if $(x_{1},...,x_{n})$ is on a trajectory of $v_{1}$ with
limit point on $Q$, then it is in $H(r_{0},\varepsilon)$ where the isotopy
$v_{t}$ is constant. Therefore, the point $(x_{1},...,x_{n})$ in this case is
the image of itself. ∎

Since $H=H(r_{3},\varepsilon)$ is diffeomorphic to
$C_{\varepsilon}(B_{r_{3}})$, we conclude that $H$ is a ball with corners
$B_{r_{3}}\times[-\varepsilon,\varepsilon]$. Consequently, $H/Q$ is a
cylindrical ball neighborhood of the critical point of $f$.

Now let us show that the cylindrical neighborhood $H/Q$ is not only
homeomorphic to a ball, but it is almost diffeomorphic to a ball. Indeed, if
$n\neq 4,5$, then every smooth manifold homeomorphic to a ball is
diffeomorphic to ball. Suppose that $n=5$. Since the boundary of $H/Q$ is
diffeomorphic to the boundary of $H$, which is almost diffeomorphic to a
sphere, we conclude that $H/Q$ is almost diffeomorphic to a ball, see [9,
Proposition C]. Suppose now that $n=4$. Then there is an ambient isotopy of
$H$ that is trivial near the boundary, and that takes the wild arc $Q$ into
the unknotted standard segment $I$. Consequently, the manifold $H/Q$ is
diffeomorphic to the manifold $H/I$, which is almost diffeomorphic to a smooth
$4$-ball. ∎

###### Proof of Theorem 5.

It is known that the critical point of a Takens function is not cone-like. On
the other hand, by Proposition 25, such a critical point admits a cylindrical
neighborhood diffeomorphic to a ball. ∎

######  Corollary 27.

The critical points of Takens functions are removable, i.e., for any
neighborhood $U$ of a critical point $x_{0}$ of a Takens function $f$, there
is a smooth function $g$ such that $g\equiv f$ outside of $U$ and $g$ has no
critical points in $U$.

###### Proof.

In the proof of Proposition 25 we may choose
$\varepsilon,r_{0},r_{1},r_{2},r_{3},$ and $r_{4}$ so small that $H/Q\subset
U$. We note that the boundary of the ball $H/Q$ is diffeomorphic to the
boundary of the ball $H$, and the function $f$ near $\partial(H/Q)$ can be
identified with the function $F$ near $\partial H$. On the other hand, the
boundary $\partial H$ consists of the upper and lower horizontal parts
$\partial_{h}H=\partial H\cap\\{F=\pm\varepsilon\\}$ and the vertical part
$\partial_{v}H=\overline{\partial H\setminus\partial_{h}H}$ such that $F$ is
constant over each component of $\partial_{h}H$ and increasing over
$\partial_{v}H$. Therefore, there is a function $f^{\prime}$ on $H/Q$ such
that $f^{\prime}$ agrees with $f$ near $\partial(H/Q)=\partial H$ and
$f^{\prime}$ has no critical points. Finally, we define $g$ to be a function
on $\mathbb{R}^{n}$ such that $g\equiv f^{\prime}$ over $H/Q$ and $g\equiv f$
otherwise. ∎

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