ANNALES

DE LA FACULTÉ
DES SCIENCES

DE
TOULOUSE
Mathématiques

FRANÇOIS LAUDENBACH

*A proof of Reidemeister-Singer's theorem by Cerf's methods*

Tome XXIII, n° 1 (2014), p. 197-221.

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# A proof of Reidemeister-Singer's theorem by Cerf's methods

FRANÇOIS LAUDENBACH(1)

**ABSTRACT.** — Heegaard splittings and Heegaard diagrams of a closed 3-manifold $M$ are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on $M$. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous *pseudo-isotopy* problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when $\dim M > 2$. The main tool that we introduce is an *elementary swallow tail lemma* which could be useful elsewhere.

**RÉSUMÉ.** — Les concepts de scindement de Heegaard et de diagramme de Heegaard d'une variété fermée $M$ de dimension 3 sont expliqués dans le langage des fonctions de Morse et des pseudo-gradients de type Morse-Smale. Nous utilisons, dans un cadre très simple, des techniques qui ont été développées par Jean Cerf pour la résolution d'un problème célèbre de *pseudo-isotopie*. Entre autres, nous montrons comment éliminer les maxima locaux surnuméraires dans un chemin générique de fonctions lorsque $\dim M > 2$. L'outil principal que nous introduisons est un *lemme élémentaire de la queue d'aronde*.

<footnote>(*) Reçu le 05/05/2013, accepté le 23/11/2013</footnote>

<footnote>(1) Laboratoire de Mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences et Techniques, Université de Nantes, 2, rue de la Houssinière, F-44322 Nantes cedex 3, France
francois.laudenbach@univ-nantes.fr</footnote>

<footnote>Work supported by the European contract ERC GEODYCON</footnote>

<footnote>Article proposé par Jean-Pierre Otal.</footnote>
# 1. Introduction

When speaking of Cerf's methods we refer to Cerf's work in [3] for the so-called *pseudo-isotopy* problem. In a few words, the method consists of reducing some isotopy problem to a problem about real functions. It was created in the setting of high dimensional manifolds. However, some parts apply in dimension three as we are going to show. The purpose of this note is to present a proof of Reidemeister-Singer's theorem (as stated below) in this way. I should say that Francis Bonahon, who like me was educated in the Orsay Topology group of the seventies-eighties, wrote such a proof; but, his notes are lost. The recent developments in Heegaard-Floer homology drove me to make this proof available. The concepts used in the next statement will be explained in the course of this introduction. We always work in the $C^\infty$ category (also called the *smooth category*), for objects, maps and families of maps.

**THEOREM 1.1** (Reidemeister [16], Singer [18]). — Let $M$ be a closed connected 3-manifold.

1) *Two Heegaard splittings become isotopic after suitable stabilizations.*

2) *More precisely, let $D_0, D_1$ be two Heegaard diagrams. Then there are stabilizations $D'_0, D'_1$ by adding pairs of cancelling handles of index 1 and 2, such that one can pass from $D'_0$ to $D'_1$ by an ambient isotopy and a finite sequence of handle slides.*

Strictly speaking, only the first item is the statement of the Reidemeister-Singer theorem. A *Heegaard splitting* consists of a closed surface $\Sigma$ of genus $g$, called *Heegaard surface*, dividing $M$ into two handlebodies $H^-, H^+$. A *Heegaard diagram* is defined by more precise data, namely, a handle decomposition of $M$ with:

- one 0-handle $B^-$ and $g$ handles of index 1 attached on the boundary $\partial B^-$, whose union forms $H^-$;

- $g$ handles of index 2 attached on $\partial H^-$ and one 3-cell $B^+$, whose union forms $H^+$.

On the common boundary $\Sigma$ of $H^+$ and $H^-$, the Heegaard diagram specifies $g$ simple curves $\beta_1, \dots, \beta_g$ in $\Sigma$, mutually disjoint, which are the cores of the attaching domains of the 2-handles; their complement in $\Sigma$ is a 2-sphere with 2g holes. It also specifies $g$ simple curves $\alpha_1, \dots, \alpha_g$ which are the boundaries of the so-called *transverse 2-cells*<fnref n="1" /> of each 1-handle; the complement in $\Sigma$ of $\cup_j \alpha_j$ is also a 2-sphere with 2g holes. The other notions involved in Theorem 1.1 will be only defined in the *functional setting* considered below.

<footnote n="1">¹ They are also called *compression discs*.</footnote>
The statement of Theorem 1.1 can be translated into the language of Morse functions as follows. Recall that a *Morse function* $f$ is a smooth function whose critical points are non-degenerate; the famous *Morse lemma* states that each critical point $p$ of $f$ belongs to a chart equipped with so-called *Morse coordinates*, meaning that $f - f(p)$ reduces to a quadratic form. Some non-classical facts concerning the choice of these coordinates will be detailed in Section 3.

A Morse function is said to be *ordered* if the order of the critical values is finer than the order of their indices, namely $f(p) < f(p')$ whenever the index of the critical point $p$ is less than the index of $p'$. In dimension 3, an ordered Morse function gives rise to a Heegaard splitting by considering a level set whose level separates the index 1 and index 2 critical values. Moreover, every Heegaard splitting is obtained this way. Along a path of ordered Morse functions the Heegaard surface moves by isotopy.

Stabilizing a Heegaard splitting consists of creating a pair of critical points of index 1 and 2 at a level keeping the ordering. Thus, item 1 of Theorem 1.1 is a consequence of Theorem 1.3, for which it is necessary to speak of *genericity*.

## 1.2. GENERICITY I

Given two Morse functions $f_0, f_1 : M \to \mathbb{R}$, the following property is generic (in Baire's sense) for the paths of functions $(f_t)_{t \in [0,1]}$ joining them:

- for all $t \in [0,1]$ apart from finitely many *exceptional* values $t_j$, the function $f_t$ is Morse;

- for $\delta > 0$ small enough, $f_{t_j+\delta}$ has one more or one less pair of critical points than $f_{t_j-\delta}$; in the first (resp. second) case, $t_j$ is called a *birth time* (resp. a *cancellation time*);

- the critical points of $f_{t_j}$ are all non-degenerate except one whose Hessian has corank 1; this point will be said a *cubic critical point*.

For short, when speaking of a *generic* path of functions, it will be understood a path as above.

In this note, all genericity argument follow from Thom's *transversality theorem in jet spaces* as it is in his article on singularities [20] (see also [7], or [9] where the generic paths of real functions are explicitly considered). In Section 2, we shall specify which transversality is involved in the above genericity of paths.
The next theorem is mainly due to Jean Cerf ([3], chap. V §1)$^2$.

**THEOREM 1.3.** — Let $M$ be a closed connected manifold of any dimension $n$ by a generic path of functions $(f_t)_{t \in [0,1]}$ such that, for every $t \in [0, 1]$ outside of a finite set $J = \{t_1, \dots, t_q, t_{q+1}, \dots, t_{q+q'}\}$, $f_t$ is an ordered Morse function. Moreover, $t_1, \dots, t_q$ are birth times and lie in $(0, \frac{1}{3})$; and $t_{q+1}, \dots, t_{q+q'}$ are cancellation (or death) times and lie in $(\frac{2}{3}, 1)$.

In particular in dimension 3, a level set of $f_{1/2}$ whose level separates the index 1 and index 2 critical values is a Heegaard splitting that is a common stabilization, up to isotopy, of those associated with $f_0$ and $f_1$.

We now turn to the second part of Theorem 1.1. In order to speak of handle decomposition and handle sliding, it is useful to consider a Morse function $f$ equipped with a *pseudo-gradient*.

**DEFINITION 1.4.** — Given a Morse function $f$, a smooth vector field $X$ on $M$ is said to be a (descending) pseudo-gradient for $f$ if the two following conditions hold:

- the Lyapunov inequality³ $X \cdot f < 0$ away from the critical locus;

- at each critical point $p$ the Hessian of $X \cdot f$ is negative definite (notice that $X \cdot f \le 0$ everywhere).

Local data of pseudo-gradients generate a global pseudo-gradient by using a partition of unity. It is easily checked that the zeroes of $X$ coincide with the critical points of $f$ and are *hyperbolic*⁴. Thus, according to the *stable/unstable manifold theorem* (see [2]), with each zero $p$ of $X$ there are associated stable and unstable manifolds, also called *invariant manifolds* and denoted respectively by $W^s(p, X)$ and $W^u(p, X)$. A point $x \in M$ belongs to $W^s(p, X)$ if $X^t(x)$ tends to $p$ as $t$ tends to $+\infty$; here, $X^t$ denotes the flow of $X$.

The unstable manifold is diffeomorphic to $\mathbb{R}^i$, where $i$ is the index of $f$ at $p$, and the stable manifold is diffeomorphic to $\mathbb{R}^{n-i}$; moreover, $p$ is a non-degenerate maximum (resp. minimum) of the restriction of $f$ to $W^u(p, X)$ (resp. $W^s(p, X)$).

<footnote>(2) Strictly speaking, only the first sentence is stated in Cerf’s article. The complement follows from his lemma about the *uniqueness of births* (valid in dimension greater than 1 only).</footnote>

<footnote>(3) This sign convention is used for instance by R. Bott p. 341 in [1].</footnote>

<footnote>(4) That is, if $p$ is a zero of $X$ the eigenvalues of the linearized vector field at $p$ have a non-zero real part.</footnote>
Given the Morse function $f$, Smale [19] proved that, generically, all invariant manifolds of a pseudo-gradient of $f$ are mutually transverse$^5$. Today, such a pseudo-gradient is said to be *Morse-Smale*.

According to Whitney [21], if $p$ is a cubic critical point of $f$, there are coordinates $(x, y) \in \mathbb{R} \times \mathbb{R}^{n-1}$, which we call *Whitney coordinates*, where $f$ reads:

$$f(x, y) = f(p) + x^3 + q(y).$$

Here, $q$ is a non-degenerate quadratic form on $\mathbb{R}^{n-1}$. For a reason which will be explained in 3.4, we require a pseudo-gradient $X$ for $f$ to coincide with $-\nabla_g f$ near the cubic critical point $p$, where $g$ is the Euclidean metric of one system of Whitney coordinates.

Given a generic path of functions $(f_t), t \in [0, 1]$, it can be enriched with a smooth path of vector fields $(X_t)$, such that $X_t$ is a pseudo-gradient of $f_t$ for all $t \in [0, 1]$.

## 1.5. GENERICITY II

The following property is generic for the paths of pairs $(f_t, X_t)_{t \in [0,1]}$:

- the path of functions is generic in the sense of 1.2;

- for every $t$, there is no $X_t$-orbit from a critical point index $j$ of $f_t$ to a critical point index $i$ if $j < i$ (briefly said: no $j/i$ connecting orbit if $j < i$);

- for every $t$ outside of a finite set $K = \{t_1, \dots, t_r\} \subset (0, 1)$ of Morse times$^6$, there is no $i/i$ connecting orbit of $X_t$;

- for each $t_k \in K$, exactly one orbit $\ell_k$ of $X_{t_k}$ connects two critical points $p$ and $p'$ having the same index; moreover, for each $x \in \ell_k$, we have:

$$T_x \ell_k = T_x W^u(p, X_{t_k}) \cap T_x W^s(p', X_{t_k}),$$

and $t \mapsto X_t$ crosses transversely at time $t_k$ the codimension-one stratum of the space of pseudo-gradients having a connecting orbit between two critical points with the same index.

For short, such a path $(f_t, X_t)_{t \in [0,1]}$ is said to be *generic*. For $t_k \in K$, one says that a *handle sliding* happens at time $t_k$. The effect of a handle sliding

<footnote>$^5$ An ordered Morse function $f$ with a Morse-Smale pseudo-gradient $X$ gives rise easily to a handle decomposition.</footnote>

<footnote>$^6$ A cubic point of index $i$ could be connected to a Morse point of index $i$ at a lower level.</footnote>
on the so-called *Morse complex* is described by J. Milnor (see Theorem 7.6 in [12]).

The argument for genericity in 1.5 is elementary once the first item is assumed. It relies on the classical transversality theorem applied to a $(j-1)$-sphere moving with $t$ with respect to a fixed $(n-i-1)$-sphere, $j \le i$, in an $(n-1)$-dimensional manifold.

Now, the statement of item 2) in Theorem 1.1 can be translated into the next one. Following M. Morse [14], a function with only two local extrema is said to be *polar*.

**THEOREM 1.6.** — Let $M$ be a closed connected manifold of dimension$^7$ $n > 2$. Given two ordered polar Morse functions $f_0, f_1$ equipped with respective Morse-Smale pseudo-gradients $X_0, X_1$, there exists a generic path of pairs $(f_t, X_t)_{t \in [0,1]}$, where the vector field $X_t$ is a pseudo-gradient for the function $f_t$, so that the following holds: for every $t \in [0, 1]$ outside of a finite set, $f_t$ is an ordered polar Morse function and $X_t$ has no $i/i$ connecting orbit. The excluded values of $t$ are the times of birth first, then the times handle sliding and finally the times of cancellation.

A direct proof of Theorem 1.3 is given in Section 2 without any reference to Cerf's work. It mainly follows from Lemma 2.1 which offers an efficient process for crossing critical values. The proof of Theorem 1.6 will be given in Section 4 and uses a few technical lemmas, including the *elementary swallow tail lemma* and the *elementary lips lemma*. Since they could be useful in a more general setting, they are written with index assumptions which are more general than necessary here. These lemmas are proved in Section 3.

## 2. Proof of Theorem 1.3

The main tool will be the next lemma.

**LEMMA 2.1** (Decrease of a critical value). — Let $f: M \to \mathbb{R}$ be a Morse function, let $X$ be a pseudo-gradient for $f$ and let $p$ be a critical point of index $k$. Assume that the unstable manifold $W^u(p, X)$ contains a closed smooth $k$-disc $D$ whose boundary lies in a level set $f^{-1}(a)$, $a < f(p)$. Then, for every $\varepsilon > 0$ with $a + \varepsilon < f(p)$, there exists a path $(f_t)_{t \in [0,1]}$ of Morse functions such that $f_0 = f$, $f_1(p) = a + \varepsilon$ and $X$ is a pseudo-gradient of $f_t$ for every $t \in [0, 1]$. Moreover, the support of the deformation may be contained in an arbitrarily small neighborhood $W$ of $D$ in $M$.

<footnote>$^{(7)}$ The statement also holds in dimension 2 with a different proof (see [8], §8). It is obvious in dimension 1.</footnote>
Note that, when $k = 0$, $W^u(p, X)$ has an empty intersection with the open sub-level set $f^{-1}((-\infty, f(p)))$. So, the condition of the lemma is fulfilled and the conclusion allows us to decrease arbitrarily the value of a local minimum.

The lemma above holds true, with the same proof, in a family whose data $(f, p, D, a)$ depend smoothly on a parameter $s \in \mathbb{R}^m$ and fulfill the same assumptions for every $s$. Moreover, $f$ only has to be a Morse function in a neighborhood of $D$. In particular, it applies to non-generic functions or pseudo-gradients.

*Proof.* — The case where $p$ has index 0 is left to the reader. Hereafter, assume $k > 0$. Set $n = \dim M$ and $c = f(p)$. For $\eta$ small enough, there exists a closed $(n-k)$-disc $D'$ in the stable manifold $W^s(p, X)$, with $D' \subset W$, whose boundary lies in $f^{-1}(c+\eta)$. Let $U$ be a tubular neighborhood of radius $\delta$ of $\partial D$ in $f^{-1}(a)$. For $\delta$ small enough with respect to $\eta$, every half-orbit of $X$ ending in $U$ is contained in $D$ or crosses $f^{-1}(c+\eta)$. Define $\mathcal{M}$ as the union of $D$, $D'$ and all segments of $X$-orbits starting from points in $f^{-1}(c+\eta)$ and ending in $U$; for a small $\delta$, we have $\mathcal{M} \subset W$. Its boundary is made of three parts, two horizontal parts $\mathcal{M} \cap f^{-1}(a)$ and $\mathcal{M} \cap f^{-1}(c+\eta)$, and the lateral boundary $\partial_\ell \mathcal{M}$ which is tangent to $X$. There are two corners in the boundary of $\mathcal{M}$, each being diffeomorphic to a product of spheres $S^{k-1} \times S^{n-k-1}$ (where $k = \text{index}(p)$); one is the boundary of $U$, trivialized as the sphere normal bundle $\partial U \to \partial D$; the other corner is $\partial_\ell \mathcal{M} \cap f^{-1}(c+\eta)$ and is diffeomorphic to the first one by the flow of $X$.

Let $N$ be a small collar neighborhood of $\partial_\ell \mathcal{M}$ in $\mathcal{M}$; it is diffeomorphic to a product

$$N \cong S^{k-1} \times S^{n-k-1} \times [0, 1] \times [a, c + \eta].$$

If $(x, y)$ are the coordinates of $R := [0, 1] \times [a, c + \eta]$, the product structure of $N$ is chosen so that the level sets of $f$ in $N$ are $\{y = \text{const.}\}$ and the vertical lines directed by $\partial_y$ are tangent to the orbits of $X$.

For constructing $f_1$ we keep $f_1 = f$ outside of $\mathcal{M}$ and change the level set foliation as said below. The level set foliation of $f_1$ coincides with the one of $f$ in the complement of $N$ in $\mathcal{M}$. Inside $N$, it is obtained by replacing the horizontal foliation of $N$ with a new one which is still transverse to the vertical lines, is still horizontal near the boundary, and puts $f^{-1}(a+\varepsilon) \cap \{x = 0\}$ on the same leaf as $f^{-1}(c) \cap \{x = 1\}$. The new foliation in $N$ is the pullback of a foliation of $R$ by the standard projection (see figure 1). The value of $f_1$ is now well-defined.
<figure><img src="image_1.png" /><figcaption>Figure 1A</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Figure 1B</figcaption></figure>

Moreover, it is easy to interpolate this construction for $t$ varying in $[0, 1]$. □

**COROLLARY 2.2.** — Let $(f_0, X)$ be a Morse function with a pseudo-gradient having no $j/i$ connecting orbit, $j < i$. Then there exists a path $(f_t)_{t \in [0,1]}$ of Morse functions issued from $f_0$ such that $f_1$ is ordered and the same vector field $X$ is a pseudo-gradient of $f_t$ for every $t \in [0, 1]$.

*Proof*. — If the function is not ordered, there is a pair of critical points $(p, q)$ with $index(p) < index(q)$ and $f(p) \ge f(q)$. Choose such a pair so that $f(p)$ is minimal among all similar unordered pairs. By this choice every orbit of $W^u(p, X)$ crosses a level set below $f(q)$; if not, one of them ends at a critical point $p'$. By assumption on $X$ we have $index(p') \le index(p) < index(q)$ and $f(p) > f(p') \ge f(q)$, contradicting the assumption on the pair $(p, q)$. Then, lemma 2.1 applies and yields a new Morse function which has the same pseudo-gradient $X$ and at least one unordered pair less than $f$. Arguing this way recursively, the corollary is proved. □

Before proving Theorem 1.3, it is useful to specify which transversality is involved in a generic path in the sense of 1.2 and what a *birth path* is. A path of functions $(f_t)$ may be thought of as a smooth function $F : [0, 1] \times M \to \mathbb{R}$, $(t, x) \mapsto f_t(x)$. We now consider the $r$-jet spaces $J^r([0, 1] \times M, \mathbb{R})$ for $r=1, 2$ and their submanifolds $\Sigma^1$ and $\Sigma^{1,1}$ defined as follows (here, we are using the so-called Thom-Boardman notation). The first one, $\Sigma^1$, is made of the 1-jets $(a, j^1g)$ where $a \in [0, 1] \times M$ and $g$ is a germ at $a$ of function $(t, x) \mapsto g(t, x)$ such that $\partial_x g(a) = 0$. The second one, $\Sigma^{1,1}$, is made of the 2-jets $(a, j^2g)$ such that:

- $dg_x g(a) = 0$ and $j^1 g$ meets $\Sigma^1$ transversely;

- the (germ of) curve $(j^1 g)^{-1}(\Sigma^1)$ passes through $a$ and is tangent to the kernel of $\partial_x g(a)$ and which is the factor $\{t = t(a)\}$.

According to Thom [20], generically $j^1 F$ is transverse to $\Sigma^1$ and $j^2 F$ is transverse to $\Sigma^{1,1}$. Thus, the critical locus of $f_t$ when $t$ runs in $[0, 1]$, which is $(j^1 F)^{-1}(\Sigma^1)$, is a smooth curve; and the isolated points $(j^2 F)^{-1}(\Sigma^{1,1})$
are the cubic critical points. By making a diffeomorphism $C^\infty$-close to $Id$
act on $[0, 1] \times M$, it is possible to move the cubic critical points so that
their $t$-coordinates are distinct. In particular, the properties in 1.2 hold true
generically.

Moreover, if $(t_0, x_0)$ is a cubic critical point, thanks to the information
on the 3-jet⁸ of $F$ at $(t_0, x_0)$, it is possible to write a normal form of $F$
on a neighborhood of $(t_0, x_0)$. This follows easily from the normal form
of cusps established by H. Whitney in [21] for generic maps from plane
to plane. Precisely, there are adapted coordinates $(t, x) = (t, y, z)$, with
$y \in \mathbb{R}^{n-1}$, $z \in \mathbb{R}$, which we call Whitney coordinates, where $F$ reads:

$$
F(t, x) = F(t_0, x_0) + z^3 \pm (t - t_0)z + q(y)
$$

Here, $q$ is a non-degenerate quadratic form on $\mathbb{R}^{n-1}$, $\pm = -$ if $t_0$ is a
birth time and $\pm = +$ if $t_0$ is a cancellation time. If $t_0$ is a birth time, we
immediately derive from the model that, for $\delta > 0$ small enough, the given
generic path of functions, restricted to $[t_0 - \delta, t_0 + \delta]$, is a birth path in the
following sense.

DEFINITION 2.3. — A birth path is a generic path of functions $(f_t)_{t \in [t_0-\delta, t_0+\delta]}$ such that there exists a path of cylinders $B_t \cong D^{n-1} \times [-1, +1]$ embedded in $M$ with the following properties for every $t \in [t_0 - \delta, t_0 + \delta]$:

- $D^{n-1} \times \{\pm 1\}$ (the top and bottom of $B_t$) lie in two level sets of $f_t$;

- the restriction of $f_t$ to $\partial D^{n-1} \times [-1, +1]$ has no critical points;

- $f_t|B_t$ is semi-conjugate to the function $c_{t_0}^t(y, z) := z^3 - (t-t_0)z + q(y)$.
Here, a semi-conjugation stands for an embedding $\varphi_t : B_t \to \mathbb{R}^n$, depending
smoothly on $t$, covering the origin of $\mathbb{R}^n$ and such that $c_{t_0}^t \circ \varphi_t = f_t|B_t$ up
to a rescaling of the values. The index of $q$ is called the index of the birth.

The function $f_t$ has no critical points in $B_t$ when $t_0 - \delta \le t < t_0$ whereas, for $t_0 < t \le t_0 - \delta$, $f_t$ has a pair of critical points in $B_t$ of respective index $i, i + 1$ if $i$ is the index of the birth.

Remarks 2.4. — 1) If $f_0$ is a Morse function given with a cylinder $B_0$ on which $f_0$ induces the height function, then $f_0$ is the beginning of a birth path with $t \in [0, 2\delta]$ which is supported in $B_0$ in the sense that the path is stationary outside of $B_0$. Indeed, $f_0|B_0$ is semi-conjugate to any function without critical point, for instance $(y, z) \mapsto z^3 + \delta z + q(y)$; thus, it is allowed to plug the functions $c_\delta^t$, $t \in [0, 2\delta]$, by taking a suitable semi-conjugation

<footnote>(8) The transversality of $j^2F$ to $\Sigma^{1,1}$ at $(t_0, x_0)$ is an open condition on the 3-jet.</footnote>
$\varphi_t : B_0 \to \mathbb{R}^n$. This birth path is said to be *elementary* (compare with a similar definition in Cerf [3] chap. III).

2) Any birth path issued from $f_0$ associated with a path of cylinders $(B_t)_{t \in [0,2\delta]}$ is homotopic to an elementary birth path among the birth paths starting from $f_0$. This is done by using an extension of the isotopy $B_0 \to B_t$.

LEMMA 2.5 (Shift of birth). —

1) Every generic path of functions on $M$ is homotopic relative to its end points to a generic path where the birth times appear before the cancellation times. More precisely, the following holds.

2) Let $(h_s)_{s \in [0,1]}$ be a generic path of functions which are Morse for all time except one cancellation time. Let $(\beta_t^1)_{t \in [0,2\delta]}$ be a birth path starting from the Morse function $h_1$ with associated cylinders $(B_t^1)_{t \in [0,2\delta]}$. Then there is a smooth family, parametrized by $s \in [0, 1]$, of birth paths $(\beta_t^s)_{t \in [0,2\delta]}$, starting from $h_s$ with associated cylinders $(B_t^s)_{t \in [0,2\delta]}$ which coincide with the given cylinders when $s=1$.

Moreover, if $\dim M > 1$, the same holds true for any generic path $(h_s)_{s \in [0,1]}$. Moreover, it is possible to choose the cylinders $B_t^0$ as neighborhoods of any given regular point of $h_0$.

*Proof of 2)$\Rightarrow$1).* — The composed path $(h_s)_{s \in [0,1]} * (\beta_t^1)_{t \in [0,2\delta]}$ is homotopic, relative to its end points, to the composed path $(\beta_t^0)_{t \in [0,2\delta]} * (\beta_{2\delta}^s)_{s \in [0,1]}$. In general, this composition is only piecewise smooth at the gluing point. But we are free to modify the parametrization of the composed path; if the two paths entering the composition are stationary near their common end point, then the composed path is smooth.

The new path from $h_0$ to $\beta_{2\delta}^1$ has one birth time appearing before one cancellation time. By arguing this way recursively one proves 1).

*Proof of 2).* — Given the cylinder $B_0^1$, one chooses a smooth family of cylinders $(B_0^s)_{s \in [0,1]}$ in $M$ ending to $B_0^1$ and so that $h_s$ induces the standard horizontal foliation $D^{n-1} \times \{pt\}$ of $B_0^s$ for every $s \in [0, 1]$. This is possible in any positive dimension since we are free to move $B_0^s$ away from the critical set of $h_s$, even at the cancellation time. Thanks to an extension of isotopies, we get a 2-parameter family of diffeomorphisms $\psi_t^s : B_0^s \to B_t^1$, $s \in [0, 1]$, $t \in [0, 2\delta]$, preserving the horizontal foliation near the boundary and such that $\psi_0^1 = \text{Id}$. Then, define

$$ \beta_t^s = \begin{cases} h_s & \text{outside of } B_0^s \\ \beta_t^1 \circ \psi_t^s & \text{in } B_0^s, \text{ up to some rescaling.} \end{cases} $$
The rescaling is needed for making the two definitions match along the boundary of $B_0^s$. When $t=1$, this is an elementary birth path issued from $h_1$. According to Remark 2.4 2), it is homotopic to $(\beta_t^1)$ relative to $h_1$. This proves the first part of 2). In case dim $M > 1$, the critical locus is non-separating and the last statement of 2) follows. $\square$

## 2.6. PROOF OF THEOREM 1.3

The case dim $M=1$ is left to the reader. Hereafter, dim $M$ is assumed to be greater than 1. Given two ordered Morse functions $f_0, f_1$, there exists a generic path $(f_t)_{t \in [0,1]}$ where $f_t$ is Morse for every $t \in [0,1]$ outside of a finite set $J$. Decompose $J = J_+ \cup J_-$ where $J_\pm$ is the set of birth/cancellation times and apply Lemma 2.5. The birth times $J_+$ can be shifted to the left, say in $[0, t_0]$, and the cylinders of birth can be located at the right level according to the index of the birth so that all Morse functions in $[0, t_0]$ are ordered. Similarly, the cancellation times can be shifted to the right, say in $[t_1, 1]$, and the cancellation cylinders can be chosen so that all Morse functions in $[t_1, 1]$ are ordered. Thus, $f_t$ is a Morse function for every $t \in [t_0, t_1]$ and is ordered for $t = t_0, t_1$.

Choose pseudo-gradients $X_t$ for $f_t$. We may assume $(X_t)_{t \in [t_0, t_1]}$ in the sense of 1.5. Thus, the pseudo-gradient $X_t$ has no $j/i$ connecting orbit with $j \le i$ for all $t \in [t_0, t_1]$ outside of a finite set $K \subset (t_0, t_1)$ (times of $i/i$ connecting orbits).

Apply corollary 2.2 to the functions $f_{t_k}, t_k \in K$, and deform the path of functions accordingly, that is: keep the same path $(X_t)$ as path of pseudo-gradients and ask the deformation to be stationary on the complement of small neighborhoods of the $t_k$'s. After that deformation, the functions $f_{t_k}, t_k \in K$, are ordered and, for every $t \in (t_k, t_{k+1})$, the vector fields $X_t$ is has no $j/i$ connecting orbit with $j \le i$. This also holds true on the intervals $(t_0, \inf K)$ and $(\sup K, t_1)$ on the left and right of $K$. So, we are reduced to reorder a path of Morse functions equipped with pseudo-gradients which have no $j/i$ connecting orbits, $j \le i$, for every time. The reordering is then obtained by applying the one-parameter version of Lemma 2.1. This finishes the proof of item 1) in Theorem 1.1. $\square$
### 3. The elementary swallow tail lemma and similar results

Before proving Theorem 1.6 and, hence, item 2) in Theorem 1.1, we need to state some lemmas: first, a very particular case of the swallow tail lemma; next, a very particular case of the lips lemma (or uniqueness of death according to [3]); finally, the cancellation theorem<fnref n="9" /> of Morse [14] (see also J. Milnor [12], Section 5).

We state them by means of Cerf graphics. Recall that the Cerf graphic of a path of functions $(f_t)_t$ is the part of $\mathbb{R}^2$ whose intersection with $\{t\} \times \mathbb{R}$ is the set of critical values of $f_t$.

The three proofs are very similar, by reduction to the one-dimensional case where they become easy. Only the proof of the elementary swallow tail lemma is detailed here since the three proofs can be performed in the same way<fnref n="10" />.

We begin with useful conjugation lemmas. The first one is likely well-known, the next ones could be less classical.

**LEMMA 3.1.** — Let $V$ be a manifold and $V'$ be a compact submanifold. Two germs of smooth functions $f$ and $g$ along $V'$ whose restrictions to $V'$ coincide and have no critical points are isotopic relative to $V'$. Moreover, if $f = g$ near a compact set $K \subset V'$, the isotopy may be stationary near $K$ in $V$. This statement holds true with parameters in a compact set.

*Proof.* — The path method of J. Moser [15] is available; it is explained below in our setting. conjugation prob. Look at the path of germs $t \in [0, 1] \mapsto f_t := (1-t)f + tg$ and search for an isotopy $(\varphi_t)_{t \in [0,1]}$ of $V$, with $\varphi_0 = \text{Id}$, satisfying the conjugation equation of germs along $V'$:

$$ (1) \qquad \begin{aligned} f_t \circ \varphi_t &= f, \\ \varphi_t(x) &= x \text{ for every } x \in V'. \end{aligned} $$

The infinitesimal generator $Z_t$ has to satisfy the derived equation:

$$ (2) \qquad \begin{aligned} df_t(x) \cdot Z_t(x) + g(x) - f(x) &= 0, \\ Z_t(x) &= 0 \text{ for every } x \in V'. \end{aligned} $$

Conversely, if $Z_t$ is a time depending vector field which is a solution of (2) near $V'$, its “flow” is defined until $t = 1$ on a small neighborhood of $V'$ and solves the conjugation problem.

<footnote n="9">⁹ Also referred simply as the cancellation lemma.</footnote>

<footnote n="10">¹⁰ Such a proof of Morse’s cancellation theorem is now available in [11].</footnote>
Here is a solution of Equation (2) by using an auxiliary Riemannian metric:

$$Z_t = (f - g) \frac{\nabla f_t}{|\nabla f_t|^2}.$$

The same proof holds for the relative statement and with parameters. $\square$

**LEMMA 3.2.** — (The $\mathfrak{M}\mathfrak{J}^2$ lemma.)<sup>11</sup> Let $\mathfrak{F}$ be the ring of germs of smooth functions at $0 \in \mathbb{R}^n$ and let $\mathfrak{M}$ be its unique maximal ideal of germs vanishing at 0. Given $f \in \mathfrak{M}$, its Jacobian ideal is the ideal $\mathfrak{J} = \mathfrak{J}(f)$ generated by the first partial derivatives of $f$. Consider a germ $h$ in the product ideal $\mathfrak{M}\mathfrak{J}^2$. Then there is a $C^\infty$ diffeomorphism $\varphi$ such that $(f+h) \circ \varphi = f$.

For instance, take a germ $f$ of Morse function with $f(0) = 0$; it reads $f = q + r$ where $q$ is a non-degenerate quadratic form and $r$ belongs to $\mathfrak{M}^3$. Since $\mathfrak{J}(q) = \mathfrak{M}$, the lemma implies that $f$ is conjugate to $q$, which is exactly the statement of Morse's lemma.

*Sketch of proof.* —<sup>12</sup> As in Lemma 3.1, we use the path method. Setting $f_t = f + th$, one searches for a family of local diffeomorphisms $\varphi_t$, $t \in [0, 1]$, such that $f_t \circ \varphi_t = f$. This amounts to find local vector fields $Z_t$ vanishing at the origin such that $df_t(x) \cdot Z_t(x) + h(x) = 0$; this consists of decomposing $h$ in the Jacobian ideal $\mathfrak{J}_t$ of $f_t$ with coefficients in $\mathfrak{M}$. The main point is that $\mathfrak{J}_t = \mathfrak{J}_o$ for all $t$. Indeed, $(\frac{\partial f_t}{\partial x_i}) = A_t(\frac{\partial f_o}{\partial x_j})$ where the matrix $A_t$ equals the Identity matrix modulo $\mathfrak{M}$. Thus, $A_t$ is invertible, and a decomposition of $h$ in $\mathfrak{J}_o$ with coefficients in $\mathfrak{M}$ yields the wanted decomposition. $\square$

The same proof works with parameters $s \in \mathbb{R}^m$ and in a relative form: Let $(f^s)_{s \in D^m}$ be a family, parametrized by the m-ball, of germs of Morse functions $(\mathbb{R}^n, 0) \to \mathbb{R}$ whose Hessians at 0 are denoted by $q^s$. Assume $f^s = q^s$ for every $s \in \partial D^m$. Then there is a family of local diffeomorphisms $\varphi^s$ such that $f^s \circ \varphi^s = q^s$ and $\varphi^s = \text{Id}$ when $s \in \partial D^m$.

In the same setting, if $f$ is given a local unstable manifold $W := W^u(0, X)$, a system of Morse coordinates $x = (y, z)$ are said to be adapted to $(f, W)$ if $f(x) = -|y|^2 + |z|^2$ and $W = \{z = 0\}$.

**COROLLARY 3.3.** — Given such data $f$ and $W$ the following holds.

1) There exist Morse coordinates adapted to $(f, W)$. (This claim also holds with parameters.)

<footnote><sup>(11)</sup> We learnt this proof of Morse's lemma from J. Mather on the occasion of a lecture in Thom's seminar at IHÉS (Bures-sur-Yvette), Dec. 1969.</footnote>

<footnote><sup>(12)</sup> A detailed proof may be found in [10].</footnote>
2) Two such systems of Morse coordinates can be joined, up to a per-mutation of the coordinates by a one-parameter family of adapted Morse coordinates<fnref n="13" />.

*Proof.* — 1) The restriction of $f$ to $W$ has a non-degenerate maximum. By Morse's lemma we have Morse coordinates $y$ of $W$ so that $f(x) = -|y|^2$ if $x \in W$. Complete the coordinates $y$ to local coordinates $(y, z')$ of $(\mathbb{R}^n, 0)$ so that $W = \{z' = 0\}$ and the $z'$-space is the orthogonal of the $y$-space with respect to $d^2f(0)$. Let $(f^y)$ be the family of the restrictions of $f$ to the slice $\{y = \text{cst}\}$. For $y = 0$, the function $f^0$ is Morse and its critical point is $z' = 0$. By the implicit function theorem, there is a smooth map $y \mapsto z' = k(y)$ such that $f^y$ is Morse with critical point at $k(y)$ (for every $y$ close to 0). Apply the change of variables $(y, z) = (y, z' - k(y))$ so that the critical point of $f^y$ becomes $z = 0$ for every $y$. By a linear transformation in each slice, we may assume the Hessian of $f^y$ to be constantly equal to $|z|^2$. The wanted Morse coordinates are now given by applying Morse's lemma with parameters to the family $(f^y)$.

2) We first connect the two given Morse coordinates by a path of coordinates which are only adapted to $W$. Then, this path is modified by applying Morse's lemma with parameters in the relative form.

### 3.4. PSEUDO-GRADIENTS FOR BIRTH PATH.

To avoid raising some problems in bifurcation theory of vector fields we adopt a still more restrictive definition of pseudo-gradients<fnref n="14" /> than in 1.4. This is allowed since we are free to choose our pseudo-gradients.

Recall from 2.3 (with slightly different notation) that a birth path at time $t_0$ consists of a generic path of functions $(f_t)_{t \in (t_0-\delta, t_0+\delta)}$, a cubic critical point $p$ of index $i$ of $f_{t_0}$ and cylinders $(B_t)$ which are neighborhoods of $p$. They are endowed with Whitney coordinates $(x, y, z) \in \mathbb{R} \times \mathbb{R}^i \times \mathbb{R}^{n-i-1}$ so that $f_t|_{B_t}$ reads:

$$f_t|_{B_t} = x^3 - (t - t_0)x - |y|^2 + |z|^2 + \text{cst.}$$

If $(X_t)_{t \in (t_0-\delta, t_0+\delta)}$ is a path of pseudo-gradients in the sense of 1.4, $X_t|_{B_t}$ is required to be the descending gradient of $f_t$ with respect to the Euclidean metric of the Whitney coordinates for every $t \in (t_0 - \delta, t_0 + \delta)$ (not only for $t = t_0$).

<footnote n="13">¹³) We are hiding some acyclicity here (compare [4]); but, the space of Morse coordinates is not acyclic, due to the isometry group $O(i, n-i)$.</footnote>

<footnote n="14">¹⁴) We could ask the path $(X_t)$ to present a bifurcation of type saddle-node along a birth/cancellation path.</footnote>
The stable/unstable manifold $W^{u/s}(p, X_{t_0})$ is described now. One checks that the $x$-axis is the kernel of the Hessian of $f_{t_0}$. The half space $\{(x, y, z) | x \le 0, z = 0\}$ is the (local) unstable manifold $W^u(p)$; its boundary is the so-called *strong-unstable* manifold. Similarly, the half space $\{(x, y, z) | x \ge 0, y = 0\}$ is the (local) stable manifold and its boundary is the *strong-stable* manifold.

Generically, $X_{t_0}$ has no $j/i$ connections where $j \le i$, except for possible $i/i$ connections from $p$ to a critical point of index $i$ at a lower level and these connections do not belong to the strong-unstable manifold of $p$. Moreover, the $i+1/i$ connections are transverse; so, this will be the case for every $t \in (t_0 - \delta, t_0 + \delta)$ if $\delta$ is small enough.

Moreover, if $\delta$ is small with respect to the “horizontal” size of the cylinders, the cubic critical point $p$ gives rise to a pair of Morse critical points $(p_t, q_t) \in B_t$ for every $t \in (0, \delta)$: the point $p_t$ has index $i+1$ and coordinates $(-\sqrt{\frac{t-t_0}{3}}, 0, 0)$; the point $q_t$ has index $i$ and coordinates $(\sqrt{\frac{t-t_0}{3}}, 0, 0)$. The closure of $W^u(p_t, X_t) \cap B_t$ reads $\{x \le x(q_t), z = 0\}$. The closure of $W^s(q_t, X_t) \cap B_t$ reads $\{x \ge x(p_t), y = 0\}$. One sees a unique connecting orbit from $p_t$ to $q_t$ and all other orbits in $W^u(p_t)$ (resp. $W^s(q_t)$) intersect the bottom (resp. the top) of $B_t$, which lies in a level set of $f_t$ according to Definition 2.3.

<figure><img src="image_3.png" /><figcaption>Figure 2. — After a birth</figcaption></figure>
LEMMA 3.5 (Elementary swallow tail lemma<fnref n="15" />). — Let $\gamma := (f_t)_{t \in [0,1]}$ be a generic path of functions on $M$. Assume that its restriction to $t \in [t_0, t_1]$ has a Cerf graphic showing a swallow tail as in figure 3A: there are three critical points, $p_t, p'_t$ of index $i+1$ and $q_t$ of index $i$, such that the pair $(p_t, q_t)$ is created at time $t_0$ and the pair $(p'_t, q_t)$ is cancelled at time $t_1$; at some $\tau \in (t_0, t_1)$ the critical values are equal: $f_\tau(p_\tau) = f_\tau(p'_\tau)$. Moreover, it is given a generic family of pseudo-gradients $X_t$ for $f_t$ satisfying the next conditions for every $t \in [t_0, t_1]$:

- $W^u(p_t)$ (resp. $W^u(p'_t)$) intersects $W^s(q_t)$ transversely along a single orbit $l_t$ (resp. $l'_t$);

- every other orbit in $W^u(p_t)$ and $W^u(p'_t)$ crosses the level set $a_t := f_t(q_t) - \epsilon$, for some $\epsilon > 0$.

Then, given $\delta > 0$, the path $\gamma$ can be deformed to a path $\gamma'$ whose Cerf graphic is trivial over $[t_0, t_1]$ as in figure 3B, the deformation being stationary on $[0, t_0 - \delta] \cup [t_1 + \delta, 1]$.

<figure><img src="image_4.png" /><figcaption>Figure 3A</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3B</figcaption></figure>

*Proof*. — There are three parts.

A) GENERAL SETUP. First, we choose birth cylinders $B_t$, $t \in (t_0 - \delta', t_0 + \delta')$ as in 3.4, the $\delta'$ being provisional. Without loss of generality, we may assume $f_t|B_t = x^3 - (t-t_0)x - |y|^2 + |z|^2$ (no additive constant). And similarly for the cancellation time $t_1$. Take $\epsilon$ as in the above statement and truncate the birth cylinders at level $\pm 2\epsilon$; from now on, $B_t$ will denote the truncated cylinder.

Set $\delta = \delta(\epsilon)$, so that, for $t = t_0 + \delta$, the two critical points of $f_t$ in $B_t$ have value $\pm \epsilon$. Decreasing $\epsilon$ if necessary, we get $\delta < \delta'$. Moreover, except the connecting orbit, every $X_t$-orbit in the invariant manifolds of $p_t$ and $q_t$ exits $B_t$ through the top or the bottom of $B_t$. And similarly for the pair $(p'_t, q_t)$ when $t \in [t_1 - \delta, t_1]$.

<footnote n="15">¹⁵ In Cerf [3] the swallow tail lemma requires no assumption about pseudo-gradient lines but there are some topological assumptions.</footnote>
Since $f_t(p_t) - f_t(q_t)$ is increasing when $t$ is close to $t_0$, by taking $\epsilon$ small enough we have $f_t(p_t) - f_t(q_t) > 2\epsilon$ for every $t \in (t_0 + \delta, t_1]$. Similarly, $f_t(p'_t) - f_t(q'_t) > 2\epsilon$ for every $t \in [t_0, t_1 - \delta)$.

For $t \in [t_0 + \delta, t_1 - \delta]$, we are going to choose Morse models $M(q_t), M(p_t)$, $M(p'_t)$ with coordinates $(x, y, z) \in \mathbb{R} \times \mathbb{R}^i \times \mathbb{R}^{n-1-i}$ so that:

$$ 
\begin{align*}
f_t|M(q_t) &= +x^2 - |y|^2 + |z|^2 + f_t(q_t), & M(q_t) \subset f_t^{-1}([f_t(q_t) - \epsilon, f_t(q_t) + \epsilon]) \\
f_t|M(p_t) &= -x^2 - |y|^2 + |z|^2 + f_t(p_t), & M(p_t) \subset f_t^{-1}([f_t(p_t) - \epsilon, f_t(p_t) + \epsilon]) \\
f_t|M(p'_t) &= -x^2 - |y|^2 + |z|^2 + f_t(p'_t), & M(q_t) \subset f_t^{-1}([f_t(p'_t) - \epsilon, f_t(p'_t) + \epsilon]).
\end{align*}
$$

The pseudo-gradient $X_t$ will be tangent to the lateral boundary of these models without specifying more. Observe that $M(q_t)$ and $M(p_t)$ are disjoint for every $t > t_0 + \delta$; and similarly for $M(q_t)$ and $M(p'_t)$ when $t < t_1 - \delta$.

We begin by fixing $M(p_t)$ and $M(q_t)$ when $t = t_0 + \delta$. We choose their $(y, z)$-coordinates to be those of $B_t$; only the $x$ coordinate has to be changed to have Morse coordinates. And similarly for $M(p'_t)$ and $M(q_t)$ when $t = t_1 - \delta$.

Then, we refer to Corollary 3.3 for extending the choice of Morse coordinates about $p_t$ to $t > t_0 + \delta$ so that they are adapted to $(f_t, W^u(p_t))$ for every $t$. The same is done for $M(p'_t)$, $t < t_1 - \delta$. For $M(q_t)$, $t \in [t_0 + \delta, t_1 - \delta]$, we do almost the same except for two differences:

1. The Morse coordinates are chosen to be adapted to the stable manifold $W^s(q_t)$.

2. Since the coordinates are already fixed for $t = t_0 + \delta$ and $t = t_1 - \delta$, item 2 of Corollary 3.3 has to be used.

Once this choice is made, nothing prevents us from modifying $X_t$ in each considered Morse model, so that it becomes tangent to the $x$-axis, the $y$-space and the $z$-space respectively, as it is the case in $B_t$ when $t \in [t_0, t_0 + \delta]$ and $t \in [t_1 - \delta, t_1]$. The unstable manifolds of $p_t$ and $p'_t$ are kept unchanged and also the stable manifold of $q_t$; but the unstable manifold of $q_t$ now satisfies

$$ (A1) \quad W^u(q_t) \cap M(q_t) = \{x = 0, z = 0\}. $$

We now recall the *cut-and-paste* construction for vector fields, which is abundantly used in [12] without using this name. Given a Morse function $f$ and a pseudo-gradient $X$, the change of $X$ by *cut-and-paste* along a regular level set $\{f = c\}$ consists of the following: cut $M$ at this level, make an isotopy of the upper part $(\psi_s)$ so that $(\psi_1)_*X$ has the same germ as $X$ along the cut, and finally glue $(\psi_1)_*X$ in the upper part to $X$ in the lower part.
The assumption for the germs guarantees the smoothness of the resulting vector field. The same construction works in a family.

By hypothesis of Lemma 3.5, the trace of $W^u(p'_t)$ in the top of $B_t$, $t \in [t_0, t_0 + \delta]$, intersects transversely the trace of $\overline{W^s}(q_t)$ in a single point $m_t$. The latter trace is a closed disc bounded by the trace of $W^s(p_t)$. Moreover, by the genericity assumption in 3.4 the point $m_t$ lies in the interior of that disc. So, we may apply cut-and-paste in the top of $B_t$ to make the part of $W^u(p'_t) \cap B_t$ lying close to $\{y=0\}$ to be contained in $\{z=0, x > x(q_t)\}$ for every $t \in [t_0, t_0 + \delta]$; this construction extends easily to $t \in (t_0 - \delta, t_0 + \delta]$. And similarly for $W^u(p_t)$ in $B_t$ for $t \in [t_1 - \delta, t_1 + \delta)$.

In the same way, when $t \in [t_0 + \delta, t_1 - \delta]$, cut-and-paste applied in the top of $M(q_t)$ makes the part of $(W^u(p_t) \cup W^u(p'_t)) \cap M(q_t)$ lying near $\{y=0\}$ to be contained in $\{z=0\}$. So, the connecting orbits cover the x-axis of $M(q_t)$. As the support of the isotopy is located near the stable manifold of $q_t$, the orbits in the unstable manifolds of $p_t$ and $p'_t$, apart from the connecting orbits, descend to the level $a_t = f_t(q_t) - \varepsilon$.

**CLAIM 1.**— There exists an arc $A_t$ in $M$ passing through $(p_t, q_t, p'_t)$ (or only one of them when a pair of critical points has disappeared), depending smoothly on $t \in (t_0 - \delta, t_1 + \delta)$ such that the Cerf graphic of $t \mapsto f_t|_{A_t}$ shows a one-variable swallow tail.

*Proof.* — Starting from the above situation of invariant manifolds, a new cut-and-paste makes $\ell_t$ (resp. $\ell'_t$) coincide with the x-axis near the bottom of $M(p_t)$ (resp. $M(p'_t)$) when $t \in [t_0 + \delta, t_1 - \delta]$.

When $t \in (t_0 - \delta, t_0 + \delta]$, $A_t$ is made of the x-axis of $B_t$, a piece of $\ell'_t$ from $B_t$ to $M(p'_t)$, the x-axis of $M(p'_t)$ and a path descending transversely to the level sets from the latter to the level $f_t(q_t) - \varepsilon$. A similar construction is performed on the other intervals of $t$. $\square$

B) *Proof of the swallow tail lemma in case $i=0$. This is the only case needed for proving Theorem 1.6.*

**CLAIM 2.**— Set $h_t := f_t|_{A_t}$. There are coordinates $(x, z) \in \mathbb{R} \times \mathbb{R}^{n-1}$ on a neighborhood $N_t$ of $A_t$, depending smoothly on $t \in (t_0 - \delta, t_1 + \delta)$, such that

$$ (i) \quad A_t = \{z=0\} $$

$$ (ii) \quad f_t(x, z) = h_t(x) + |z|^2. $$

*Proof.* — Indeed, it is true on a neighborhood $U_t$ of the set of critical points $\{p_t, p'_t, q_t\}$ by the choice we made of the Morse models in A). First, extend this coordinates arbitrarily so that (i) holds. As $h_t$ restricted to
$A_t \setminus U_t$ has no critical points, Lemma 3.1 applies with one parameter $t \in (t_0 - \delta, t_1 + \delta)$ and the following correspondence of notation: $V = M$, $V' = A_t \setminus U_t$, $K = \partial V'$, $f = f_t$, $g = h_t + |\cdot|^2$. $\square$

Now, choose a function $h_t^1$ coinciding with $h_t$ near the boundary of $A_t$ with a single critical point, indeed a maximum, and satisfying $h_t^1(x) \le h_t(x)$ for every $x \in A_t$. For $s \in [0, 1]$, set $k_t^s(x) = s(h_t^1(x) - h_t(x))$ and consider the deformation of path of functions $s \mapsto (h_t^s)_t$ given by

$$ (*) \quad h_t^s(x) = h_t(x) + k_t^s(x). $$

Note that the path $(h_t^1)$ has a "trivial" Cerf graphic. So, the formula (*) solves the one-dimensional elementary swallow tail lemma.

Using the coordinates given by Claim 2, the deformation extends to the neighborhoods $N_t$ thanks to the formula

$$ s \mapsto h_t(x) + \omega(|z|)k_t^s(x) + |z|^2, $$

where $\omega$ is a bump function with a small support, centered at 0. The $z$-derivative vanishes at $z=0$ only and the critical points are those of the one-dimensional case. Moreover, the deformation is stationary on the boundary of $N_t$ and, hence, extends to $M$ as a family $s \mapsto (f_t^s)_{t \in (t_0-\delta, t_1+\delta)}$. When $s=1$, the Cerf graphic of $(f_t^s)_{t \in [t_0-\delta, t_1+\delta]}$ is trivial and the swallow tail lemma is proved when $i=0$. $\square$

### C) Proof of Lemma of the swallow tail lemma when $i > 0$. We continue with the birth cylinders and the Morse models we introduced in part A).

**CLAIM 3.** — There exists a smooth one-parameter family $(W_t)_{t \in (t_0-\delta, t_1+\delta)}$ of smooth compact $(i+1)$-submanifolds, such that:

- $A_t \subset W_t$,

- $\partial W_t$ lies at level $a_t$ of the end points of $A_t$,

- the only critical points of $f_t|W_t$ are $p_t, q_t, p'_t$ and are non-degenerate except for the cubic points when $t$ equals $t_0$ or $t_1$.

**Proof.** — As a consequence of the cut-and-paste we have made, the closure of $W^u(p_t)$ in the upper level set $\{f_t \ge a_t\}$ and the one of $W^u(p'_t)$ intersect precisely the part of $W^u(q_t)$ lying in that upper level set. Moreover, both match smoothly along this common part of their boundary. This is given for free by the last choice of pseudo-gradients (see Formula (A1)). So, we set

$$ W_t = [W^u(p_t) \cup W^u(q_t) \cup W^u(p'_t)] \cap \{f_t \ge a_t\}. $$
□

CLAIM 4. — There are coordinates $(x, y, z) \in \mathbb{R} \times \mathbb{R}^i \times \mathbb{R}^{n-i-1}$ on a neighborhood $N_t$ of $A_t$, depending smoothly on $t \in (t_0 - \delta, t_1 + \delta)$, such that

$$ (i) \quad A_t = \{y = 0, z = 0\} \text{ and } W_t = \{z = 0\}, $$

$$ (ii) \quad f_t(x, y, z) = h_t(x) - |y|^2 + |z|^2. $$

*Proof*. — This is similar to Claim 2, except that here Lemma 3.1 has to be applied twice: firstly in a neighborhood $\mathcal{V}_t$ of $A_t$ in $W_t$ and secondly in a neighborhood of $\mathcal{V}_t$ in $M$. □

The radial vector field $Y_t := \sum_{j=1}^i y_j \partial_{y_j}$ in $N_t$ is transverse to the level sets of $f_t$ in $(N_t \setminus A_t) \cap \{z=0\}$. Keeping its notation, it extends to $W_t$ as a Lyapunov vector field (meaning that the Lyapunov inequality holds) for $f_t|(W_t \setminus A_t)$ since $f_t$ has no critical points on $W_t \setminus A_t$. So, by following the trajectories of $-Y_t$ we get a fibration of $W_t$ over $A_t$ in $i$-discs, pinched at the end points of $A_t$ (the diameter of the fibre vanishes there). The fibre $D_x$ over $x \in A_t$ is equipped with a Morse function, namely $g_{t,x} := f_t|_{D_x}$, which has one critical point, a maximum indeed, at $x \in A_t$.

Extend $Y_t$ to some neighborhood $\tilde{N}_t$ of $W_t$ in $M$ as a Lyapunov vector field $\tilde{Y}_t$ of $f_t|(\tilde{N}_t \setminus A_t)$. Choosing $\tilde{N}_t$ to be invariant by the positive semi-flow of $\tilde{Y}_t$ gives $\tilde{N}_t$ a structure of bundle over $A_t$ whose fibre $\tilde{D}_x$, $x \in \tilde{A}_t$, is diffeomorphic to $D_x \times D^{n-i-1}$. The restriction $\tilde{g}_{t,x}$ of $f_t$ to the fibre $\tilde{D}_x$, $x \in A_t$, is a Morse function with the single critical point $x \in A_t$. It is equipped with the pseudo-gradient $\tilde{Y}_t$, whose unstable manifold is $D_x$.

We apply Lemma 2.1 to the function $\tilde{g}_{t,x}$, where $(t, x)$ is a parameter. This lemma allows us to decrease the critical value $f_t(x)$ as we want, without introducing new critical points, as long as this value remains greater than $f_t(\partial W_t) = a_t$. This process yields a deformation of $(f_t)$ which extends the solution $(*)$ of the one-dimensional swallow tail lemma without introducing new critical points, and solves the general case. □

LEMMA 3.6 (Elementary lips lemma). — Let $\gamma := (f_t)_{t \in [0,1]}$ be a generic path of functions on the manifold $M$. Assume that its restriction to $t \in [t_0, t_1]$ has a Cerf graphic as in figure 4 (lips): for $t \in (t_0, t_1)$, there are two critical points $p_t, q_t$ of respective indices $i+1$ and $i$ such that the pair $(p_t, q_t)$ is created at time $t_0$ and is cancelled at time $t_1$. Moreover, a smooth family of pseudo-gradients $X_t$ for $f_t$ is given satisfying the next conditions for all $t \in [t_0, t_1]$:

- $W^u(p_t)$ intersects $W^s(q_t)$ transversely along a single orbit $\ell_t$;
- all the other orbits in $W^u(p_t)$ cross the level set $f(q_t) - \varepsilon$, for some $\varepsilon > 0$.

Then $\gamma$ can be deformed to a path $\gamma'$ so that the corresponding lips are removed from the Cerf graphic, the deformation being stationary on $[0, t_0 - \delta] \cup [t_1 + \delta, 1]$ for any $\delta > 0$.

<figure><img src="image_6.png" /><figcaption>Figure 4A</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4B</figcaption></figure>

**LEMMA 3.7** (Morse's cancellation theorem). — Let $f : M \to \mathbb{R}$ be a Morse function equipped with a pseudo-gradient $X$. Let $(p, q)$ be a pair of critical points of consecutive indices whose invariant manifolds satisfy the next conditions:

- $W^u(p)$ intersects $W^s(q)$ transversely and along a single orbit;

- all the other orbits in $W^u(p)$ cross the level set $f(q) - \varepsilon$ for some $\varepsilon > 0$.

Then, for every small neighborhood $U$ of the closure of the intersection $W^u(p) \cap \{f \ge f(q) - \varepsilon\}$, there is a Morse function which has no critical points in $U$ and coincides with $f$ away from $U$.

# 4. Path of polar functions

## 4.1. *Proof of Theorem 1.6*

According to Theorem 1.3, there is a path $\gamma := (f_t)$ fulfilling all requirements of Theorem 1.6 (birth times before cancellation times and order of critical values) except the one min/one max condition. So, the matter is to kill the appearance of extra local minima or maxima. We are looking at the local minima only.

First, we make the assumption (H) that one can follow continuously a minimum $m_t$ of $f_t$ from $t=0$ to $t=1$. By permuting the birth times if necessary (since $\dim M > 1$, the last claim of Lemma 2.5 applies) and cancelling by pairs the crossings of index 0 critical values (Lemma 2.1), we may assume that the index 0 part of the Cerf graphic shows no crossings (see figure 5A).
Let $\mu$ be the maximal number of extra minima along $\gamma$; we are going to
decrease $\mu$ by 1. Denote $(t'_0, t'_1)$ the interval where $f_t$ has $\mu$ extra minima.
For $t \in (t'_0, t'_1)$, denote the upper local minimum of $f_t$ by $m'_t$.

Without loss of generality we may assume that 3/2 separates the index
1 critical values from those of index 2; the same is true for the value 3/2 – η,
if η > 0 is small. Set L_t := f_t⁻¹(3/2 – η). Since M is connected and L_t lies
above all the critical points of index 1, L_t is connected.

If $X_t$ is a pseudo-gradient of $f_t$, we see in $L_t$ the trace $S_t$ of the stable manifold $W^s(m_t, X_t)$ and, when $t \in (t'_0, t'_1)$, the trace $S'_t$ of the stable manifold $W^s(m'_t, X_t)$. Both are changing when handle slides of index 1 happen. But, due to $n \ge 3$, they remain connected; indeed, each one is always an $(n-1)$-sphere with holes.

<figure><img src="image_8.png" /><figcaption>Figure 5A</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 5B</figcaption></figure>

So, choose smoothly points $x_t \in S_t$ and $x'_t \in S'_t$ linked by a simple arc $\alpha_t$
in $L_t$. We introduce a cancelling pair of critical points $(s_t, r_t)$ of respective
index (2, 1) in a collar neigborhood above $L_t$; the birth time is chosen less
than $t'_0$, the cancellation time greater than $t'_1$ (compare figure 5B), and the
base of the birth cylinder is a $(n-1)$-disc in $L_t$ centered at $x_t$. Denote by
$\gamma' := (f'_t)$ this new path from $f_0$ to $f_1$. After choosing a suitable pseudo-
gradient $X'_t$, we have for every $t \in [t'_0 + \varepsilon, t'_1 - \varepsilon]$:

$$
W^u(r_t, X'_t) \cap L_t = \{x_t, x'_t\}, \quad W^u(s_t, X'_t) \cap L_t = \alpha_t.
$$

In particular, there are no $X'_t$-connecting orbits form $r_t$ to another critical
point of index 1. Therefore, Lemma 2.1 applies and a new deformation of
the path $\gamma'$ puts the critical value of $r_t$ below the other critical values of
index 1 when $t \in [t'_0 + 2\varepsilon, t'_1 - 2\varepsilon]$ (compare the Cerf graphic in figure 6A).
By the choice of $x'_t$, there is exactly one connecting orbit from $r_t$ to $m'_t$ for
every $t \in [t'_0 + 2\varepsilon, t'_1 - 2\varepsilon]$. One makes cancellations at times $t'_0 + 2\varepsilon$ and
$t'_1 - 2\varepsilon$. These cancellations may be viewed as a new deformation of the
path $\gamma'$; the final Cerf graphic looks like figure 6B, with two swallow tails
separated by lips. Lemma 3.5 and 3.6 apply and yield some deformation
of the path of functions so that the swallow tails and lips vanish. The final path of this last deformation has $\mu - 1$ extra minima. This finishes the proof in case of (H).

<figure><img src="image_10.png" /><figcaption>Figure 6A</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 6B</figcaption></figure>

I am indebted to the anonymous referee who made me observe that the general case easily reduces to assumption (H). Indeed, a suitable isotopy of $M$ makes the minima (resp. maxima) of $f_0$ and $f_1$ coincide. Since the germ of smooth function is unique at a non-degenerate extremum, up to isotopy and rescaling, we may assume that $f_0$ and $f_1$ coincide on small discs $d$ and $d'$ about these extrema. Then, by connecting $f_0$ to $f_1$ in the space of smooth functions having a given restriction to $d$ and $d'$, (H) is fulfilled. $\square$

## 4.2. Final comments

1) The Reidemeister-Singer theorem, that is, item 1 in Theorem 1.1, is also proved by R. Craggs in the piecewise linear category (see [6]). His proof relies of previous results on collapsings, due to Chillingworth [5]. But the original proof was revisited and explained by L. Siebenmann in [17].

2) It is worth noticing that both parts of Theorem 1.1 are consequence of two statements (Theorems 1.3 and 1.6) about functions which hold true in any dimension. These two theorems should be known to specialists. Maybe, the proof of Theorem 1.3 that is given here is almost the simplest one. I did not find any written proof of Theorem 1.6.

3) The proof of the latter theorem is not very elementary, due to the use of the swallow tail lemma. So, the classical 3-dimensional proof of item 2 in Theorem 1.1 remains competitive. The statement reads as this: *Let $H$ be a 3-dimensional handlebody of genus $g$, and let $D, D'$ be two minimal systems of $g$ compression discs of $H$ whose complement is a 3-ball. Then, one can pass from $D$ to $D'$ by finitely many handle slides.* This can be proved by a very standard *cut-and-past* technique.

I am grateful to Francis Bonahon, Jean Cerf, Alexis Marin and Patrick Massot for comments on first versions of this note. I am indebted to the ref-
eree who suggested me several improvements. Carlos Moraga Ferrándiz [13]
is the first who used of the techniques introduced in this note; I thank him
for valuable suggestions. I am also grateful to Marc Chaperon for discussions
about the saddle-node bifurcation.

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