Vanishing polyhedron and collapsing map

Abstract

In this paper we give a detailed proof of the fact that the Milnor fiber $X_t$ of an analytic complex isolated singularity function defined on a reduced n-equidimensional analytic complex space X is a regular neighborhood of a polyhedron $P_t \subset X_t$ of real dimension $n-1$. Moreover, we describe the degeneration of $X_t$ onto the special fiber $X_0$, by giving a continuous collapsing map $\psi _t: X_t \rightarrow X_0$ which sends $P_t$ to $\{0\}$ and which restricts to a homeomorphism $X_t {\backslash } P_t \rightarrow X_0 {\backslash } \{0\}$.

1 Introduction

Let $f{:}~ (X,x) \rightarrow ({\mathbb {C}},0)$ be a germ of complex analytic function f at a point x of a reduced equidimensional complex analytic space $X \subset {\mathbb {C}}^N$ (with arbitrary singularity). In [11] the first author proved that there exist sufficiently small positive real numbers ${\epsilon }$ and $\eta$ with $0<\eta \ll {\epsilon }\ll 1$ such that the restriction:

$$\begin{aligned} f_|: {\mathbb {B}}_{\epsilon }(x) \cap X \cap f^{-1}({\mathbb {D}}_\eta ^*) \rightarrow {\mathbb {D}}_\eta ^* \end{aligned}$$

is a locally trivial topological fibration, where ${\mathbb {B}}_{\epsilon }(x)$ is the closed ball of radius ${\epsilon }$ around $x \in {\mathbb {C}}^N$, ${\mathbb {D}}_\eta$ is the closed disk of radius $\eta$ around $0 \in {\mathbb {C}}$ and ${\mathbb {D}}_\eta ^* := {\mathbb {D}}_\eta {\backslash } \{0\}$.

The topology of the fiber $X_t := {\mathbb {B}}_{\epsilon }(x) \cap X \cap f^{-1}(t)$ does not depend on ${\epsilon }$ small enough (see Theorem 2.3.1 of [14]). We call $X_t$ the Milnor fiber of f, with boundary $\partial X_t := X_t \cap {\mathbb {S}}_{\epsilon }(x)$. We also set $X_0 := {\mathbb {B}}_{\epsilon }(x) \cap X \cap f^{-1}(0)$.

The first author sketched a proof of the following theorem in [13]:

Theorem 1

Let $X \subset {\mathbb {C}}^N$ be a reduced equidimensional complex analytic space and let ${\mathcal S} = (S_{\alpha })_{{\alpha }\in A}$ be a Whitney stratification of X. Let $f{:}~ (X,x) \rightarrow ({\mathbb {C}},0)$ be a germ of complex analytic function at a point $x \in X$. If f has an isolated singularity at x relatively to ${\mathcal S}$ and if ${\epsilon }$ and $\eta$ are sufficiently small positive real numbers as above, then for each $t \in {\mathbb {D}}_\eta ^*$ there exist:

• (i) a polyhedron $P_t$ of real dimension $\dim _{\mathbb {C}}X_t$ in the Milnor fiber $X_t$, compatible with the Whitney stratification $\mathcal {S}$, and a continuous simplicial map:

  $$\begin{aligned} \tilde{\xi }_t: \partial X_t \rightarrow P_t \end{aligned}$$

  compatible with $\mathcal {S}$, such that $X_t$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_t$;

• (ii) a continuous map $\psi _t: X_t \rightarrow X_0$ that sends $P_t$ to $\{0\}$ and that restricts to a homeomorphism $X_t {\backslash } P_t \rightarrow X_0 {\backslash } \{0\}$.

The purpose of this paper is to give a complete and detailed proof of Theorem 1, following the strategy proposed in [13].

That theorem was conjectured by Thom in a seminar, in the early 70’s, when X is smooth. He noticed that Pham gave an explicit construction of such a vanishing polyhedron in [19] when $f{:}~ {\mathbb {C}}^n \rightarrow {\mathbb {C}}$ is a polynomial of the form:

$$\begin{aligned} f(z_1, \ldots , z_n) = z_1^{\nu _1} + \cdots + z_n^{\nu _n}, \end{aligned}$$

with $\nu _i \ge 2$ integer, for $i=1, \ldots , n$.

In this paper, we are going to prove the following stronger version of Theorem 1. Let ${\mathbb {D}}^+$ be a closed semi-disk in ${\mathbb {D}}_{\eta _2}$ as in Fig. 1 (with $0 \in \partial {\mathbb {D}}^+$).

Fig. 1

The semi-disk $\mathbb {D}^+$

So our main theorem is:

Theorem 2

Let $X \subset {\mathbb {C}}^N$ be a n-dimensional reduced equidimensional complex analytic space and let ${\mathcal S} = (S_{\alpha })_{{\alpha }\in A}$ be a Whitney stratification of X. Let $f{:}~ (X,x) \rightarrow ({\mathbb {C}},0)$ be a germ of complex analytic function with an isolated singularity at x, relatively to ${\mathcal S}$. Let ${\epsilon }$ and $\eta$ be sufficiently small positive real numbers as above, and let ${\mathbb {D}}^+$ be a closed semi-disk in ${\mathbb {D}}_\eta \subset {\mathbb {C}}$ such that 0 is in its boundary. Then there exist:

• (i) A polyhedron $P^+$ in $X^+ := X \cap f^{-1}({\mathbb {D}}^+) \cap {\mathbb {B}}_{\epsilon }(x)$ of real dimension $n +1$, compatible with the Whitney stratification $\mathcal {S}$, such that for each $t \in {\mathbb {D}}^+ {\backslash } \{0\}$ the intersection $P^+ \cap X_t$ is a polyhedron $P_t$ of real dimension $n-1$, compatible with the Whitney stratification $\mathcal {S}$.

• (ii) A continuous simplicial map:

  $$\begin{aligned} \tilde{\xi }_+: \partial X^+ \rightarrow P^+ \end{aligned}$$

  compatible with $\mathcal {S}$, such that $X^+$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_+$, and such that for each $t \in {\mathbb {D}}^+ {\backslash } \{0\}$ the map $\tilde{\xi }_+$ restricts to a continuous simplicial map:

  $$\begin{aligned} \tilde{\xi }_t: \partial X_t \rightarrow P_t \end{aligned}$$

  compatible with $\mathcal {S}$, such that $X_t$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_t$, where $\partial X^+ := X^+ \cap {\mathbb {S}}_{\epsilon }(x)$, $\partial X_t := X_t \cap {\mathbb {S}}_{\epsilon }(x)$ and ${\mathbb {S}}_{\epsilon }(x)$ is the boundary of ${\mathbb {B}}_{\epsilon }(x)$.

• (iii) A continuous map $\psi _t: X_t \rightarrow X_0$ that sends $P_t$ to $\{0\}$ and that restricts to a homeomorphism $X_t {\backslash } P_t \rightarrow X_0 {\backslash } \{0\}$, for any $t \in {\mathbb {D}}^+ {\backslash } \{0\}$.

In Sect. 2 we recall some classical definitions and results. In Sect. 3 we construct the relative polar curve of f, which is the main tool to prove Theorem 2. In Sect. 4 we prove Theorem 2 when X is two-dimensional. Then in Sect. 5 we present two propositions (Propositions 29 and 30) and we use them to prove Theorem 2 in the general case. In Sect. 6 we prove those Propositions by finite induction on the dimension of X. Finally, in Sect. 7 we make the detailed construction of a vector field (Lemma 35) that is used in Sect. 6.

The authors are grateful to the reviewer’s valuable comments and suggestions, which improved the readability and quality of the manuscript.

2 Background

In this section we recall some definitions, references and theorems that will be used in this paper.

2.1 Whitney stratification

Following [14] (Section 1, p. 67), we have:

Definition 3

Let X be a subanalytic set (resp. a reduced complex analytic space). We say that a locally finite family of non-singular subanalytic connected subsets $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ of X is a subanalytic stratification (resp. complex analytic stratification) of X if:

• (i) the family $\mathcal {S}$ is a partition of X; and

• (ii) the closure $\bar{S}_{\alpha }$ of $S_{\alpha }$ in X and $\bar{S}_{\alpha }{\setminus } S_{\alpha }$ are subanalytic (resp. complex analytic) subspaces of X, for any ${\alpha }\in A$.

The subsets $S_{\alpha }$ are called strata of the stratification $\mathcal {S}$ of X.

In this paper we will use both subanalytic and complex analytic stratifications. Although we work with a complex analytic space $X \subset {\mathbb {C}}^N$, when we intersect it with a closed ball ${\mathbb {B}}_{\epsilon }$ in ${\mathbb {C}}^N$ we obtain a subanalytic set.

We say that a (subanalytic or complex analytic) stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ as above satisfies the frontier condition if for any $({\alpha },{\beta }) \in A \times A$ such that $S_{\alpha }\cap \bar{S}_{\beta }\ne \emptyset$ ones has that $S_{\alpha }\subset \bar{S}_{\beta }$. In this case, $\bar{S}_{\alpha }$ and $\bar{S}_{\alpha }{\setminus } S_{\alpha }$ are union of strata of the stratification $\mathcal {S}$, for any ${\alpha }\in A$.

We have:

Definition 4

Let X be a subanalytic set (resp. reduced complex analytic space). We say that a subanalytic (resp. complex analytic) stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ of X is a subanalytic (resp. complex analytic) Whitney stratification if:

• (i) the stratification $\mathcal {S}$ satisfies the frontier condition; and

• (ii) for any $({\alpha },{\beta }) \in A \times A$ such that $S_{\alpha }\cap \bar{S}_{\beta }\ne \emptyset$ the pair of strata $(S_{\alpha }, S_{\beta })$ satisfies the Whitney condition, which is the following: for any $y \in S_{\alpha }$ there exists a local embedding of (X, y) in $({\mathbb {R}}^N,0)$ such that for any sequence $(x_n,y_n)_{n \in \mathbb {N}}$ in $S_{\beta }\times S_{\alpha }$ that converges to (y, y) and such that the limit T of the tangent spaces $T_{x_n}S_{\beta }$ and the limit $\lambda$ of the real secants $\overline{x_n y_n}$ in ${\mathbb {R}}^N$ exist, one has the inclusion $\lambda \subset T$.

One can verify that, for any $y \in S_{\alpha }$ fixed, if the condition above is satisfied for some local embedding, then it is satisfied for any local embedding.

In [23] and in [7] (for the complex case and for the subanalytic case respectively) it is proved the following:

Theorem 5

Let X be a subanalytic set (resp. reduced complex analytic space) and let $(\Phi _i)_{i \in I}$ be a locally finite family of subanalytic (resp. complex analytic) closed subsets of X. There exists a subanalytic (resp. complex analytic) Whitney stratification of X such that each $\Phi _i$ is a union of strata, for $i \in I$.

Next we give a lemma due to Cheniot [4] (see also Lemma 4.2.2. of Chapter III of [21]) that will be implicitly used many times in the paper:

Lemma 6

Let X and Y be two subanalytic sets in ${\mathbb {R}}^N$ (or reduced complex analytic spaces in ${\mathbb {C}}^N$, in the complex case) such that X has a subanalytic (resp. complex analytic) Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ and such that Y is non-singular. If Y intersects each stratum $S_{\alpha }$ transversally in ${\mathbb {R}}^N$ (resp. in ${\mathbb {C}}^N$), then the Whitney stratification of X induces a subanalytic (resp. complex analytic) Whitney stratification $\mathcal {P} = (P_{\alpha })_{{\alpha }\in A}$ of $X \cap Y$, where $P_{\alpha }:= S_{\alpha }\cap Y$, for each ${\alpha }\in A$.

Next we will present a stronger version of the Whitney stratification. But first we need the following definition. Given two vector subspaces A and B of ${\mathbb {R}}^N$ set:

$$\begin{aligned} \delta (A,B) := sup_{\begin{array}{c} x \in A \\ \Vert x\Vert =1 \end{array}} d(x,B), \end{aligned}$$

where d(x, B) is the distance between x and B. Then we have:

Definition 7

Let $X \subset {\mathbb {R}}^N$ be a subanalytic set (resp. reduced complex analytic space) with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$. We say that $\mathcal {S}$ has the property (w) if for any $({\alpha },{\beta }) \in A \times A$ such that $S_{\alpha }\cap \bar{S}_{\beta }\ne \emptyset$ the pair of strata $(S_{\alpha }, S_{\beta })$ satisfies the Kuo–Verdier condition below:

For any $y^{\prime } \in S_{\alpha }$ there exists a neighborhood U of $y^{\prime }$ in ${\mathbb {R}}^N$ and a real constant $C > 0$ such that for any $(x, y) \in (S_{\beta }\cap U, S_{\alpha }\cap U)$ one has that:

$$\begin{aligned} \delta \left( T_x (S_{\beta }) , T_y (S_{\alpha }) \right) \le C \Vert x-y\Vert . \end{aligned}$$

In Theorem 1.2 of chapter V of [21], Teissier proved the following:

Lemma 8

Let $X \subset {\mathbb {C}}^N$ be a reduced complex analytic space with a complex analytic Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$. Then $\mathcal {S}$ has the property (w).

The analogous of this result in the subanalytic case is not true (see Example 1 of [3] for instance).

Definition 9

Let X and Y be subanalytic sets and let $A \subset X$ be endowed with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$. We say that a morphism $f{:}~ X \rightarrow Y$ is transversal to $\mathcal {S}$ if, for any ${\alpha }\in A$, f induce a smooth morphism $f_{\alpha }: S_{\alpha }\rightarrow Y$.

According to Remark (3.7) of [22], we have:

Lemma 10

Let X be a subanalytic set (resp. complex analytic set) and let $A \subset X$ be endowed with a subanalytic (resp. complex analytic) Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ with the property (w). If Y is smooth and if $f{:}~ X \rightarrow Y$ is transversal to $\mathcal {S}$, then for any smooth and locally closed subset $Z \subset Y$ one has that $f^{-1}(Z) \cap \mathcal {S}$ is a subanalytic (resp. complex analytic) Whitney stratification of $A \cap f^{-1}(Z)$ with the property (w).

Considering $f{:}~ X \rightarrow {\mathbb {R}}$ the square of the distance function to a point $x \in X$ and Z a closed interval $[0, {\epsilon }^2]$, we have:

Corollary 11

If $X \subset {\mathbb {C}}^N$ is a reduced complex analytic space with a complex analytic Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ and if ${\mathbb {B}}_{\epsilon }(x)$ is a ball around $x \in X$ in ${\mathbb {C}}^N$ of small enough radius ${\epsilon }>0$, then $\mathcal {S}$ induces a subanalytic Whitney stratification on $X \cap {\mathbb {B}}_{\epsilon }(x)$ with the property (w).

Let X be a subanalytic set endowed with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ that has the property (w). We say that a Whitney stratification $\mathcal {S}^{\prime } = (S_{\beta }^{\prime })_{{\beta }\in A^{\prime }}$ with the property (w) is finer then $\mathcal {S}$ (or that $\mathcal {S}^{\prime }$ is a refinement for $\mathcal {S}$) if for any ${\beta }\in A^{\prime }$ there exists ${\alpha }\in A$ such that $S_{\beta }^{\prime } \subset S_{\alpha }$.

As in Remark (3.6) of [22], we have:

Remark 12

Let X and Y be subanalytic sets and let $A \subset X$ be a closed subset. Let $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ be a Whitney stratification of A and let $\mathcal {Z} = (Z_{\beta })_{{\beta }\in B}$ be a Whitney stratification of Y, both of them with the property (w). If $f{:}~ X \rightarrow Y$ is a morphism such that the restriction $f_{|A}: A \rightarrow Y$ is proper, then we can consider a refinement $\mathcal {S} = (S_{\alpha }^{\prime })_{{\alpha }\in A^{\prime }}$ of $\mathcal {S}$ and a refinement $\mathcal {Z^{\prime }} = (Z_{\beta }^{\prime })_{{\beta }\in B^{\prime }}$ of $\mathcal {Z}$ such that, for any ${\beta }\in B^{\prime }$, one has that:

• (i) $f^{-1}(Z_{\beta }^{\prime }) \cap A$ is a union of strata of $\mathcal {S}^{\prime }$;

• (ii) the restriction $f_|: f^{-1}(Z_{\beta }^{\prime }) \rightarrow Z_{\beta }^{\prime }$ is transversal to $\mathcal {S}^{\prime } \cap f^{-1}(Z_{\beta }^{\prime })$.

Now let $f{:}~ X \rightarrow {\mathbb {C}}$ be a complex analytic function defined on a complex analytic space X with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$. Following [12], we have:

Definition 13

We say that f has an isolated singularity at $x \in X$ relatively to the stratification $\mathcal {S}$ if:

• (i) the restriction of f to $S_{\alpha }$ is a submersion, for any ${\alpha }\in A$ such that $S_{\alpha }$ does not contain x;

• (ii) the restriction of f to $S_{{\alpha }(x)}$ has an isolated critical point at x, where $S_{{\alpha }(x)}$ is the stratum that contains x.

2.2 Rugose vector fields

Following [5], we will briefly define a rugose vector field on a subanalytic set $X \subset {\mathbb {R}}^N$ endowed with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ with the property (w). See [22] for the detailed definitions.

We say that a real-valued function $f{:}~ X \rightarrow {\mathbb {R}}$ is a rugose function if for any ${\alpha }\in A$ one has that:

• (i) the restriction of f to $S_{\alpha }$ is smooth;

• (ii) for any $x \in S_{\alpha }$ there exists a neighborhood U of x in ${\mathbb {R}}^N$ and a real constant $C >0$ such that for any $x^{\prime } \in U \cap S_{\alpha }$ and for any $y \in U \cap X$ one has that:

  $$\begin{aligned} \Vert f(x^{\prime })-f(y)\Vert \le C \Vert x-y\Vert . \end{aligned}$$

We say that a map $f{:}~ X \rightarrow {\mathbb {R}}^M$ is a rugose map if each of its coordinate functions is a rugose function. We say that a vector bundle F over X is rugose if its chart change maps are rugose.

A rugose vector bundle F on X tangent to the stratification $\mathcal {S}$ is a vector bundle over X such that, for every stratum $S_{\alpha }$ there is an injection $i_{\alpha }: F|_{S_{\alpha }} \rightarrow T S_{\alpha }$ such that the vector bundle morphism $F \rightarrow i^* T {\mathbb {R}}^N |_X$ induced by $i_{\alpha }$ is rugose.

A stratified vector field $\vec {\nu }$ on X is a section of the tangent bundle $T {\mathbb {R}}^N |_X$ such that at each $x \in X$, the vector $\vec {\nu }(x)$ is tangent to the stratum that contains x.

A stratified vector field $\vec {\nu }$ on X is called rugose near $y \in S_{\alpha }$ if there exists a neighborhood U of y in ${\mathbb {R}}^N$ and a real constant $C > 0$ such that:

$$\begin{aligned} \Vert \vec {\nu }(y^{\prime }) - \vec {\nu }(x)\Vert \le C \Vert y^{\prime } - x\Vert , \end{aligned}$$

for every $(x,y^{\prime }) \in (U \cap S_{\beta }, U \cap S_{\alpha })$ with $S_{\alpha }\subset \bar{S}_{\beta }$.

On the other hand, we say that a rugose stratified vector field $\vec {\nu }$ on X is integrable if there exists an open neighborhood U of $X \times \{0\}$ in $X \times {\mathbb {R}}$ and a rugose map $\theta : U \rightarrow X$ such that for any ${\alpha }\in A$ one has that:

• (i) $\theta \left( (S_{\alpha }\times {\mathbb {R}}) \cap U \right) \subset S_{\alpha }$;

• (ii) for any $x \in S_{\alpha }$ such that $(x,t) \in (S_{\alpha }\cap {\mathbb {R}}) \cap U$ one has that $\frac{\partial }{\partial t} \theta (x,t) = \vec {\nu } \left( \theta (x,t) \right)$.

We say that the map $\theta$ is the flow associated to the vector field $\vec {\nu }$.

For each $x_0 \in X$ we say that the restriction $\theta |_{(\{x_0\} \times {\mathbb {R}}) \cap U} : (\{x_0\} \times {\mathbb {R}}) \cap U \rightarrow X$ is an integral curve for the vector field $\vec {\nu }$ with initial condition $x_0$. Condition (i) above assures that if $x_0$ is in a stratum $S_{\alpha }$ then the image of the integral curve for $\vec {\nu }$ with initial condition $x_0$ is contained in $S_{\alpha }$.

We have (Proposition 4.8 of [22]):

Proposition 14

Any rugose stratified vector field on a closed subanalytic set X of ${\mathbb {R}}^N$ is integrable. Moreover, given a rugose vector field, if $\theta$ and $\theta ^{\prime }$ are rugose maps defined in open neighborhoods U and $U^{\prime }$ of $X \times \{0\}$ in $X \times {\mathbb {R}}$, satisfying the properties (i) and (ii) above, then $\theta$ and $\theta ^{\prime }$ coincide in $U \cap U^{\prime }$.

2.3 Stratified maps

We have:

Definition 15

Let X and Y be subanalytic sets (or reduced complex analytic spaces) with Whitney stratifications $(X_{\alpha })_{{\alpha }\in A}$ and $(Y_{\beta })_{{\beta }\in B}$ respectively. A real analytic (resp. complex analytic) morphism $h: X \rightarrow Y$ is a stratified map if:

• (i) h sends each stratum $X_{\alpha }$ to a unique stratum $Y_{{\beta }({\alpha })}$, for some ${\beta }({\alpha }) \in B$;

• (ii) the restriction of h to each stratum $X_{\alpha }$ induces a smooth map $h_{\alpha }: X_{\alpha }\rightarrow Y_{{\beta }({\alpha })}$.

We say that a stratified map h as above is a stratified submersion if each $h_{\alpha }$ is a (surjective) submersion.

We say that a stratified map h as above is a stratified homeomorphism if h is a homeomorphism and each $h_{\alpha }$ is a smooth diffeomorphism.

Let $h: X \rightarrow Y$ be a stratified map as above and let $\vec {\nu }$ be a stratified vector field on Y. We say that a stratified vector field $\vec {\mu }$ on X lifts $\vec {\nu }$ if for each $x \in X$ one has that $dh \left( \vec {\mu }(x) \right) = \vec {\nu } \left( h(x) \right)$.

We have (see [22], Proposition 4.6):

Proposition 16

Let X be a real analytic space endowed with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ with the property (w), and let Z be a locally closed subset of X which is union of strata $S_{\alpha }$. Let Y be a non-singular real analytic space and let $h : X \rightarrow Y$ be a stratified submersion. If $\vec {\nu }$ is a smooth vector field in Y, then there exists a rugose stratified vector field $\vec {\mu }$ on Z that lifts $\vec {\nu }$.

There is a more general version of the theorem above, which includes the case when Y is not smooth or the case when the vector field $\vec {\nu }$ is non-everywhere smooth, as we present below (see [22], Remark 4.7):

Proposition 17

Let X and Y be real analytic spaces with Whitney stratifications $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$ and $\mathcal {W} = (W_{\beta })_{{\beta }\in B}$, respectively, both of them with the property (w). Let $h : X \rightarrow Y$ be a stratified submersion. Also, let Z be a locally closed subset of X which is a union of strata $S_{\alpha }$. Suppose that each restriction:

$$\begin{aligned} f_{| W_{\beta }}: Z \cap f^{-1}(W_{\beta }) \rightarrow W_{\beta }, \end{aligned}$$

with ${\beta }\in B$, is transversal to $\mathcal {S} \cap Z \cap f^{-1}(W_{\beta })$ (that is, $f_{| W_{\beta }}$ induces a smooth map on each stratum of $f^{-1}(W_{\beta }) \cap Z$). If $\vec {\nu }$ is a rugose stratified vector field on f(Z), then there exists a rugose stratified vector field $\vec {\mu }$ on Z that lifts $\vec {\nu }$.

2.4 Simplicial maps

Let $\mathcal {X}$ be a topological space. A triangulation for $\mathcal {X}$ is a pair $(\mathcal {K}, h)$, where $\mathcal {K}$ is a simplicial complex and h is a homeomorphism $h: \mathcal {K} \rightarrow \mathcal {X}$. We say that $\mathcal {X}$ is triangulable if there exists a triangulation $(\mathcal {K}, h)$ for $\mathcal {X}$.

Hironaka proved in [7] that any subanalytic set is triangulable.

In this paper, a polyhedron is a compact topological space that is triangulable. We are only interested in the existence of a simplicial structure; a particular decomposition into faces is not important in this work.

We say that a map $f{:}~ P \rightarrow P^{\prime }$ between two polyhedra is a simplicial map if there exist triangulations $(\mathcal {K}, h)$ and $(\mathcal {K}^{\prime }, h^{\prime })$ for P and $P^{\prime }$, respectively, such that the induced map $\tilde{f} : \mathcal {K} \rightarrow \mathcal {K}^{\prime }$ is a simplicial map in the usual sense (that is, $\tilde{f}$ has the property that whenever the vertices $v_0, \ldots , v_n$ of $\mathcal {K}$ span a simplex of $\mathcal {K}$, the points $f(v_0), \ldots , f(v_n)$ are vertices of a simplex of $\mathcal {K^{\prime }}$).

Notice that our definition is slightly different from the usual definition of a simplicial map, since it relates spaces which do not have fixed simplicial structures (compare with Lemma 2.7 of [17]).

Finally, we have:

Definition 18

Let P be a polyhedron contained in a subanalytic space X endowed with a Whitney stratification $\mathcal {S} = (S_{\alpha })_{{\alpha }\in A}$. We say that P is adapted to the stratification $\mathcal {S}$ if the interior of each simplex of P is contained in $S_{\alpha }$ for some ${\alpha }\in A$.

2.5 Some useful results

Now we will state two results that will be used later. The first one is Thom–Mather’s first isotopy lemma (Proposition 11.1 of [15]):

Lemma 19

(Thom–Mather’s first isotopy lemma) Let M and P be smooth manifolds and let X be a closed subset of M with a Whitney stratification (real or complex). If $f{:}~ X \rightarrow P$ is proper stratified map and if it is a submersion on each stratum, then f is a locally trivial fibration.

The second one is Remmert’s theorem (see Corollary 1.68 of [6] for instance):

Theorem 20

(Finite mapping theorem) Let $f{:}~ X \rightarrow Y$ be a finite morphism of complex analytic spaces and $Z \subset X$ a closed analytic complex subspace of X. Then $f(Z) \subset Y$ is an analytic subset of Y.

3 Polar curves

In the rest of the paper, X will be a fixed reduced equidimensional complex analytic space in ${\mathbb {C}}^N$ such that $0 \in X$, and $f{:}~ (X,0) \rightarrow ({\mathbb {C}},0)$ will be the germ of a complex analytic function. Notice that if we prove Theorem 2 with this assumption, then we clearly prove it in the general case $f{:}~ (X,x) \rightarrow ({\mathbb {C}},0)$, and we do so just to simplify the notation.

Moreover, we will endow the germ (X, 0) with a fixed complex analytic Whitney stratification ${\mathcal S} = (S_{\alpha })_{{\alpha }\in A}$ such that $f^{-1}(0)$ is a union of strata (see Lemma 5).

The notion of polar curve for a complex analytic function defined on an open neighborhood of ${\mathbb {C}}^N$ relatively to a linear form $\ell$ was introduced by Teissier and by the first author in [20] and [9], respectively. Later, in [10] the first author extended this concept to a germ of complex analytic function $f{:}~ (X,0) \rightarrow ({\mathbb {C}},0)$ relatively to the Whitney stratification ${\mathcal S} = (S_{\alpha })_{{\alpha }\in A}$. We are going to recall that.

Notice that by now we are not supposing that f has an isolated singularity (in the stratified sense). This hypothesis will be asked after the lemma below.

Let $f{:}~ X \rightarrow {\mathbb {C}}$ be a representative of the germ of function f such that X is closed in an open neighborhood U of 0 in ${\mathbb {C}}^N$. For any linear form:

$$\begin{aligned} \ell : {\mathbb {C}}^N \rightarrow {\mathbb {C}}\end{aligned}$$

the function f and the restriction of $\ell$ to X induce the analytic morphism:

$$\begin{aligned} \phi _\ell : X \rightarrow {\mathbb {C}}^2 \end{aligned}$$

defined by $\phi _\ell (z) = \big ( \ell (z), f(z) \big )$, for any $z \in X$.

We have the following lemma:

Lemma 21

There is a representative X of (X, 0) in an open neighborhood U of $0 \in {\mathbb {C}}^N$ and a non-empty Zariski open set $\Omega$ in the space of non-zero linear forms of ${\mathbb {C}}^N$ to ${\mathbb {C}}$ such that, for any $\ell \in \Omega$ and for any stratum $S_{\alpha }$ which is disjoint from $f^{-1}(0)$, the analytic morphism $\phi _\ell : X \rightarrow {\mathbb {C}}^2$ satisfies:

• (i) The critical locus of the restriction of $\phi _\ell$ to $S_{\alpha }$ is either empty or a smooth reduced complex curve, whose closure in X is denoted by $\Gamma _{\alpha }$.

• (ii) The image $(\Delta _{\alpha },0)$ of $(\Gamma _{\alpha },0)$ by $\phi _\ell$ is the germ of a complex curve.

Proof

Let us choose an open neighborhood U of $0 \in {\mathbb {C}}^N$ such that the intersection $U \cap S_{\alpha }$ is not empty for finitely many indices ${\alpha }$. Furthermore, we may assume that the closure $\overline{S}_{\alpha }$ in U is defined by an ideal $I(\overline{S}_{\alpha })$ generated by complex analytic functions $g_1, \ldots , g_m$ defined on U, that is, $I(\overline{S}_{\alpha }) = (g_1, \ldots , g_m)$.

Now consider a linear form $\ell = a_1 x_1 + \cdots + a_N x_N$ and a stratum $S_{\alpha }$ that is not contained in $f^{-1}(0)$ and such that $0 \in \overline{S}_{\alpha }$. Let $C_{\ell ,{\alpha }}$ be the critical set of the restriction of $\phi _\ell$ to $S_{\alpha }$. Consider the matrix:

$$\begin{aligned} J_{\alpha }= \begin{pmatrix} \partial g_1 / \partial x_1 &{} \quad \dots &{} \quad \partial g_1 / \partial x_N \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ \partial g_m / \partial x_1 &{} \quad \dots &{} \quad \partial g_m / \partial x_N \\ \end{pmatrix}. \end{aligned}$$

A point z of $S_{\alpha }$ is a point where the rank of $J_{\alpha }$ at z is $\rho := \displaystyle \max \nolimits _{z \in \overline{S}_{\alpha }} \ \text {rank} \big ( J_{\alpha }(z) \big )$, since it is a non-singular point of $\overline{S}_{\alpha }$. A point of $C_{\ell ,{\alpha }}$ is a point of $S_{\alpha }$ where the matrix:

$$\begin{aligned} J_{\phi ,{\alpha }} = \begin{pmatrix} \partial g_1 / \partial x_1 &{}\quad \dots &{}\quad \partial g_1 / \partial x_N \\ \vdots &{}\quad \ddots &{}\quad \vdots \\ \partial g_m / \partial x_1 &{}\quad \dots &{}\quad \partial g_m / \partial x_N \\ \partial f / \partial x_1 &{}\quad \dots &{}\quad \partial f / \partial x_N \\ a_1 &{}\quad \dots &{}\quad a_N \\ \end{pmatrix} \end{aligned}$$

has rank at most $\rho +1$. So the determinants of the $(\rho +2)$-minors:

$$\begin{aligned} \begin{pmatrix} \partial g_{i_1} / \partial x_{j_1} &{}\quad \dots &{}\quad \partial g_{i_1} / \partial x_{j_{\rho +2}} \\ \vdots &{}\quad \ddots &{}\quad \vdots \\ \partial g_{i_\rho } / \partial x_{j_1} &{}\quad \dots &{}\quad \partial g_{i_\rho } / \partial x_{j_{\rho +2}} \\ \partial f / \partial x_{j_1} &{}\quad \dots &{}\quad \partial f / \partial x_{i_{\rho +2}} \\ a_{i_1} &{}\quad \dots &{}\quad a_{i_{\rho +2}} \\ \end{pmatrix} \end{aligned}$$

are zero, that is:

$$\begin{aligned} \sum _{k=1}^{\rho +2} (-1)^{k+1} \cdot a_{i_k} \cdot \text {det} \begin{pmatrix} \partial g_{i_1} / \partial x_{j_1} &{}\quad \dots &{}\quad \partial g_{i_1} / \partial x_{j_{k-1}} &{}\quad \partial g_{i_1} / \partial x_{j_{k+1}} &{}\quad \dots &{}\quad \partial g_{i_1} / \partial x_{j_{\rho +2}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \partial g_{i_\rho } / \partial x_{j_1} &{}\quad \dots &{}\quad \partial g_{i_\rho } / \partial x_{j_{k-1}} &{}\quad \partial g_{i_\rho } / \partial x_{j_{k+1}} &{}\quad \dots &{}\quad \partial g_{i_\rho } / \partial x_{j_{\rho +2}} \\ \partial f / \partial x_{j_1} &{}\quad \dots &{}\quad \partial f / \partial x_{j_{k-1}} &{}\quad \partial f / \partial x_{j_{k+1}} &{}\quad \dots &{}\quad \partial f / \partial x_{j_{\rho +2}} \\ \end{pmatrix} =0. \end{aligned}$$

An analytic version of a classical theorem of Bertini (see [1] and [2]) states that if $h_1, \ldots , h_r$ are holomorphic functions defined on a complex analytic space Y and if the complex numbers $\lambda _i$ are sufficiently generic, for $i=1, \ldots , r$, then the singular locus of the subvariety $\left\{ \sum _{i=1}^r \lambda _i h_i =0 \right\}$ is contained in the union of the singular set of Y and the set:

$$\begin{aligned} \{h_1 = \cdots = h_r =0 \}. \end{aligned}$$

So it follows from the analytic theorem of Bertini that there exists a non-empty Zariski open set $\Omega _{\alpha }$ in the space of non-zero linear forms from ${\mathbb {C}}^N$ to ${\mathbb {C}}$ such that for any $\ell \in \Omega _{\alpha }$ one has that the singular points $\Sigma _{C_{\ell ,{\alpha }}}$ of $C_{\ell ,{\alpha }}$ are contained in the union of the set of the points where the determinants above are zero and of the singular locus of $S_{\alpha }$. That is:

$$\begin{aligned} \Sigma _{C_{\ell ,{\alpha }}} \subset \left( \text {Crit} \ \left( f_{| \overline{S}_{\alpha }}\right) \cup \Sigma _{\overline{S}_{\alpha }} \right) \cap S_{\alpha }. \end{aligned}$$

Since this intersection is contained in $f^{-1}(0) \cap S_{\alpha }$, which is empty, it follows that $C_{\ell ,{\alpha }}$ is either smooth or empty.

Now, since $\phi _\ell ^{-1}(0,0) \cap (\Gamma _{\alpha },0) \subset \{0\}$, it follows from the geometric version of the Weierstrass preparation theorem given in [8] that the restriction of $\phi _\ell$ to the germ $(\Gamma _{\alpha },0)$ is finite. So the finite mapping theorem (Theorem 20) implies that the image $\Delta _{\alpha }$ of the analytic set $\Gamma _{\alpha }$ by $\phi _\ell$ is a complex curve.

Finally, notice that there is a finite number of indices ${\alpha }\in A$ such that $S_{\alpha }$ is not contained in $f^{-1}(0)$ and such that 0 is contained in $\overline{S}_{\alpha }$. Let $A_0$ be the finite subset of A formed by such indices. So the set:

$$\begin{aligned} \Omega := \bigcap _{{\alpha }\in A_0} \Omega _{\alpha }\end{aligned}$$

is the desired non-empty Zariski open set in the space of non-zero linear forms from ${\mathbb {C}}^N$ to ${\mathbb {C}}$. $\square$

For any $\ell \in \Omega$ we say that the germ of curve at 0 given by:

$$\begin{aligned} \Gamma _\ell := \bigcup _{{\alpha }\in A} \Gamma _{\alpha }\end{aligned}$$

is the polar curve of f relatively to $\ell$ at 0 and that the germ of curve at 0 given by:

$$\begin{aligned} \Delta _\ell := \bigcup _{{\alpha }\in A} \Delta _{\alpha }\end{aligned}$$

is the polar discriminant of f relatively to $\ell$ at 0.

From now on, we fix a linear form $\ell \in \Omega$ and we set $\phi := \phi _\ell$, $\Gamma := \Gamma _\ell$ and $\Delta := \Delta _\ell$.

Notice that there exists an open neighborhood U of 0 in ${\mathbb {C}}^2$ and a representative X of (X, 0) such that the map $\phi = (\ell , f): X \rightarrow U$ is stratified and such that it induces a stratified submersion $X {\setminus } \phi ^{-1}(\Delta ) \rightarrow \phi (X) {\setminus } \Delta$.

From now on we will assume that f has an isolated singularity at 0 relatively to the stratification ${\mathcal {S}}$. So by an analytic version of Corollary 2.8 of [16], there exist ${\epsilon }$ and $\eta _2$ small enough positive real numbers with $0< \eta _2 \ll {\epsilon }\ll 1$ such that, for any $t \in {\mathbb {D}}_{\eta _2}$, the sphere ${\mathbb {S}}_{\epsilon }$ of radius ${\epsilon }$ around 0 intersects $f^{-1}(t) \cap S_{\alpha }$ transversally, for any ${\alpha }\in A$.

We can also choose the linear form $\ell$ in such a way that there exists $\eta _1$ sufficiently small, with $0< \eta _2 \ll \eta _1 \ll {\epsilon }\ll 1$, such that $\phi ^{-1}(s,t) \cap S_{\alpha }= \ell ^{-1}(s) \cap f^{-1}(t) \cap S_{\alpha }$ intersects ${\mathbb {S}}_{\epsilon }$ transversally, for any $(s,t) \in {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2}$, where ${\mathbb {D}}_{\eta _1}$ and ${\mathbb {D}}_{\eta _2}$ are the closed disks in ${\mathbb {C}}$ centered at 0 and with radii $\eta _1$ and $\eta _2$, respectively.

So we have:

Proposition 22

The map $\phi =(\ell ,f)$ induces a stratified submersion:

$$\begin{aligned} \phi _|: {\mathbb {B}}_{\epsilon }\cap X \cap \phi ^{-1}({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta ) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta . \end{aligned}$$

In particular, the first isotopy lemma of Thom–Mather (Lemma 19) gives:

Corollary 23

The restriction $\phi _|$ above is a topological locally trivial fibration.

Therefore the curve $\Delta$ plays the role of a local topological discriminant for the stratified map $\phi$.

For any t in the disk ${\mathbb {D}}_{\eta _2}$ set:

$$\begin{aligned} D_t := {\mathbb {D}}_{\eta _1} \times \{t\}. \end{aligned}$$

If $t \ne 0$, the Milnor fiber $f^{-1}(t) \cap {\mathbb {B}}_{\epsilon }$ of f is homeomorphic to $\phi ^{-1}(D_t) \cap {\mathbb {B}}_{\epsilon }$ (see Theorem 2.3.1 of [14]). So, in order to simplify notation, we reset:

$$\begin{aligned} X_t := \phi ^{-1}(D_t) \cap {\mathbb {B}}_{\epsilon }. \end{aligned}$$

Notice that with this notation, the boundary $\partial X_t$ of $X_t$ is given by the union of $\phi ^{-1}(\mathring{D}_t) \cap {\mathbb {S}}_{\epsilon }$ and $\phi ^{-1}(\partial D_t) \cap {\mathbb {B}}_{\epsilon }$.

By Lemma 10 together with Corollary 11, the (complex analytic) Whitney stratification ${\mathcal S}$ of X induces a (subanalytic) Whitney stratification $\mathcal {S}(t)$ of $X_t$. Precisely, the strata of such stratification are the following intersections, for ${\alpha }\in A$:

• (i) $\mathcal {S}_{\alpha }\cap (X_t {\setminus } \partial X_t)$

• (ii) $\mathcal {S}_{\alpha }\cap \phi ^{-1}(\mathring{D}_t) \cap {\mathbb {S}}_{\epsilon }$

• (iii) $\mathcal {S}_{\alpha }\cap \phi ^{-1}(\partial D_t) \cap \mathring{{\mathbb {B}}}_{\epsilon }$

• (iv) $\mathcal {S}_{\alpha }\cap \phi ^{-1}(\partial D_t) \cap {\mathbb {S}}_{\epsilon }$

Now, for any $t \in {\mathbb {D}}_{\eta _2}^*$ one has that $\phi$ induces a stratified map:

$$\begin{aligned} \ell _t: X_t \rightarrow D_t. \end{aligned}$$

By construction, the restriction of $\ell _t$ to each stratum of $X_t$ is a submersion at any point that is not in $\Gamma$. Therefore it induces a locally trivial fibration over $D_t {\backslash } (\Delta \cap D_t)$. That is, if we set:

$$\begin{aligned} \Delta \cap D_t = \{y_1(t), \ldots , y_k(t) \} \end{aligned}$$

then the restriction of $\ell _t$ given by:

$$\begin{aligned} \varphi _t: X_t {\setminus } \ell _t^{-1} \big ( \{y_1(t), \ldots , y_k(t) \} \big ) \rightarrow D_t {\setminus } \{y_1(t), \ldots , y_k(t) \} \end{aligned}$$

is a stratified submersion (see Definition 15) and a locally trivial fibration, by Thom–Mather first isotopy lemma (Lemma 19).

We notice that at this moment the parameter t is fixed, and the points $y_j(t)$ are numbered in an arbitrary way, as there is no natural way to do it. But in Sect. 4.2 and 6.2 those points will be defined in a continuous manner for t varying on a closed semi-disk ${\mathbb {D}}^+ \subset {\mathbb {D}}_{\eta _2}$ as in Fig. 1 above.

Remark 24

In the case that $\Gamma$ is empty, one has that:

$$\begin{aligned} \phi _|: {\mathbb {B}}_{\epsilon }\cap X \cap \phi ^{-1} ({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2}) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} \end{aligned}$$

is a locally trivial topological fibration, which implies a locally trivial topological fibration $\ell _t: X_t \rightarrow D_t$. Hence in this case the Milnor fiber $X_t$ is homeomorphic to the product of $D_t$ and the general fiber of $\ell _t$.

So from now on we shall assume that the polar curve $\Gamma$ is not empty.

4 The two-dimensional case

We shall prove Theorem 2 by induction on the dimension n of the analytic space X. We could start by proving the theorem for $n=1$ and then proceed by induction for $n\ge 2$, but we choose to start with the 2-dimensional case, in order to provide the reader a better intuition of the constructions.

So in this section we prove Theorem 2 when (X, 0) is a 2-dimensional reduced equidimensional germ of complex analytic space and $f{:}~ (X,0) \rightarrow ({\mathbb {C}},0)$ has an isolated singularity at 0 in the stratified sense.

One particularity of this 2-dimensional case is that the singular set $\Sigma$ of X has dimension at most one. If $\Sigma$ has dimension one, we can put it inside the polar curve $\Gamma$. More precisely, only in this section we denote by $\Gamma$ the union of the polar curve of f with $\Sigma$. We also denote by $\Delta$ the union of the polar discriminant of f with $\phi (\Sigma )$. Notice that if $\Delta$ is not empty, then it is a complex curve.

In order to make the constructions easier to understand, we will proceed in three steps. The first step will be to construct a polyhedron $P_t$ in $X_t$, which we call a vanishing polyhedron, and a simplicial map $\tilde{\xi }_t$ in $X_t$, for any $t \in {\mathbb {D}}_{\eta _2} {\backslash } \{0\}$ fixed. In the second step, we do the construction of $P_t$ and $\tilde{\xi }_t$ simultaneously, for t varying in a closed semi-disk ${\mathbb {D}}^+$ of ${\mathbb {D}}_{\eta _2}$ as in Fig. 1 above (with $0 \in \partial {\mathbb {D}}^+$). This will give a polyhedron $P^+$ in $X^+ := X \cap f^{-1}({\mathbb {D}}^+) \cap {\mathbb {B}}_{\epsilon }$. In the third step, we will construct the map $\psi _t : X_t \rightarrow X_0$, for any $t \in {\mathbb {D}}^+ {\backslash } \{0\}$. We call $\psi _t$ a collapsing map for f.

4.1 First step: constructing the vanishing polyhedron $P_t$ and the map $\tilde{\xi }_t$

Let us fix $t \in {\mathbb {D}}_{\eta _2}^*$ fixed. We recall that:

$$\begin{aligned} \Delta \cap D_t = \{y_1(t), \ldots , y_k(t) \}. \end{aligned}$$

Let $\lambda _t$ be a point in $D_t {\setminus } \{y_1(t), \ldots , y_k(t)\}$ and for each $j = 1, \ldots , k$, let $\delta (y_j(t))$ be the line segment in $D_t$ starting at $\lambda _t$ and ending at $y_j(t)$. We can choose $\lambda _t$ in such a way that any two of these line segments intersect only at $\lambda _t$.

Set:

$$\begin{aligned} Q_t := \bigcup _{j=1}^k \delta (y_j(t)) \end{aligned}$$

and

$$\begin{aligned} P_t := \ell _t^{-1}(Q_t). \end{aligned}$$

Since $\ell _t$ is finite, one can see that $P_t$ is a one-dimensional polyhedron in $X_t$ (see Sect. 2.4). And since the map $\varphi _t: X_t {\setminus } \ell _t^{-1} \big ( \{y_1(t), \ldots , y_k(t) \} \big ) \rightarrow D_t {\setminus } \{y_1(t), \ldots , y_k(t) \}$ is a stratified submersion, the interior of each 1-simplex of $P_t$ is contained in some stratum $X_t \cap \mathcal {S}_{\alpha }$ of $X_t$, so $P_t$ is adapted to the stratification $\mathcal {S}$ (see Definition 18).

We shall call $P_t$ a vanishing polyhedron for f.

Recall that in this 2-dimensional case, by definition, the curve $\Gamma$ contains the singular set $\Sigma$ of X, so $P_t$ contains the intersection $\Sigma \cap X_t$. Hence $X_t {\backslash } P_t$ is a smooth manifold.

Lemma 25

There exists a subanalytic Whitney stratification $\mathcal {Z}= (Z_{\beta })_{{\beta }\in B}$ of $D_t$ with the property (w), and a continuous vector field $\vec {\nu }_t$ on $D_t$ such that:

• 1. It is non-zero on $D_t {\setminus } Q_t$;

• 2. It vanishes on $Q_t$;

• 3. It is transversal to $\partial D_t$ and points inwards;

• 4. It restricts to a rugose stratified vector field on the interior $\mathring{D}_t$ of $D_t$ (relatively to the stratification $\mathcal {Z}$);

• 5. The associated flow $q_t: [0, \infty ) \ \times \ (D_t {\backslash } Q_t) \rightarrow D_t {\backslash } Q_t$ defines a map:

  $$\begin{aligned} \begin{aligned}\xi _t \ :\, \partial D_t&\longrightarrow {Q_t} \\ u&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } q_t(\tau ,u), \end{aligned} \end{aligned}$$

  such that $\xi _t$ is continuous, simplicial (as defined in Sect. 2.4) and surjective.

Proof

Let $d_t: D_t \rightarrow {\mathbb {R}}$ be the function given by the distance to the set $Q_t$, that is $d_t(x) := d(x, Q_t)$. Consider the small closed neighborhood of $Q_t$ in $D_t$ given by:

$$\begin{aligned} \mathcal {R}_t:= (d_t)^{-1}\big ( [0,r] \big ) \end{aligned}$$

for some small $r>0$. Since both $Q_t$ and $d_t$ are subanalytic, it follows that $\mathcal {R}_t$ is subanalytic.

Since we have k-many points $y_1(t), \ldots , y_k(t)$, it follows that there exist exactly k-many points in $\partial \mathcal {R}_t$ whose distance to the point $\lambda _t$ is r. Let us call them $p_1, \ldots , p_k$. For each $p_i$, let $N_i$ be the closed line-segment in $\mathcal {R}_t$ that joins the points $p_i$ and $\lambda _t$, and let $\tilde{N}_i$ be the closed line-segment in $D_t$ that contains both the points $\lambda _t$ and $p_i$, and also some point of the boundary of $D_t$. See Fig. 2.

Fig. 2

The line-segments $\tilde{N}_i$

We consider the Whitney stratification $\mathcal {Z}$ of $D_t$ with the property (w) that has the smallest number of strata, such that $Q_t$ and the line-segments $\tilde{N}_i$ are union of strata.

We will prove the lemma in two steps.

First step: We will endow $\mathcal {R}_t$ with a vector field $\vec {v}_1$, as follows:

For each point $y_j(t)$, let $L_j$ be the line-segment in $\mathcal {R}_t$ that is ortogonal to the line-segment $\delta (y_j(t))$ at $y_j(t)$. Notice that the line-segments $L_1, \ldots , L_k$ together with the line-segments $N_1, \ldots , N_k$ give a decomposition of $\mathcal {R}_t$ in 2k-many polygons $R_1, \ldots , R_{2k}$ that contain the point $\lambda _t$ and k-many semi-disks $M_1, \ldots , M_k$ such that $y_j(t) \in M_j$, for $j=1, \ldots , k$. See Fig. 3.

Fig. 3

Decomposition of $\mathcal {R}_t$

Now we endow each polygon $R_m$, for $m=1, \ldots , 2k$, with a vector field $\omega _m$ as follows:

Let $h_1$ be the combination of the rotation and the translation in ${\mathbb {R}}^2$ that takes $\lambda _t$ to the origin and that takes the line-segment $\delta (y_j(t))$ contained in $R_m$ to a line-segment $[x_0,0]$ in the first coordinate-axis. Notice that $h_1$ takes the line-segment $N_i$ contained in $R_m$ to a line-segment contained in the real line of equation $y= {\alpha }x$, for some ${\alpha }\in {\mathbb {R}}$. See Fig. 4.

Also, let $h_2$ be the diffeomorphism from the rectangle $[x_0,0] \times [0, r]$ onto $h_1(R_m)$ given by:

$$\begin{aligned} h_2(x,y) := \left( x +{\alpha }\frac{(x_0-x)}{x_0} y, y \right) . \end{aligned}$$

Setting $h:= h_1^{-1} \circ h_2$, we have that $h: [x_0,0] \times [0, r] \rightarrow R_m$ is a diffeomorphism such that:

• h takes $[x_0,0] \times \{0\}$ onto $\delta (y_j(t))$;

• h takes $\{x_0\} \times [0, r]$ onto $L_j$;

• h takes $\{0\} \times [0, r]$ onto $N_i$.

Fig. 4

The diffeomorphism h

Now let $\rho : [x_0,0] \times [0, r] \rightarrow {\mathbb {R}}$ be the function given by the square of the distance to $[x_0,0] \times \{0\}$, that is, $\rho (x,y) := y^2$, and let $\vec {\omega }$ be the vector field in $[x_0,0] \times [0, r]$ given by the opposite of the gradient vector field associated to $\rho$. Notice that $\vec {\omega }$ is continuous, smooth outside $[x_0,0] \times \{0\}$, non-zero outside $[x_0,0] \times \{0\}$ and zero on $[x_0,0] \times \{0\}$.

The differential of h takes $\vec {\omega }$ to a vector field $\vec {\omega }_m$ on $R_m$ that is smooth and non-zero outside $\delta (y_j(t))$ and zero on $\delta (y_j(t))$.

Moreover, the restriction of $\vec {\omega }_m$ to the line-segment $L_j$ coincides with the vector field on $L_j$ given by the opposite of the gradient of the square of the function given by the distance to the point $y_j(t)$. Also, the restriction of $\vec {\omega }_m$ to the line-segment $N_i$ coincides with the vector field on $N_i$ given by the opposite of the gradient of the square of the function given by the distance to the point $\lambda _t$. See Fig. 5.

Fig. 5

The vector field $\vec {\omega }_m$

On the other hand, we endow each semi-disk $M_j$, for $j=1, \ldots , k$, with the vector field $\vec w_j$ given by the opposite of the gradient of the square of the function given by the distance to the point $y_j(t)$.

So putting all the polygons $R_m$ and all the semi-disks $M_j$ together, each of them endowed with the corresponding vector field $\vec {\omega }_m$ or $\vec w_j$, we get a continuous vector field $\vec {v}_1$ on $\mathcal {R}_t$ such that:

• $\vec {v}_1$ is smooth on each stratum of the stratification of $\mathcal {R}_t$ induced by $\mathcal {Z}$;

• $\vec {v}_1$ is non-zero outside $Q_t$;

• $\vec {v}_1$ is zero on $Q_t$;

• $\vec {v}_1$ never points in the direction of the gradient of the function given by the square of the distance to the point $\lambda _t$.

Let us show that:

• $\vec {v}_1$ is a rugose vector field.

Set $W_1 := D_t {\setminus } (Q_t \cup \tilde{N}_1 \cup \dots \cup \tilde{N}_k)$ and set $W_2 := Q_t \cup \tilde{N}_1 \cup \dots \cup \tilde{N}_k$.

If $z \in W_2 \cap \mathcal {R}_t$ is not $\lambda _t$, let U be a neighborhood of z in $D_t$ that is contained in the union of two polygons $R_m$ and $R_{m^{\prime }}$. Then for any $z_1 \in W_1 \cap U$ and $z_2 \in W_2 \cap U$, we have that the line-segment $\overline{z_1 z_2}$ that joints $z_1$ and $z_2$ is contained either in $R_m$ or in $R_{m^{\prime }}$. Let us suppose that it is contained in $R_m$. Then we have:

$$\begin{aligned} \Vert \vec {\omega }_m(z_1) - \vec {\omega }_m(z_2) \Vert \le K_m \Vert z_1 - z_2 \Vert , \end{aligned}$$

where $K_m$ is the Lipschitz constant of the Lipshitz vector field $\vec {\omega }_m$.

If $z = \lambda _t$, let U be the open ball around $\lambda _t$ of radius r. Given $z_1 \in W_1 \cap U$ and $z_2 \in W_2 \cap U$, consider the line-segment $\overline{z_1 z_2}$, starting from $z_1$ and going to $z_2$. It intersects the set Q at points $q_1, \ldots , q_s$, and it intersects the line-segments $N_i$ at the points $t_1, \ldots , t_{s^{\prime }}$, in this order. That is:

$$\begin{aligned} \overline{z_1 z_2} = \overline{z_1 q_1 t_1 q_2 t_2 \dots q_{s-1} t_{s-1} q_s z_2} \ \ \text {or} \ \ \overline{z_1 z_2} = \overline{z_1 t_1 q_1 t_2 q_2 \dots t_s q_s z_2}. \end{aligned}$$

Let us consider the first case ($\overline{z_1 z_2} = \overline{z_1 q_1 t_1 q_2 t_2 \dots q_{s-1} t_{s-1} q_s z_2}$); the second case is analogous.

Let re-order the polygons $R_m$ in such a way that, for each $i=1, \ldots , s$, we have:

• $R_0$ is the polygon that contains $z_1$;

• $R_{2i-1}$ is the polygon that contains the line-segment $\overline{q_i t_i}$;

• $R_{2i}$ is the polygon that contains the line-segment $\overline{t_i q_{i+1}}$.

Setting $K := \max _{m= 1, \ldots , 2k} \{K_m\}$, we have:

$$\begin{aligned}&\Vert \vec {\omega }_0(z_1) - \vec {\omega }_s(z_2) \Vert \\&\quad \le \Vert \vec {\omega }_0(z_1) - \vec {\omega }_0(q_1) \Vert + \Vert \vec {\omega }_1(q_1) - \vec {\omega }_1(t_1) \Vert + \Vert \vec {\omega }_2(t_1) - \vec {\omega }_2(q_2) \Vert + \Vert \vec {\omega }_3(q_2) - \vec {\omega }_3(t_2) \Vert + \cdots \\&\quad \dots + \Vert \vec {\omega }_{2s-2}(t_{s-1}) - \vec {\omega }_{2s-2}(q_s) \Vert + \Vert \vec {\omega }_{2s-1}(q_s) - \vec {\omega }_{2s-1}(z_2) \Vert \\&\quad \le K_0 \Vert z_1 - q_1\Vert + K_1 \Vert q_1 - t_1\Vert + K_2 \Vert t_1 - q_2\Vert + \cdots + K_{2s-2} \Vert t_{s-1} - q_s\Vert + K_{2s-1} \Vert q_s - z_2\Vert \\&\quad \le 2s K \Vert z_1 - z_2 \Vert \le 2k K \Vert z_1 - z_2 \Vert . \end{aligned}$$

Second step: Constructing the vector field $\vec {\nu }_t$ on $D_t$.

Let $r^{\prime }$ be a small real number with $0<r^{\prime }<r$ and with $r-r^{\prime } \ll 1$, and set $\mathcal {R}_t^{\prime }:= (d_t)^{-1}\big ( [0,r^{\prime }] \big )$, so $\mathcal {R}_t^{\prime } \subset \mathcal {R}_t$. We endow $D_t {\backslash } int(\mathcal {R}_t^{\prime })$ with the vector field $\vec {v}_2$ given by the opposite of the gradient vector field of the function on $D_t {\backslash } int(\mathcal {R}_t^{\prime })$ given by the square of the distance to the point $\lambda _t$.

Since the vector fields $\vec {v}_1$ and $\vec {v}_2$ never have opposite directions, the vector field $\vec {\nu }_t$ is obtained by gluing the vector fields $\vec {v}_1$ and $\vec {v}_2$, using a partition of unity. That is, we consider a pair $(\rho _1, \rho _2)$ of continuous functions from the compact disk $D_t$ to the closed unit interval [0, 1] such that:

• for every point $p \in D_t$ one has that $\rho _1(p) + \rho _2(p) = 1$,

• the support of $\rho _1$ is contained in $int(\mathcal {R}_t)$,

• the support of $\rho _2$ is contained in $D_t {\backslash } \mathcal {R}_t^{\prime }$.

Hence for any $p \in \mathcal {R}_t^{\prime }$ we have that $\big ( \rho _1(p), \rho _2(p) \big ) = (1,0)$ and for any $p \in D_t {\backslash } int(\mathcal {R}_t)$ we have that $\big ( \rho _1(p), \rho _2(p) \big ) = (0,1)$. So we set $\vec {\nu }_t := \rho _1 \vec {v}_1 + \rho _2 \vec {v}_2$ .

Clearly, $\vec {\nu }_t$ is a continuous vector field on $D_t$ that is non-zero on $D_t {\setminus } Q_t$, zero on $Q_t$ and transversal to $\partial D_t$, pointing inwards, and that restricts to a rugose stratified vector field on the interior of $D_t$ (relatively to the stratification $\mathcal {Z}$ of $D_t$). Moreover, each orbit associated to $\vec {\nu }_t$ has a limit point in $Q_t$.

So the flow $q_t: [0, \infty ) \ \times \ (D_t {\backslash } Q_t) \rightarrow D_t {\backslash } Q_t$ associated to $\vec {\nu }_t$ defines a continuous, simplicial and surjective map:

$$\begin{aligned} \begin{aligned} \xi _t \ : \, \partial D_t&\longrightarrow Q_t \\ u&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } q_t(\tau ,u). \end{aligned} \end{aligned}$$

$\square$

Remark 26

Lemma 25 is still true if the set $Q_t$ is taken as the union of simple paths that intersect only at a point $\lambda _t$, instead of considering line-segments. It is enough to consider a suitable homeomorphism of the disk $D_t$ onto itself which is a diffeomorphism outside the point $\lambda _t$ and which sends the union of those paths to a union of line-segments.

Now recall the proper map $\ell _t: X_t \rightarrow D_t$ and recall that $X_t$ has a subanalytic Whitney stratification $\mathcal {S}(t)$ with the property (w) induced by the Whitney stratification $\mathcal {S}$ of X. By Remark 12, we can consider a refinement $\mathcal {S}^{\prime }(t)$ of $\mathcal {S}(t)$ and a refinement $\mathcal {Z^{\prime }}= (Z_{\beta }^{\prime })_{{\beta }\in B^{\prime }}$ of $\mathcal {Z}$ such that:

• (i) $\ell _t^{-1}(Z_{\beta }^{\prime })$ is a union of strata of $\mathcal {S}^{\prime }(t)$;

• (ii) the restriction $\ell _t{_|}: \ell _t^{-1}(Z_{\beta }^{\prime }) \rightarrow Z_{\beta }^{\prime }$ is transversal to $\mathcal {S}^{\prime }(t) \cap f^{-1}(Z_{\beta }^{\prime })$.

One has:

Proposition 27

We can choose a lifting of the vector field $\vec {\nu }_t$ of Lemma 25 to a continuous vector field $\vec \vartheta _t$ on $X_t$ so that:

• 1. It is non-zero outside $P_t$ and zero on $P_t$;

• 2. It is transversal to $\partial X_t$ and points inwards;

• 3. It restricts to a stratified rugose vector field on the interior of $X_t$ (relatively to the stratification $\mathcal {S}^{\prime }(t)$);

• 4. The flow $\tilde{q}_t: [0, \infty ) \ \times \ (X_t {\backslash } P_t) \rightarrow X_t$ associated to $\vec \vartheta _t$ defines a map:

  $$\begin{aligned} \begin{aligned} \tilde{\xi }_t \ : \, \partial X_t&\longrightarrow P_t \\ z&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } \tilde{q}_t(\tau ,z) \end{aligned} \end{aligned}$$

  such that $\tilde{\xi }_t$ is continuous, simplicial and surjective;

• 5. The fiber $X_t$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_t$.

Proof

Recall from Proposition 22 that the restriction $\ell _t$ of the linear form $\ell$ to the Milnor fiber $X_t$ induces a stratified submersion:

$$\begin{aligned} \varphi _t: X_t {\setminus } \ell _t^{-1} \big ( \{y_1(t), \ldots , y_k(t) \} \big ) \rightarrow D_t {\setminus } \{y_1(t), \ldots , y_k(t) \}. \end{aligned}$$

So, by Proposition 17, we can lift the vector field $\vec {\nu }_t$ to a continuous vector field $\vec \vartheta _t$ in $X_t$ that satisfies properties (1), (2) and (3).

Let us show that we can choose $\vec \vartheta _t$ satisfying also condition (4). Fix $z \in \partial X_t$. We want to show that $\lim _{\tau \rightarrow \infty } \tilde{q}_t(\tau ,z)$ exists, that is, that there exists a point $\tilde{p} \in P_t$ such that for any open neighborhood $\tilde{U}$ of $\tilde{p}$ in $X_t$ there exists $\theta >0$ such that $\tau >\theta$ implies that $\tilde{q}_t(\tau ,z) \in \tilde{U}$.

From Lemma 25 we know that there exists $p \in Q_t$ such that $\lim _{\tau \rightarrow \infty } q_t(\tau ,\ell _t(z)) = p$, where $q_t: [0, \infty ) \ \times \ (D_t {\backslash } Q_t) \rightarrow D_t$ is the flow associated to the vector field $\vec {\nu }_t$. So for any small open neighborhood U of p in $D_t$ there exists $\theta >0$ such that $\tau >\theta$ implies that $q_t(\tau ,\ell _t(z)) \in U$. Setting $\{ \tilde{p}_1, \ldots , \tilde{p}_r \} := \ell _t^{-1}(p)$, we can consider U sufficiently small such that there are disjoint connected components $\tilde{U}_1, \ldots , \tilde{U}_r$ of $\ell _t^{-1}(U)$ such that each $\tilde{U}_j$ contains $\tilde{p}_j$.

Since $\vec \vartheta _t$ is a lifting of $\vec {\nu }_t$, we have that $q_t(\tau ,\ell _t(z)) = \ell _t(\tilde{q}_t(\tau ,z))$ for any $\tau \ge 0$. So $\tau >\theta$ implies that $\ell _t^{-1} \big ( \ell _t(\tilde{q}_t(\tau ,z)) \big ) \subset \ell _t^{-1}(U)$. Hence for some $j \in \{1, \ldots , r\}$ we have that $\tilde{q}_t(\tau ,z) \in \tilde{U}_j$. Therefore $\lim _{\tau \rightarrow \infty } \tilde{q}_t(\tau ,z) = \tilde{p}_j$. This proves (4).

Now we show that $X_t$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_t$. In fact, the integration of the vector field $\vec \vartheta _t$ on $X_t$ gives a surjective continuous map:

$$\begin{aligned} {\alpha }: [0, \infty ] \ \times \ \partial X_t \rightarrow X_t \end{aligned}$$

that restricts to a homeomorphism:

$$\begin{aligned} {\alpha }_|: [0, \infty ) \ \times \ \partial X_t \rightarrow X_t {\backslash } P_t. \end{aligned}$$

Since the restriction ${\alpha }_{\infty }: \{\infty \} \times \partial X_t \rightarrow P_t$ is equal to $\tilde{\xi }_t$, which is surjective, it follows that the induced map:

$$\begin{aligned}{}[{\alpha }_\infty ]: \big ( (\{\infty \} \times \partial X_t ) \big / \sim \big ) \rightarrow P_t \end{aligned}$$

is a homeomorphism, where $\sim$ is the equivalence relation given by identifying $(\infty , z) \sim (\infty ,z^{\prime })$ if ${\alpha }_\infty (z) = {\alpha }_\infty (z^{\prime })$. Hence the map:

$$\begin{aligned}{}[{\alpha }]: \big ( ([0,\infty ] \times \partial X_t) \big / \sim \big ) \rightarrow X_t \end{aligned}$$

induced by ${\alpha }$ defines a homeomorphism between $X_t$ and the mapping cylinder of $\tilde{\xi }_t$. This proves (5). $\square$

4.2 Second step: constructing $P^+$ and $\tilde{\xi }_+$

Recall from Lemma 21 that the polar curve $\Gamma$ is not contained in $f^{-1}(0)$. Hence $\Delta$ is not contained in ${\mathbb {D}}_{\eta _1} \times \{0\}$, and so the natural projection $\pi : {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} \rightarrow {\mathbb {D}}_{\eta _2}$ restricted to $\Delta$ induces a ramified covering:

$$\begin{aligned} {\pi }_|: \Delta \rightarrow {\mathbb {D}}_{\eta _2} \end{aligned}$$

of degree k, whose ramification locus is $\{0\} \subset \Delta$.

So the intersection of the polar discriminant $\Delta$ with the product ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ give semi-disks $Y_1, \ldots , Y_k$ in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ such that $Y_j$ projects differentially onto ${\mathbb {D}}^+$ outside $0 \in Y_j$, for each $j=1, \ldots , k$. The set $\Lambda := \{0\} \times {\mathbb {D}}^+$ is also a semi-disk in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$, which can be supposed to intersect $Y_j$ only at $0 \in {\mathbb {C}}^2$, for any $j=1, \ldots , k$.

We can choose the simple paths $\delta (y_1(t)), \ldots , \delta (y_k(t))$ for each $t \in {\mathbb {D}}^+$ in such a way that $\delta (y_j(t))$ depends continuously on the parameter $t \in {\mathbb {D}}^+$, for each $j=1, \ldots , k$; and it forms a 3-dimensional triangle $\mathcal {T}_j$ in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ bounded by the semi-disks $Y_j$ and $\Lambda$ and by the union of paths $\bigcup _{t \in {\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}} \delta _j(y_j(t))$.

The Fig. 6 below represents the 2-dimensional triangles $T_j := \mathcal {T}_j \cap ({\mathbb {D}}_{\eta _1} \times \gamma )$, for $j=1, \ldots , k$, where $\gamma$ is a simple path in ${\mathbb {D}}^+$ going joining some $t_0 \in {\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}$ to $0 \in {\mathbb {C}}^2$. It helps the reader to understand the construction of each $\mathcal {T}_j$.

Fig. 6

The triangles $T_j$

Setting $Q^+ := \bigcup \nolimits _{j=1}^k \mathcal {T}_j$, we define:

$$\begin{aligned} P^+ := \phi ^{-1}(Q^+), \end{aligned}$$

which is contained in $X^+ := X \cap f^{-1}({\mathbb {D}}^+) \cap {\mathbb {B}}_{\epsilon }$. It is a polyhedron adapted to the stratification $\mathcal {S}^+$ of $X^+$ induced by $\mathcal {S}$, and the intersection $P^+ \cap X_t$ is a vanishing polyhedron $P_t$ as in Sect. 4.1, for any $t \in {\mathbb {D}}^+ {\backslash } \{0\}$. Moreover, $P^+ \cap X_0 = \{0\}$.

Notice that the complex Whitney stratification $\mathcal {S}$ of X in fact induces a subanalytic Whitney stratification $\mathcal {S}^+$ of $X^+$ with the property (w). This is because $\mathcal {S}$ has the property (w), by Lemma 8, and hence one can use Lemma 10.

Now, analogously to Lemma 25, we can construct a Whitney stratification $\mathcal {Z}^+$ of ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ with the property (w), and an integrable vector field $\vec \nu _+$ in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ that deformation retracts ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ onto $Q^+$. That is, we can consider a continuous vector field $\vec \nu _+$ in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ such that:

• It is non-zero outside $Q^+$ and it is zero on $Q^+$;

• It is transversal to $\partial {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$;

• It restricts to a rugose stratified vector field on the interior of ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ (with respect to $\mathcal {Z}^+$);

• The projection of $\vec \nu _+$ onto ${\mathbb {D}}^+$ is zero;

• The flow $q_+: [0, \infty ) \ \times \ \big ( ({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+) {\backslash } Q^+ \big ) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ associated to $\vec \nu _+$ defines a map:

  $$\begin{aligned} \begin{aligned} \xi _+ \ : \, \partial {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+&\longrightarrow Q^+ \\ z&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } q_+(\tau ,z) \end{aligned} \end{aligned}$$

  that is continuous, simplicial and surjective.

As we did above, we can use Remark 12 to obtain a refinement $(\mathcal {S}^+)^{\prime }$ of $\mathcal {S}^+$ and a refinement $(\mathcal {Z}^+)^{\prime }= (Z_{\beta }^{+^{\prime }})_{{\beta }\in B^{\prime }}$ of $\mathcal {Z}^+$ such that:

• (i) $\phi ^{-1}(Z_{\beta }^{+^{\prime }})$ is a union of strata of $(\mathcal {S}^+)^{\prime }$;

• (ii) the restriction $\phi _|: \phi ^{-1}(Z_{\beta }^{+^{\prime }}) \rightarrow (Z_{\beta }^{+^{\prime }})$ is transversal to $(\mathcal {S}^+)^{\prime } \cap f^{-1}(Z_{\beta }^{+^{\prime }})$.

So by Proposition 16 we have:

Proposition 28

The vector field $\vec \nu _+$ can be lifted to an integrable vector field $\vec \vartheta _+$ in $X^+$ such that:

• (i) For any $t \in {\mathbb {D}}^+ {\backslash } \{0\}$, the restriction of $\vec \vartheta _+$ to $X_t$ gives a vector field $\vec \vartheta _t$ as in Proposition 27, relatively to the polyhedron $P_t = P^+ \cap X_t$.

• (ii) The vector field $\vec \vartheta _+$ is non-zero on $X^+ {\backslash } P^+$, zero on $P^+$, transversal to $\partial X^+$, pointing inwards, and it restricts to a rugose stratified vector field on the interior of $X^+$ (relatively to the refinement $(\mathcal {S}^+)^{\prime }$).

Analogously to the proof of (6) of Proposition 27, one can show that the flow associated to the vector field $\vec \vartheta _+$ defines a map $\tilde{\xi }_+$ with the desired properties.

This proves (i) and (ii) of Theorem 2 in the 2-dimensional case.

4.3 Third step: constructing the collapsing map $\psi _t$

First, let us recall that $X_t := {\mathbb {B}}_{\epsilon }\cap \phi ^{-1}({\mathbb {D}}_{\eta _1} \times \{t\})$ and that the map:

$$\begin{aligned} \phi = (\ell ,f): X \rightarrow {\mathbb {C}}^2 \end{aligned}$$

induces a stratified submersion $\phi _|: {\mathbb {B}}_{\epsilon }\cap X \cap \phi ^{-1}({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta ) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta$ (see Proposition 22).

Let $\gamma$ be a simple path in ${\mathbb {D}}_{\eta _2}$ joining 0 and some $t_0 \in \partial {\mathbb {D}}_{\eta _2}$, such that $\gamma$ is transverse to $\partial {\mathbb {D}}_{\eta _2}$. See Fig. 7. We want to describe the collapsing of f along $\gamma$, that is, how $X_t$ degenerates to $X_0$ as $t \in \gamma$ goes to 0.

Fig. 7

The path $\gamma$

Recall the sets $Q^+$ and $P^+$ defined above, and set $Q_\gamma := Q^+ \cap ({\mathbb {D}}_{\eta _1} \times \gamma )$. It is a union of triangles $T_1, \ldots , T_k$, as in Fig. 6. Also set $P_\gamma := P^+ \cap f^{-1}(\gamma )$.

The vector field $\vec \nu _+$ constructed above restricts to a continuous stratified vector field $\vec \nu _\gamma$ in ${\mathbb {D}}_{\eta _1} \times \gamma$ such that:

• It is non-zero outside $Q_\gamma$ and it is zero on $Q_\gamma$;

• It is transversal to $\partial {\mathbb {D}}_{\eta _1} \times \gamma$;

• It restricts to a rugose stratified vector field on the interior of ${\mathbb {D}}_{\eta _1} \times \gamma$ (relatively to the Whitney stratification $\mathcal {Z}^+ \cap ({\mathbb {D}}_{\eta _1} \times \gamma )$ (with the property (w)) induced by $\mathcal {Z}^+$);

• The projection of $\vec \nu _\gamma$ onto $\gamma$ is zero;

• The flow $q_\gamma : [0, \infty ) \ \times \ \big ( ({\mathbb {D}}_{\eta _1} \times \gamma ) {\backslash } Q_\gamma \big ) \rightarrow {\mathbb {D}}_{\eta _1} \times \gamma$ associated to $\vec \nu _\gamma$ defines a map:

  $$\begin{aligned} \begin{aligned} \xi _\gamma \ : \, \partial {\mathbb {D}}_{\eta _1} \times \gamma&\longrightarrow Q_\gamma \\ z&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } q_\gamma (\tau ,z) \end{aligned} \end{aligned}$$

  that is continuous, simplicial and surjective.

Now, for any real number $A>0$ set:

$$\begin{aligned} V_A(Q_\gamma ):= ({\mathbb {D}}_{\eta _1} \times \gamma ) {\backslash } q_\gamma \big ( [0,A) \times \partial {\mathbb {D}}_{\eta _1} \times \gamma \big ), \end{aligned}$$

which is a closed neighborhood of $Q_\gamma$ in ${\mathbb {D}}_{\eta _1} \times \gamma$. This gives a system of closed neighborhoods of $Q_\gamma$ in ${\mathbb {D}}_{\eta _1} \times \gamma$, such that:

• (i) The boundary $\partial V_A(Q_\gamma )$ of $V_A(Q_\gamma )$ is a stratified topological manifold, for any $A \ge 0$, since it is the image of $\partial {\mathbb {D}}_{\eta _1} \times \gamma$ by $\xi _\gamma$;

• (ii) $V_0(Q_\gamma ) = {\mathbb {D}}_{\eta _1} \times \gamma$;

• (iii) For any $A_1 > A_2$ one has $V_{A_1}(Q_\gamma ) \subset V_{A_2}(Q_\gamma )$;

• (iv) For any open neighborhood U of $Q_\gamma$ in ${\mathbb {D}}_{\eta _1} \times \gamma$, there exists $A_U \ge 0$ sufficiently big such that $V_{A_U}(Q_\gamma )$ is contained in U.

Notice that $\partial V_A(Q_\gamma )$ fibers over $\gamma$ with fiber a circle, by the restriction of the projection $\pi : {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} \rightarrow {\mathbb {D}}_{\eta _2}$.

Now, setting:

$$\begin{aligned} X_\gamma := X \cap f^{-1}(\gamma ) \cap {\mathbb {B}}_{\epsilon }, \end{aligned}$$

we can finally construct a stratified rugose vector field $\vec \zeta _\gamma$ in $X_\gamma {\setminus } P_\gamma$ (relatively to the stratification $\mathcal {S}^{\prime }(\gamma )$ of $X_\gamma$ induced by the stratification $(\mathcal {S}^+)^{\prime }$ of $X^+$) whose flow gives the degeneration of $X_{t_0}$ to $X_0$. We will say that $\vec \zeta _\gamma$ is a collapsing vector field that gives the degeneration of f along the path $\gamma$.

Recall that the restriction:

$$\begin{aligned} \phi _|: \phi ^{-1}\big ( ({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2}) {\backslash } Q \big ) \rightarrow ({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2}) {\backslash } Q \end{aligned}$$

is a stratified submersion (relatively to the stratification $\mathcal {S}$), and that the restriction of $\pi \circ \phi$ to $\phi ^{-1}\big ( \partial V_A(Q_\gamma ) \big ) \cap {\mathbb {B}}_{\epsilon }$ is a proper locally trivial fibration over $\gamma$.

Let $\vec \theta$ be a vector field on $\gamma$ that goes from $t_0$ to 0 in time $a>0$ and fix $A>0$. We are going to construct a smooth and integrable vector field $\vec \zeta _\gamma$ in $X_\gamma {\backslash } P_\gamma$ that lifts $\vec \theta$ outside $\{0\}$, and such that $\vec \zeta _\gamma$ is tangent to $\phi ^{-1} \big ( \partial V_A^{\prime }(Q_\gamma ) \big )$, for any$A^{\prime } \ge A$. We will construct it locally, that is, for each point $p \in X_\gamma {\backslash } P_\gamma$ we will construct a vector field $\vec \zeta _p$ in some neighborhood $U_p$ of p, and then we will glue all of them using a partition of unity associated to the covering given by the neighborhoods $U_p$ (see Lemma 41.6 of [18] for the proof of the existence of a partition of unity associated to an infinite covering, since $X_\gamma {\backslash } P_\gamma$ is not compact).

Each $\vec \zeta _p$ is constructed in the following way:

• (a) If $p \notin \phi ^{-1}\big ( V_A(Q_\gamma ) \big ) \cap {\mathbb {B}}_{\epsilon }$, there is an open neighborhood $U_p$ of p in $X_\gamma$ that does not intersect the closed set $\phi ^{-1}\big ( V_A(Q_\gamma ) \big ) \cap {\mathbb {B}}_{\epsilon }$. Then we define a smooth vector field $\vec \zeta _p$ on $U_p$ that lifts $\vec \theta$.

• (b) If $p \in \left[ \phi ^{-1}\big ( V_A(Q_\gamma ) \big ) \cap {\mathbb {B}}_{\epsilon }\right] {\setminus } P_\gamma$, there is an open neighborhood $U_p$ of p in $X_\gamma$ that does not intersect $P_\gamma$. We define a smooth vector field $\vec \zeta _p$ on $U_p$ that lifts $\vec \theta$ and that is tangent to each stratum of $\phi ^{-1}\big ( \partial V_{A^{\prime }}(Q_\gamma ) \big ) \cap {\mathbb {B}}_{\epsilon }$, for any $A^{\prime } \ge A$. This is possible because the restriction of $\pi \circ \phi$ to $\phi ^{-1}\big ( V_A(Q_\gamma ) {\setminus } Q_\gamma \big ) \cap {\mathbb {B}}_{\epsilon }$ is a stratified submersion, which restricts to a locally trivial fibration $\phi ^{-1}\big ( \partial V_{A^{\prime }}(Q_\gamma ) \big ) \cap {\mathbb {B}}_{\epsilon }\rightarrow \gamma$, for each $A^{\prime } \ge A$.

Then the collapsing vector field $\vec \zeta _\gamma$ is obtained by gluing the vector fields $\vec \zeta _p$ using a partition of unity. Notice that $\vec \zeta _\gamma$ lifts $\vec \theta$ outside $\{0\}$.

Hence the flow $h: [0,a] \times X_\gamma {\backslash } P_\gamma \rightarrow X_\gamma {\backslash } P_\gamma$ associated to $\vec \zeta _\gamma$ defines a stratified homeomorphism $\tilde{\psi }_{t_0}$ from $X_{t_0} {\backslash } P_{t_0}$ to $X_0 {\backslash } \{0\}$.

Moreover, we can show that the extension $\psi _{t_0}: X_{t_0} \rightarrow X_0$ given by:

$$\begin{aligned} \psi _{t_0} (x) := {\left\{ \begin{array}{ll} \tilde{\psi }_{t_0}(x) &{} \text {if}\quad x \in X_{t_0} {\backslash } P_{t_0} \\ 0 &{} \text {if}\quad x \in P_{t_0} \\ \end{array}\right. } \end{aligned}$$

is continuous. It is enough to show that if $(x_r)_{r \in {\mathbb {R}}}$ is a sequence of points in $X_{t_0} {\backslash } P_{t_0}$ such that $x_r \in \phi ^{-1} \left( \partial V_r(Q_\gamma ) \right)$, for r sufficiently large (so $x_r$ converges to a point $x \in P_{t_0}$), then the sequence of points $\left( \tilde{\psi }_{t_0}(x_r) \right) _{r \in {\mathbb {R}}}$ in $X_0$ converges to 0. Since the collapsing vector field $\vec \zeta _\gamma$ is tangent to $\phi ^{-1} \left( \partial V_r(Q_\gamma ) \right)$, for each r sufficiently large, it follows that:

$$\begin{aligned} \tilde{\psi }_{t_0} (x_r) \in \phi ^{-1} \left( \partial V_r(Q_\gamma ) \right) \cap X_0. \end{aligned}$$

On the other hand, by the condition (iv) above, we have that the system of neighborhoods $V_r(Q_\gamma )$ is such that $\cap _{r \in {\mathbb {N}}} V_r(Q_\gamma ) = Q_\gamma$, so $\phi ^{-1} \left( \partial V_r(Q_\gamma ) \right) \cap X_0$ goes to $\{0\}$ when r goes to infinity, in the sense of condition (iv) above. Precisely, for any open neighborhood U of 0 in $X_0$ there exists $R>0$ sufficiently large such that $\phi ^{-1} \left( \partial V_R(Q_\gamma ) \right) \cap X_0$ is contained in U. Therefore, the sequence of points $( \tilde{\psi }_{t_0}(x_r))_{r \in {\mathbb {R}}}$ in $X_0$ converges to 0.

This finishes the proof of Theorem 2 in the 2-dimensional case.

5 Elements of the proof of the main theorem

Now we go back to the general case of a germ of complex analytic function

$$\begin{aligned} f: (X,x) \rightarrow ({\mathbb {C}},0) \end{aligned}$$

at a point x of a reduced equidimensional complex analytic space $X \subset {\mathbb {C}}^N$ of any dimension. Let ${\mathcal S} = (S_{\alpha })_{{\alpha }\in A}$ be a Whitney stratification of X and suppose that f has an isolated singularity at x in the stratified sense.

In order to simplify the notations, suppose further that x is the origin in ${\mathbb {C}}^N$.

Recall the polar curve $\Gamma$ of f relatively to a generic linear form $\ell$, as well as the polar discriminant $\Delta := \phi (\Gamma )$, where $\phi$ is the stratified map:

$$\begin{aligned} \phi := (\ell ,f): (X,0) \rightarrow ({\mathbb {C}}^2,0). \end{aligned}$$

As before, we assume that the polar curve $\Gamma$ is non-empty.

In the next section, we will prove the following proposition:

Proposition 29

For any $t \in {\mathbb {D}}_{\eta _2}^*$, there exists a refinement $\mathcal {S}^{\prime }(t)$ of the Whitney stratification $\mathcal {S}(t)$ of $X_t$ (with the property (w)) induced by $\mathcal {S}$, such that there are:

• (i) A polyhedron $P_t$ of dimension $\dim _{\mathbb {C}}X -1$ that is contained in the Milnor fiber $X_t := {\mathbb {B}}_{\epsilon }\cap f^{-1}(t)$ and that is adapted to the stratification $\mathcal {S}^{\prime }(t)$;

• (ii) A continuous vector field $\vec \vartheta _t$ in $X_t$ so that:

  • 1. It is non-zero outside $P_t$ and it is zero on $P_t$;

  • 2. It is transversal to $\partial X_t$ (in the stratified sense) and pointing inwards;

  • 3. It restricts to a rugose stratified vector field on the interior of $X_t$ (relatively to the stratification $\mathcal {S}^{\prime }(t)$)

  • 4. The flow $\tilde{q}_t: [0, \infty ) \ \times \ (X_t {\backslash } P_t) \rightarrow X_t$ associated to $\vec \vartheta _t$ defines a map:

    $$\begin{aligned} \begin{aligned} \tilde{\xi }_t \ : \, \partial X_t&\longrightarrow P_t \\ z&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } \tilde{q}_t(\tau ,z) \end{aligned} \end{aligned}$$

    such that $\tilde{\xi }_t$ is continuous, stratified, simplicial and surjective;

  • 5. The Milnor fiber $X_t$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_t$.

We say that the polyhedron $P_t$ above is a vanishing polyhedron for f.

The idea of the construction of $P_t$ is quite simple and we will briefly describe it here. First recall the stratified map $\ell _t: X_t \rightarrow D_t$ given by the restriction of $\phi$ to $X_t$.

By induction hypothesis, we have a vanishing polyhedron $P_t^{\prime }$ for the restriction of f to the hyperplane section $X \cap \{ \ell =0 \}$.

For each point $y_j(t)$ in the intersection of the polar discriminant $\Delta$ with the disk $D_t := {\mathbb {D}}_{\eta _1} \times \{t\}$ as above, let $x_j(t)$ be a point in the intersection of the polar curve $\Gamma$ with $\ell _t^{-1}\big (y_j(t)\big )$. To simplify, we can assume that $x_j(t)$ is the only point in such intersection (see Conjecture 34 below).

Also by the induction hypothesis, we have a collapsing cone $P_j$ for the restriction of the map $\ell _t$ to a small neighborhood of $x_j(t)$. The “basis” of a such cone is the polyhedron $P_j(a_j) := P_j \cap \ell _t^{-1}(a_j)$, where $a_j$ is a point in $\delta (y_j(t)) {\backslash } y_j(t)$ close to $y_j(t)$.

Since $\ell _t$ is a locally trivial fiber bundle over $\delta (y_j(t)) {\backslash } y_j(t)$, we can “extend” the cone $P_j$ until it reaches the “central” polyhedron $P_t^{\prime }$. This gives a polyhedron $C_j$. The union of all the polyhedra $C_j$ together with $P_t^{\prime }$ gives our vanishing polyhedron $P_t$ (see Fig. 8).

Fig. 8

The vanishing polyhedron $P_t$

The detailed construction of $P_t$ will be given in the next section. We will also show in the next section that the construction of the polyhedron $P_t$ and the vector field $\vec \vartheta _t$ can be done simultaneously, for any t in a closed semi-disk ${\mathbb {D}}^+$ of ${\mathbb {D}}_{\eta _2}$ such that 0 is in its boundary, as in Fig. 1.

Precisely, we will prove the following:

Proposition 30

Let ${\mathbb {D}}^+$ be a closed semi-disk in ${\mathbb {D}}_{\eta _2}$ such that 0 is in its boundary, and set $X^+ := X \cap f^{-1}({\mathbb {D}}^+) \cap {\mathbb {B}}_{\epsilon }$. There exists a refinement $(\mathcal {S}^+)^{\prime }$ of the Whitney stratification $\mathcal {S}^+$ of $X^+$ (with the property (w)) induced by the stratification $\mathcal {S}$ of X such that there are a polyhedron $P^+$, adapted to the stratification $(\mathcal {S}^+)^{\prime }$, and an integrable vector field $\vec \vartheta _+$ in $X^+$, so that:

• (i) the intersection $P^+ \cap X_t$ is a vanishing polyhedron $P_t$ as in Proposition 29, for any $t \in {\mathbb {D}}^+ {\backslash } \{0\}$, and $P^+ \cap X_0 = \{0\}$;

• (ii) for any $t \in {\mathbb {D}}^+ {\backslash } \{0\}$, the restriction of $\vec \vartheta _+$ to $X_t$ gives a vector field $\vec \vartheta _t$ as in Proposition 29, relatively to the polyhedron $P_t = P^+ \cap X_t$;

• (iii) the vector field $\vec \vartheta _+$ is non-zero outside $P^+$, zero on $P^+$, transversal to $\partial X^+ := X^+ \cap {\mathbb {S}}_{\epsilon }$ in the stratified sense, pointing inwards, and it restricts to a rugose stratified vector field on the interior of $X^+$ (relatively to the stratification $(\mathcal {S}^+)^{\prime }$).

We say that the polyhedron $P^+$ is a collapsing cone for f along the semi-disk ${\mathbb {D}}^+$.

As an immediate corollary, we have:

Corollary 31

Let $\gamma$ be a simple path in ${\mathbb {D}}_{\eta _2}$ joining 0 and some $t_0 \in {\mathbb {D}}_{\eta _2}$ (as in Fig. 7), and set $X_\gamma := X \cap f^{-1}(\gamma ) \cap {\mathbb {B}}_{\epsilon }$. There exists a refinement $\mathcal {S}^{\prime }(\gamma )$ of the Whitney stratification $\mathcal {S}(\gamma )$ of $X_\gamma$ (with the property (w)) induced by the stratification $\mathcal {S}$ of X such that there are a polyhedron $P_\gamma$ in $X_\gamma$, adapted to the stratification $\mathcal {S}^{\prime }(\gamma )$, and an integrable vector field $\vec \vartheta _\gamma$ in $X_\gamma$, so that:

• (i) the intersection $P_\gamma \cap X_t$ is a vanishing polyhedron $P_t$ as in Proposition 29, for any $t \in \gamma {\backslash } \{0\}$, and $P_\gamma \cap X_0 = \{0\}$;

• (ii) for any $t \in \gamma {\backslash } \{0\}$, the restriction of $\vec \vartheta _\gamma$ to $X_t$ gives a vector field $\vec \vartheta _t$ as in the proposition above, relatively to the polyhedron $P_t = P_\gamma \cap X_t$;

• (iii) the vector field $\vec \vartheta _\gamma$ is non-zero outside $P_\gamma$, zero on $P_\gamma$, transversal to $\partial X_\gamma := X_\gamma \cap {\mathbb {S}}_{\epsilon }$ in the stratified sense, pointing inwards, and it restricts to a rugose stratified vector field on the interior of $X_\gamma$.

We say that the polyhedron $P_\gamma$ above is a collapsing cone for f along the path $\gamma$.

One can check that the flow $\tilde{q}_\gamma : [0,\infty ) \times (X_\gamma {\backslash } P_\gamma ) \rightarrow X_\gamma$ given by the integration of the vector field $\vec \vartheta _\gamma$ on $X_\gamma {\setminus } P_\gamma$ defines a continuous, simplicial and surjective map:

$$\begin{aligned} \begin{aligned} \tilde{\xi }_\gamma \ : \, \partial X_\gamma&\longrightarrow P_\gamma \\ z&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } \tilde{q}_\gamma (\tau ,z) \end{aligned} \end{aligned}$$

such that $X_\gamma$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_\gamma$ (see Proposition 27).

Remark 32

In order to prove Theorem 1, we just need Corollary 31. Nevertheless, we will need Proposition 29 to prove Proposition 30 and we will need Proposition 30 when $\dim _{\mathbb {C}}X =n-1$ to prove Proposition 29 when $\dim _{\mathbb {C}}X=n$.

So let us assume now that Proposition 30 is true. Itens (i) and (ii) of Theorem 2 follow easily from Proposition 30 (the proof that $X^+$ is homeomorphic to the mapping cylinder of the map $\tilde{\xi }_+$ given by the vector field $\vec \vartheta _+$ is analogous to the proof of item (7) of Proposition 27). Then we can easily prove (iii) of Theorem 2 as follows:

Fix $t \in {\mathbb {D}}_{\eta _2}^*$ and let $\gamma$ be a simple path in ${\mathbb {D}}_{\eta _2}$ connecting t and 0. Consider the polyhedron $P_\gamma$ and the vector field $\vec \vartheta _\gamma$ in $X_\gamma$ given by Corollary 31, as well as the flow $\tilde{q}_\gamma$ given by the integration of $\vec \vartheta _\gamma$.

For any positive real $A>0$ set:

$$\begin{aligned} \tilde{V}_A(P_\gamma ):= X_\gamma {\setminus } \tilde{q}_\gamma \big ( [0,A) \times \partial X_\gamma \big ), \end{aligned}$$

which is a closed neighborhood of $P_\gamma$ in $X_\gamma$. Notice that using the first isotopy lemma of Thom–Mather (Lemma 19), the boundary $\partial \tilde{V}_A(P_\gamma )$ of $\tilde{V}_A(P_\gamma )$ is a locally trivial topological fibration over $\gamma$.

Following the steps (a) and (b) of the end of Sect. 4.3 and using Proposition 16, we can construct a collapsing vector field $\vec \zeta _\gamma$ on $X_\gamma {\backslash }P_\gamma$ such that:

• it is a rugose stratified vector field (relatively to the stratification $\mathcal {S}^{\prime }(\gamma )$ of $X_\gamma$):

• it projects on a smooth vector field $\vec \theta$ on $\gamma$ that goes from $t_0$ to 0 in a time $a>0$;

• it is tangent to $\partial \tilde{V}_A(P_\gamma )$, for any $A > 0$.

So the flow $g: [0,a] \times X_\gamma {\backslash } P_\gamma \rightarrow X_\gamma {\backslash } P_\gamma$ associated to the collapsing vector field $\vec \zeta _\gamma$ defines a homeomorphism $\psi _t$ from $X_t {\backslash } P_t$ to $X_0 {\backslash } \{0\}$ that extends to a continuous map from $X_t$ to $X_0$ and that sends $P_t$ to $\{0\}$, for any $t \in \gamma {\backslash } \{0\}$. This proves (iii) of Theorem 2.

Remark 33

Notice that the collapsing vector field $\vec \zeta _\gamma$ on $X_\gamma {\backslash } P_\gamma$ that gives the collapsing of f along the path $\gamma$ can be extended to a collapsing vector field $\vec \zeta _+$ on $X^+ {\setminus } P^+$ that gives the collapsing of f along the semi-disk ${\mathbb {D}}^+$.

6 Proof of Propositions 29 and 30

We will prove Propositions 29 and 30 by induction on the dimension of X, in the following way: we will prove that if Proposition 30 is true whenever $\dim _{\mathbb {C}}X = n-1$, then Proposition 29 is true whenever $\dim _{\mathbb {C}}X = n$, and this implies that Proposition 30 is true whenever $\dim _{\mathbb {C}}X = n$.

Notice that in Sects. 4.1 and 4.2 we have proved Propositions 29 and 30, respectively, when $\dim _{\mathbb {C}}X = 2$.

6.1 Proof of Proposition 29: constructing the vanishing polyhedron

As we said above, the polyhedron $P_t$ will consist of a “central” polyhedron $P_t^{\prime }$ on which we will attach the polyhedra $C_j$. The first step will be to construct the central polyhedron $P_t^{\prime }$, and then we will construct the polyhedra $C_j$.

Recall that we have fixed a linear form $\ell : {\mathbb {C}}^N \rightarrow {\mathbb {C}}$ that satisfies the conditions of Lemma 21. Then $\Gamma$ is the polar curve of f relatively to $\ell$ at 0 and $\Delta$ is the polar discriminant of f relatively to $\ell$ at 0.

Also recall from Proposition 22 that the map $\phi =(\ell ,f)$ induces a stratified submersion (relatively to the stratification induced by $\mathcal {S}$):

$$\begin{aligned} \phi _|: {\mathbb {B}}_{\epsilon }\cap X \cap \phi ^{-1}({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta ) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta \end{aligned}$$

and that for each $t \in {\mathbb {D}}_{\eta _2}^*$ fixed, the restriction $\ell _t$ of $\ell$ to the Milnor fiber $X_t$ induces a topological locally trivial fibration:

$$\begin{aligned} \varphi _t: X_t {\backslash } \ell _t^{-1}\big (\{y_1(t), \ldots , y_k(t)\}\big ) \rightarrow D_t {\backslash } \{y_1(t), \ldots , y_k(t)\}, \end{aligned}$$

where $D_t = {\mathbb {D}}_{\eta _1} \times \{t\}$ and $\{y_1(t), \ldots , y_k(t)\} = \Delta \cap D_t$.

For any $t \in {\mathbb {D}}_{\eta _2}$ set $\lambda _t := (0,t)$. Since the complex line $\{0\} \times {\mathbb {C}}$ is not a component of $\Delta$, we can suppose that $\lambda _t \notin \{y_1(t), \ldots , y_k(t)\}$.

For each $j = 1, \ldots , k$, let $\delta (y_j(t))$ be a smooth simple path in $D_t$ starting at $\lambda _t$ and ending at $y_j(t)$, such that two of them intersect only at $\lambda _t$.

First step: constructing the central polyhedron ${P_t}$’:

Consider the restriction $f^{\prime }$ of f to the intersection $X \cap \{\ell =0\}$, which has complex dimension $n-1$. Then we can apply the induction hypothesis to $f^{\prime }$ to obtain a vanishing polyhedron $P_t^{\prime }$ in the fiber $X_t \cap \{ \ell =0 \}$ and an integrable vector field $\vec \vartheta _t^{\prime }$ in $X_t \cap \{ \ell =0 \}$ that deformation retracts $X_t \cap \{ \ell =0 \}$ onto $P_t^{\prime }$.

Second step: constructing the polyhedra ${C_j}$:

First of all, in order to make it easier for the reader to understand the constructions, we will suppose that $\Gamma$ intersects $\ell _t^{-1}\big (y_j(t)\big )$ in only one point, which we call $x_j(t)$. The proof of the general case follows the same steps. In fact, we make the following conjecture:

Conjecture 34

For $\ell$ general enough, the map-germ $\phi _\ell = (\ell ,f): (X,x) \rightarrow ({\mathbb {C}}^2,0)$ induces a bijective morphism from $\Gamma$ onto $\Delta$.

Now recall that $\ell _t$ induces a locally trivial fibration over $\delta (y_j(t)) {\backslash } \{y_j(t)\}$. If we look at the local situation at $x_j(t)$, we can apply the induction hypothesis to the germ ${\ell _t}_|: (X_t, x_j(t)) \rightarrow (D_t, y_j(t))$, which has an isolated singularity at $x_j(t)$ in the stratified sense, in lower dimension. That is, considering a small ball $B_j$ in ${\mathbb {C}}^{N}$ centered at $x_j(t)$; a small disk $D_s$ in $D_t$ centered at $y_j(t)$ and a semi-disk $D_s^+$ of $D_s$ which contains $\delta (y_j(t)) \cap \mathring{D}_s$ in its interior, we obtain:

• a collapsing cone $P_j^+$ for $\ell _t$ along the semi-disk $D_s^+$;

• a collapsing cone $P_j$ for $\ell _t$ along the path $D_s \cap \delta (y_j(t))$, of real dimension $n-1$;

which give the collapsing of the map ${\ell _t}_|: B_j \cap \ell _t^{-1}(D_s) \rightarrow D_s$ along the path $D_s \cap \delta (y_j(t))$. See Fig. 9.

Fig. 9

The restriction $\ell _t$

Now we are going to extend the cone $P_j$ until it hits $P_t^{\prime }$, as follows.

First we need to construct the following vector fields on $A_j := \ell _t^{-1}\big (\delta (y_j(t)) {\backslash } \{y_j(t)\}\big )$ that will be used to extend the cone $P_j$ and glue it on the central polyhedron $P_t^{\prime }$:

• Vector field ${\vec \Xi }$: let $\vec \xi$ be a smooth non-singular vector field on $\delta (y_j(t)) {\backslash } \{y_j(t)\}$ that goes from $y_j(t)$ to $\lambda _t = (0,t)$. Since the restriction of $\ell _t$ to each Whitney stratum of $\mathcal {S}$ has maximum rank over $\delta (y_j(t)) {\backslash } \{y_j(t)\}$, we can lift $\vec \xi$ to an rugose (and hence integrable) stratified vector field $\vec \Xi$ on $A_j$ (see Proposition 16). In particular, for any $u \in \delta (y_j(t)) {\backslash } \{y_j(t)\}$ we can use the vector field $\vec \Xi$ to obtain a stratified homeomorphism ${\alpha }_u: \ell _t^{-1}(\lambda _t) \rightarrow \ell _t^{-1}(u)$, which takes $P_t^{\prime }$ to a polyhedron ${\alpha }_u(P_t^{\prime })$ in $\ell _t^{-1}(u)$. See Fig. 10.

Fig. 10

The vector field $\vec {\Xi }$

• Vector field ${\vec \kappa }$: we can transport the vector field $\vec \vartheta _t^{\prime }$ of $\ell _t^{-1}(\lambda _t) = X_t \cap \{ \ell =0 \}$ given by the induction hypothesis to all the fibers $\ell _t^{-1}(u)$, for any $u \in \delta (y_j(t)) {\backslash } \{y_j(t)\}$. The transportation of $\vec \vartheta _t^{\prime }$ to $\ell _t^{-1}(u)$ is the vector field on $\ell _t^{-1}(u)$ given by the flow obtained as image by ${\alpha }_u$ of the flow given by $\vec \vartheta _t^{\prime }$. So we obtain a vector field $\vec \kappa$ on $A_j$ whose restriction to $\ell _t^{-1}(\lambda _t)$ is $\vec \vartheta _t^{\prime }$. The flow associated to $\vec \kappa$ takes a point $z \in \ell _t^{-1}(u)$ to the polyhedron ${\alpha }_u(P_t^{\prime })$. See Fig. 11.

Fig. 11

The vector field $\vec {\kappa }$

• Vector field ${\vec \kappa _1}$: let $\theta$ be a smooth function on $\delta (y_j(t))$ such that $\theta (\lambda _t)=0$ and such that $\theta$ is non-singular and positive on $\delta (y_j(t)) {\backslash } \{\lambda _t \}$. It induces a function $\tilde{\theta } := \theta \circ \ell _t$ defined on $A_j$. Set:

  $$\begin{aligned} \vec \kappa _1 := \vec \kappa + \tilde{\theta } \cdot \vec \Xi , \end{aligned}$$

  which is an integrable vector field, tangent to the strata of the interior of $A_j$ induced by $\mathcal {S}$. Furthermore, this vector field $\vec \kappa _1$ is pointing inwards on the boundary $\partial A_j$, i.e. transversal in $A_j$ to the strata of $\partial A_j$ induced by $\mathcal {S}$. See Fig. 12.

Fig. 12

The vector field $\vec {\kappa }_1$

Since the vectors $\vec \kappa (z)$ and $\vec \Xi (z)$ are not parallel for any $z \in A_j {\backslash } P_t^{\prime }$, the vector field $\vec \kappa _1$ is zero only on the vanishing polyhedron $P_t^{\prime }$ of $\ell _t^{-1}(\lambda _t)$. Then if z is a point in $A_j {\backslash } \ell _t^{-1}(\lambda _t)$, the orbit of $\vec \kappa _1$ that passes through z has its limit point $z_1^{\prime }$ in $P_t^{\prime }$.

Moreover, since the integral curve associated to $\vec \kappa$ that contains $z \in A_j$ has its limit point $z^{\prime }$ in the polyhedron ${\alpha }_{\ell _t(z)}(P_t^{\prime })$ (which is the transportation of $P_t^{\prime }$ to $\ell _t^{-1}\big (\ell _t(z)\big )$ by the flow associated to $\vec \Xi$), it follows that $z_1^{\prime }$ is the point corresponding to $z^{\prime }$ by $\vec \Xi$, that is, $z_1^{\prime } = {\alpha }_{\ell _t(z)}(z^{\prime })$. In fact, if $u:= \ell _t(z)$ and if $w := ({\alpha }_u)^{-1}(z)$ is the corresponding point in $\ell _t^{-1}(\lambda _t)$, then by construction the integral curve $\mathcal {C}_\kappa (z)$ associated to $\vec \kappa$ that contains z is given by ${\alpha }_u(\mathcal {C}(w))$, where $\mathcal {C}(w)$ is the integral curve associated to $\vec \vartheta _t^{\prime }$ that contains w.

Set $a_j:= \partial D_s \cap \delta (y_j(t))$ and $P_j(a_j) := P_j \cap \ell _t^{-1}(a_j)$, where $P_j$ is the collapsing cone for $\ell _t$ at $x_j(t)$ along the path $D_s \cap \delta (y_j(t))$, as defined above. By the previous paragraph, $\vec \kappa _1$ takes $P_j(a_j)$ to $P_t^{\prime }$.

Since the action of the flow given by $\vec \kappa$ is simplicial, we can assume that the action of the flow given by $\vec \kappa _1$ is simplicial. Then the image of $P_j(a_j)$ by the flow of $\vec \kappa _1$ is a subpolyhedron $P_j^{\prime }$ of $P_t^{\prime }$. Moreover, the orbits of the points in $P_j(a_j)$ give a polyhedron $R_j$. See Fig. 13.

Fig. 13

Decomposition of the vanishing polyhedron $P_t$

Set:

$$\begin{aligned} C_j := P_j \cup R_j \cup P_j^{\prime }. \end{aligned}$$

It is a polyhedron in $P_t$ of real dimension $n-1$. We call $C_j$ a wing of the polyhedron $P_t$. In the case when X is smooth, it corresponds to a Lefschetz thimble.

Then the polyhedron we are going to consider is:

$$\begin{aligned} P_t := P_t^{\prime } \bigcup _{j=1}^k C_j. \end{aligned}$$

It is adapted to the stratification $\mathcal {S}$, since $P_j$ is adapted to $\mathcal {S}$ and the vector field $\vec \kappa _1$ is tangent to the strata of $\mathcal {S}$.

Now we have:

Lemma 35

There exists a refinement $\mathcal {S}^{\prime }(t)$ of the Whitney stratification $\mathcal {S}(t)$ of $X_t$ (with the property (w)) induced by $\mathcal {S}$, such that there is a continuous vector field $\vec \vartheta _t$ on $X_t$ such that:

• (i) It is non-zero on $X_t {\backslash } P_t$ and it is zero on $P_t$;

• (ii) It is transversal to the strata of $\partial X_t$, pointing inwards;

• (iii) It restricts to a rugose stratified vector field on the interior of $X_t$ (relatively to the stratification $\mathcal {S}^{\prime }(t)$);

• (iv) The orbits associated to $\vec \vartheta _t$ have a limit point at $P_t$ when the parameter goes to infinity.

The vector field $\vec \vartheta _t$ is obtained by gluing several vector fields on $X_t$ given by the lifting by $\varphi _t$ of suitable vector fields on $D_t$. The detailed proof of Lemma 35 is quite involved since it contains too many technical steps and constructions, so we present it separately in Sect. 7.

The flow defined by the vector field $\vec \vartheta _t$ of Lemma 35 gives a continuous, surjective and simplicial map $\tilde{\xi }_t$ from $\partial X_t$ to $P_t$ such that $X_t$ is homeomorphic to the mapping cylinder of $\tilde{\xi }_t$ (see the proof of Proposition 27). This proves Proposition 29.

We remark that although we have used just Corollary 31 (in lower dimension) in the construction of the polyhedron $P_t$, we will need the stronger Proposition 30 (in lower dimension) for the construction of the vector field $\vec \vartheta _t$ of Lemma 35, in Sect. 7.

6.2 Proof of Proposition 30: constructing the polyhedron $P^+$

Given a closed semi-disk ${\mathbb {D}}^+$ in ${\mathbb {D}}_{\eta _2}$ as in Fig. 1 above (with 0 in its boundary), we want to construct a polyhedron $P^+$ in $X^+ := X \cap f^{-1}({\mathbb {D}}^+) \cap {\mathbb {B}}_{\epsilon }$, adapted to the stratification induced by $\mathcal {S}$, and a continuous vector field $\vec \vartheta _+$ in $X^+$, tangent to each stratum of $X^+$, satisfying the conditions (i), (ii) and (iii) of Proposition 30.

Recall that in Sect. 4.2 we already did that when X has complex dimension 2. Also recall the 3-dimensional triangles $\mathcal {T}_1, \ldots , \mathcal {T}_k$ in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$, bounded by the semi-disks $Y_j$ and $\Lambda$, where $Y_1, \ldots , Y_k$ are the semi-disks given by the intersection of the polar discriminant $\Delta$ with ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$ and $\Lambda := \{0\} \times {\mathbb {D}}^+$. For each $j=1, \ldots , k$ and for each $t \in {\mathbb {D}}^+$, the intersection $\mathcal {T}_j \cap D_t$ gives a simple path $\delta (y_j(t))$ used to construct a vanishing polyhedron $P_t$ as in Sect. 6.1.

Finally, also recall from Proposition 22 that the map $\phi = (\ell ,f)$ induces a stratified submersion:

$$\begin{aligned} \phi _|: {\mathbb {B}}_{\epsilon }\cap X \cap \phi ^{-1}({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta ) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta , \end{aligned}$$

where $\Delta \subset {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2}$ is the polar discriminant of f relatively to the linear form $\ell$.

We are going to construct $P^+$. The construction of the vector field $\vec \vartheta _+$ is analogous to the construction of the vector field $\vec \vartheta _t$ of Lemma 35, which is described with details in the next section, so we leave it to the reader.

The construction of the polyhedron $P^+$ will be made in three steps:

First step: fixing an initial polyhedron ${P_{t_0}}$:

Fix some $t_0 \in {\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}$. By Proposition 29, we can choose a vanishing polyhedron $P_{t_0}$ in the Milnor fiber:

$$\begin{aligned} X_{t_0} = X \cap f^{-1}(t_0) \cap {\mathbb {B}}_{\epsilon }, \end{aligned}$$

which has the form:

$$\begin{aligned} P_{t_0} = P_{t_0}^{\prime } \bigcup _{j=1}^k C_j, \end{aligned}$$

where each $C_j$ is a wing glued to the central polyhedron $P_{t_0}^{\prime }$ along a subpolyhedron $(P_j)_{t_0}^{\prime }$ of $P_{t_0}^{\prime }$ (recall that $P_{t_0}^{\prime }$ is a vanishing polyhedron for the restriction $f^{\prime }$ of f to $X \cap \{\ell =0\}$).

Second step: extending ${P_{t_0}}$ over ${{\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}}$.

By the induction hypothesis, the vanishing polyhedron $P_{t_0}^{\prime } \subset X_{t_0} \cap \{\ell =0\}$ can be extended to a collapsing cone $(P^+)^{\prime } \subset X^+ \cap \{\ell =0\}$ for the restriction of f to $\{\ell =0\}$ along the semi-disk ${\mathbb {D}}^+$.

Now set $\tilde{X} := \phi ^{-1} \big ({\mathbb {D}}_{\eta _1} \times ({\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}) \big )$ and recall that $\phi$ induces a topological trivial fibration from $\tilde{X}$ onto ${\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}$ (see Fig. 7).

So we can extend the polyhedron $P_{t_0}$ constructed above for t varying in ${\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}$. That is, we consider a polyhedron $\tilde{P}$ in $\tilde{X}$ such that $\tilde{P} \cap X_{t_0} = P_{t_0}$ and such that for any $t \in {\mathbb {D}}^+ \cap \partial {\mathbb {D}}_{\eta _2}$ one has that $\tilde{P} \cap X_t$ is a vanishing polyhedron $P_t$ in $X_t$ with central $P_t^{\prime } := (P^+)^{\prime } \cap X_t$ and k-many wings, each one of them conic from the corresponding point $x_j(t) \in \Gamma \cap X_t$.

Third step: constructing some suitable neighborhoods:

For each $x_j(t)$ over $y_j(t)$, with $t \in {\mathbb {D}}^+$, choose a small radius r(t) such that the set:

$$\begin{aligned} {\mathcal {B}}_j := \bigcup _{t \in {\mathbb {D}}^+} {\mathbb {B}}_{r(t)}(x_j(t)) \end{aligned}$$

is a neighborhood of:

$$\begin{aligned} \bigcup _{t \in {\mathbb {D}}^+ {\backslash } \{0\}}\{ x_j(t) \} \end{aligned}$$

conic from 0, where r(t) can be taken as a real analytic function of t with $r(0)=0$, by Puiseux’s theorem.

To each ${\mathcal {B}}_j$ one can associate a neighborhood:

$$\begin{aligned} {\mathcal {A}}_j := \bigcup _{t \in {\mathbb {D}}^+} {\mathbb {D}}_{s(t)}(y_j(t)) \end{aligned}$$

in ${\mathbb {D}}_{\eta _1} \times {\mathbb {D}}^+$, where s(t) is an analytic function of $t \in {\mathbb {D}}^+$ with $0<s(t) \ll r(t)$, if $t \ne 0$, and $s(0)=0$.

Finally, let ${\mathcal {U}}$ be a neighborhood of $\Lambda {\backslash } \{0\}$, conic from 0, that meets all the ${\mathcal {A}}_j$’s, but not containing any $y_j(t)$. See Fig. 14.

Fig. 14

Some suitable neighborhoods

Fourth step: constructing a suitable vector field:

By the induction hypothesis, we have a collapsing cone $(P^+)^{\prime }$ in $X^+ \cap \{\ell =0\}$ and a collapsing vector field $\vec \zeta _+^{\prime }$ in $X^+ \cap \{\ell =0\}$ that give the degeneration of $f^{\prime }$ along ${\mathbb {D}}^+$. That is, $\vec \zeta _+^{\prime }$ is a rugose (and hence integrable) vector field, tangent to each stratum of the interior of $X^+ \cap \{\ell =0\}$, whose associated flow defines a homeomorphism from $(X_{t_0} \cap \{\ell =0\}) {\backslash } P_{t_0}^{\prime }$ to $(X_{t_0} \cap \{\ell =0\}) {\backslash } \{0\}$ that extends to a continuous map from $X_{t_0} \cap \{\ell =0\}$ to $X_0 \cap \{\ell =0\}$ and that sends $P_{t_0}^{\prime }$ to $\{0\}$ (see Remark 33). Notice that the vector field $\vec \zeta _+^{\prime }$ lifts a radial vector field in $\{0\} \times {\mathbb {D}}^+$ that goes to 0.

Set:

$$\begin{aligned} \tilde{{\mathcal {U}}} := \phi ^{-1}({\mathcal {U}}) \cap X^+. \end{aligned}$$

Since ${\mathcal {U}}$ is a cone over a contractible space, we can extend the collapsing vector field $\vec \zeta _+^{\prime }$ to a integrable vector field $\vec \zeta _{\mathcal {U}}$ on $\tilde{{\mathcal {U}}}$, tangent to each stratum of its interior, and that lifts a radial vector field in $\{0\} \times {\mathbb {D}}^+$ that goes to 0. Notice that the flow given by the vector field $\vec \zeta _{\mathcal {U}}$ sends the intersection $P_{t_0}^{\prime } \cap \tilde{{\mathcal {U}}}$ to $\{0\}$, where $P_{t_0}^{\prime } = P_{t_0} \cap \{\ell =0\}$.

Since each ${\mathcal {B}}_j {\backslash } \{0\}$ is a stratified topological locally trivial fibration over ${\mathbb {D}}^+ {\backslash } \{0\}$, one can also construct a rugose vector field $\vec \sigma _j$ on ${\mathcal {B}}_j {\backslash } \{0\}$ that trivializes it over ${\mathbb {D}}^+ {\backslash } \{0\}$ and that is tangent to the intersection of the polar curve $\Gamma$ with ${\mathcal {B}}_j {\backslash } \{0\}$ (which is the set of the points $x_j(t) \in {\mathcal {B}}_j$ for $t \in {\mathbb {D}}^+ {\setminus } \{0\}$).

Then, using a partition of unity $(\rho _{\mathcal {U}}, \rho _1, \ldots , \rho _k)$ adapted to $\tilde{{\mathcal {U}}}, {\mathcal {B}}_1, \ldots , {\mathcal {B}}_k$, we glue all the vector fields $\vec \sigma _j$’s and $\vec \zeta _{\mathcal {U}}$ together. We obtain a continuous trivializing vector field:

$$\begin{aligned} \vec \sigma := \rho _{\mathcal {U}}\vec \zeta _{\mathcal {U}}+ \sum _{j=1}^k \rho _j \vec \sigma _j \end{aligned}$$

in $X^+ \cap (\tilde{{\mathcal {U}}} \cup _{j=1}^k {\mathcal {B}}_j)$ such that:

• it is tangent on each stratum of the interior of $X^+ \cap (\tilde{{\mathcal {U}}} \cup _{j=1}^k {\mathcal {B}}_j)$;

• it is rugose and hence integrable;

• it projects to a radial vector field in ${\mathbb {D}}^+$ that converges to 0.

So the flow associated to the vector field $\vec \sigma$ goes from $\tilde{X} \cap (\tilde{{\mathcal {U}}} \cup _{j=1}^k {\mathcal {B}}_j)$ to 0, and the action of this flow over $\tilde{P}$ give the polyhedron $P^+$.

7 Proof of Lemma 35: constructing the vector field $\vec \vartheta _t$

In this section we give the detailed construction of the vector field $\vec \vartheta _t$ on $X_t := \phi ^{-1}({\mathbb {D}}_{\eta _1} \times \{t\}) \cap {\mathbb {B}}_{\epsilon }$ of Lemma 35, whose flow gives a continuous, surjective and simplicial map $\tilde{\xi }_t$ from the boundary of the Milnor fiber $\partial X_t := \phi ^{-1}({\mathbb {D}}_{\eta _1} \times \{t\}) \cap {\mathbb {S}}_{\epsilon }$ to the polyhedron $P_t$ constructed in the previous section, such that $X_t$ is homeomorphic to the simplicial map cylinder of $\tilde{\xi }_t$.

Recall that we have fixed a linear form $\ell : {\mathbb {C}}^N \rightarrow {\mathbb {C}}$ that satisfies the conditions of Lemma 21. Then $\Gamma$ is the polar curve of f relatively to $\ell$ at 0 and $\Delta$ is the polar discriminant of f relatively to $\ell$ at 0.

Also recall from Proposition 22 that the map $\phi =(\ell ,f)$ induces a stratified submersion (relatively to the stratification induced by the Whitney stratification $\mathcal {S}$ of X):

$$\begin{aligned} \phi _|: {\mathbb {B}}_{\epsilon }\cap X \cap \phi ^{-1}({\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta ) \rightarrow {\mathbb {D}}_{\eta _1} \times {\mathbb {D}}_{\eta _2} {\setminus } \Delta . \end{aligned}$$

As before, fix $t \in {\mathbb {D}}_{\eta _2} {\setminus } \{0\}$ and take a point $\lambda _t$ in $D_t := {\mathbb {D}}_{\eta _1} \times \{t\}$ such that $\lambda _t \notin \{y_1(t), \ldots , y_k(t)\}$, where $\{y_1(t), \ldots , y_k(t)\} := \Delta \cap D_t$. Also, for each $j = 1, \ldots , k$, let $\delta (y_j(t))$ be a simple path in $D_t$ starting at $\lambda _t$ and ending at $y_j(t)$, such that two of them intersect only at $\lambda _t$. We defined the set $Q_t := \cup _{j=1}^k \delta (y_j(t))$.

Recall that in Lemma 25 we constructed a Whitney stratification $\mathcal {Z}= (Z_{\beta })_{{\beta }\in B}$ of $D_t$ (with the property (w)) and a continuous vector field $\vec {\nu }_t$ on $D_t$ such that:

• 1. It is non-zero on $D_t {\setminus } Q_t$;

• 2. It vanishes on $Q_t$;

• 3. It is transversal to $\partial D_t$ and points inwards;

• 4. It restricts to a rugose stratified vector field on the interior $\mathring{D}_t$ of $D_t$ (relatively to the stratification $\mathcal {Z}$);

• 5. The associated flow $q_t: [0, \infty ) \ \times \ (D_t {\backslash } Q_t) \rightarrow D_t {\backslash } Q_t$ defines a map:

  $$\begin{aligned} \begin{aligned} \xi _t \ : \, \partial D_t&\longrightarrow Q_t \\ u&\longmapsto \displaystyle \lim _{\tau \rightarrow \infty } q_t(\tau ,u), \end{aligned} \end{aligned}$$

  such that $\xi _t$ is continuous, simplicial (as defined in Sect. 2.4) and surjective.

After that, we considered a refinement $\mathcal {S}^{\prime }(t)$ of the Whitney stratification $\mathcal {S}(t)$ (with the property (w)) induced by $\mathcal {S}$, and we considered a refinement $\mathcal {Z^{\prime }}= (Z_{\beta }^{\prime })_{{\beta }\in B^{\prime }}$ of $\mathcal {Z}$ such that the restriction $\ell _t: X_t \rightarrow D_t$ of $\ell$ to the Milnor fiber $X_t$ is a stratified map. So $\ell _t$ induces a stratified submersion:

$$\begin{aligned} \varphi _t: X_t {\backslash } \ell _t^{-1}\big (\{y_1(t), \ldots , y_k(t)\}\big ) \rightarrow D_t {\backslash } \{y_1(t), \ldots , y_k(t)\}. \end{aligned}$$

Finally, recall that we can apply the induction hypothesis to the restriction $f^{\prime }$ of f to the intersection $X \cap \{\ell =0\}$, which has complex dimension $n-1$. We obtain a vanishing polyhedron $P_t^{\prime }$ in the intersection $X_t \cap \{ \ell =0 \}$ and a vector field $\vec \vartheta _t^{\prime }$ that deformation retracts it onto $P_t^{\prime }$.

The vector field $\vec \vartheta _t$ is obtained by gluing several vector fields on $X_t$ given by the lift of suitable vector fields on the disk $D_t$ by $\varphi _t$. By Proposition 16, the resulting vector fields are rugose, and hence integrable.

Recall that the polyhedron $P_t$ is the union of the wings $C_j$ and that the polyhedron $P_t^{\prime }$ is given by the induction hypothesis (as in Sect. 6.1). Moreover, each wing $C_j$ consists of a collapsing cone $P_j$, a product $R_j$ and the gluing polyhedron $P_j^{\prime }$ on $P_t^{\prime }$, that is:

$$\begin{aligned} C_j = P_j \cup R_j \cup P_j^{\prime }. \end{aligned}$$

See Fig. 13.

Then it is natural that the construction of the vector field $\vec \vartheta _t$ concerns at least three subsets of the Milnor fiber $X_t$: the points that are taken to $P_t^{\prime } {\backslash } P_j^{\prime }$ by the flow associated to $\vec \vartheta _t$; the points that are taken to $P_j^{\prime }$ and the points that are taken to $C_j {\backslash } P_j^{\prime }$. This justifies the complexity of the construction given below.

7.1 First step: decomposing $D_t$

Let $q_t: [0,\infty ) \times \partial D_t \rightarrow D_t$ be the flow associated to the vector field $\vec {\nu }_t$ defined in Lemma 25. Set:

$$\begin{aligned} V := D_t - q_t([0,A) \times \partial D_t), \end{aligned}$$

for some $A \gg 0$, which is a closed neighborhood of $Q_t$ whose boundary:

$$\begin{aligned} \partial V = q_t(\{A\} \times \partial D_t) \end{aligned}$$

is transversal to each $\partial D_s(y_j(t))$, that is, the vector field $\vec {\nu }_t$ is transversal to the boundary $\partial D_s(y_j(t))$ of each disk $D_s(y_j(t))$. See Fig. 15.

Fig. 15

The neighborhood V

Then we will construct the vector field $\vec \vartheta _t$ on $X_t$ by gluing a vector field $\vec \tau$ in $\ell _t^{-1}(D_t {\backslash } V^{\prime })$, where $V^{\prime } := D_t - q_t([0,A^{\prime }[ \times \partial D_t)$, with $A^{\prime } > A$, $A^{\prime }-A \ll 1$; and a vector field $\vec \upsilon$ in $\ell _t^{-1}(V)$, using a partition of unity.

The vector field $\vec \tau$ in $\ell _t^{-1}(D_t {\backslash } V^{\prime })$ is a lifting of the vector field $\vec {\nu }_t$. It is transversal to the boundary of $X_t$, pointing inwards, and it restricts to a rugose stratified vector field on $\phi ^{-1}(D_t {\backslash } V^{\prime }) \cap \mathring{{\mathbb {B}}}_{\epsilon }$.

The construction of the vector field $\vec \upsilon$ in $\ell _t^{-1}(V)$ is much more complicated. We are going to do it in the rest of this subsection.

7.2 Second step: decomposing V

We first decompose V into “branches” $V_j$ as follows: each “branch” $V_j$ is a closed neighborhood of $\delta (y_j(t)) {\backslash } \{0\}$ whose boundary is composed by $\partial V \cap V_j$ and two simple paths that one can suppose to be orbits of the vector field $\vec {\nu }_t$ constructed above. See Fig. 16.

Fig. 16

The branch $V_j$

We will construct the vector field $\vec \upsilon$ by gluing the vector fields $\vec \upsilon _j$ that we are going to construct on each $\ell _t^{-1}(V_j)$. In other words, we will construct a vector field $\vec \upsilon _j$ on $\ell _t^{-1}(V_j)$, for each j fixed, which is continuous, integrable, tangent to the strata of $\mathcal {S}$, non-zero and smooth on $\ell _t^{-1}(V_j) {\backslash } C_j$, and zero on $C_j$, where $C_j$ is the polyhedron defined in Sect. 6.1.

7.3 Third step: covering $\ell _t^{-1}(V_j)$ by open sets $W_{j,i}$

Fix $j \in \{1, \ldots , k\}$. The approach of the construction of each vector field $\vec \upsilon _j$ will be the following: we will cover $\ell _t^{-1}(V_j)$ by open sets $W_{j,1}$, $W_{j,2}$, $W_{j,3}$ and $W_{j,4}$. Then we will construct the vector fields $\vec \upsilon _{j,i}$ on $W_{j,i}$, for $i=1, \ldots , 4$, in such a way that each orbit of the vector field $\vec \upsilon _j$ obtained by gluing them with a partition of unity has a limit point in $P_t$.

As before, given positive real numbers r and s, let $B_r$ denote the ball around $x_j(t)$ in ${\mathbb {C}}^N$ of radius r and let $D_s$ denote the disk around $y_j(t)$ in $D_t$ of radius s.

Let r and $r^{\prime }$ be small enough positive real numbers such that $r^{\prime }<r$ and $r-r^{\prime } \ll 1$. Let us cover $\ell _t^{-1}(V_j)$ by the open sets $W_{j,1}$, $W_{j,2}$, $W_{j,3}$ and $W_{j,4}$ defined as follows:

$$\begin{aligned} \bullet \ \ W_{j,1} := \ell _t^{-1}(\mathring{D}_s \cap \mathring{V_j}) \cap \mathring{B_r} \, \end{aligned}$$

and

$$\begin{aligned} \bullet \ \ W_{j,2} := \ell _t^{-1}(\mathring{D}_s \cap \mathring{V_j}) {\setminus } B_{r^{\prime }}. \end{aligned}$$

To define $W_{j,3}$ and $W_{j,4}$ we have to do a construction first. Set:

$$\begin{aligned} W_{j,3}^{\prime }:= \ell _t^{-1}(\mathring{V_j} {\backslash } D_{s^{\prime }}), \end{aligned}$$

where $s^{\prime }<s$ and $s-s^{\prime } \ll 1$.

We can construct a vector field $\vec \omega _j$ in $V_j {\backslash } \mathring{D}_{s^{\prime }}$ which is smooth, non zero outside $\{0\}$, zero on $\{0\}$, with trajectories transversal to $\partial V \cap (V_j {\backslash } D_{s^{\prime }})$ and to $\partial D_{s^{\prime }} \cap V_j$, as in Fig. 17.

Fig. 17

The vector field $\vec {\omega }_j$

Since the Whitney stratification $\mathcal {S}^{\prime }(t)$ of $X_t$ with the property (w) induces a Whitney stratification on $\ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }})$ with the property (w), and since the restriction of $\ell _t$ to $\ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }}) \cap S_{\alpha }$ is a submersion for each ${\alpha }\in A$, we can lift $\vec \omega _j$ to a continuous vector field $\vec \Omega _j$ in $\ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }})$ that is rugose, smooth and tangent to each stratum, and that trivializes $\ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }})$ over $V_j {\backslash } \mathring{D}_{s^{\prime }}$.

Recall that the induction hypothesis applied to the restriction of f to $X \cap \{\ell =0\}$ gives a vanishing polyhedron $P_t^{\prime }$ in $X_t \cap \{ \ell =0 \}$ and a vector field $\vec \vartheta _t^{\prime }$ that deformation retracts $X_t \cap \{ \ell =0 \}$ onto $P_t^{\prime }$.

Then one can transport the vector field $\vec \vartheta _t^{\prime }$ on $\ell _t^{-1}(0)$ to all the fibers $\ell _t^{-1}(u)$ for $u \in V_j {\backslash } \mathring{D}_{s^{\prime }}$. This way we obtain a vector field $\vec {\mathcal {V}}_j$ in $\ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }})$ which is integrable, tangent to each stratum of $\ell _t^{-1}(u) \cap \mathring{{\mathbb {B}}}_{\epsilon }$ and transversal to each stratum of $\ell _t^{-1}(u) \cap {\mathbb {S}}_{\epsilon }$, for any $u \in V_j {\backslash } \mathring{D}_{s^{\prime }}$.

Now consider the vector field $\vec \Upsilon _j$ in $\ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }})$ given by:

$$\begin{aligned} \vec \Upsilon _j := \vec {\mathcal {V}}_j + \vec \Omega _j, \end{aligned}$$

which is integrable, tangent to the strata of the stratification $\mathcal {S}^{\prime }(t)$, transversal to the strata of ${\mathbb {S}}_{\epsilon }\cap \ell _t^{-1}(V_j {\backslash } \mathring{D}_{s^{\prime }})$, non-zero outside $P_t^{\prime }$ and zero on $P_t^{\prime }$.

One can see (as in the case of the vector field $\vec \kappa _1$ of Sect. 6.1) that each orbit of the vector field $\vec \Upsilon _j$ has a limit point in $P_t^{\prime }$.

The orbits by the action of $\vec \Upsilon _j$ which intersect $\ell _t^{-1}(\mathring{V}_j \cap \partial D_s) \cap \mathring{B}_r$ define a set that we call $A(V_j,r)$. We set $W_{j,4}^{\prime } := A(V_j,r)$ and:

$$\begin{aligned} \bullet \ \ W_{j,4} := W_{j,4}^{\prime } \cup W_{j,1}. \end{aligned}$$

Finally, the set $W_{j,3}$ is given by:

$$\begin{aligned} \bullet \ \ W_{j,3} := W_{j,3}^{\prime } {\backslash } A(V_j^{\prime },r^{\prime }), \end{aligned}$$

where $r^{\prime }<r$, with $r-r^{\prime } \ll 1$, and $V_j^{\prime } := D_t {\backslash } q_t([0, A^{\prime }[ \times \partial D_t)$, with $A^{\prime }<A$ and $A-A^{\prime } \ll 1$. One can check that both $W_{j,3}$ and $W_{j,4}$ are open sets.

7.4 Fourth step: constructing the vector fields $\vec \upsilon _{j,i}$

• 1. Construction of ${\vec \upsilon _{j,1}}$: We can consider a continuous vector field $\vec \nu _j$ on $V_j$ which is smooth and non-zero outside$\delta (y_j(t))$, zero on $\delta (y_j(t))$, transversal to $\partial V_j$ and tangent to $\partial D_s \cap V_j$, like in Fig. 18 (see the construction of the vector field $\vec \nu _t$ of Lemma 25).

Fig. 18

The vector field $\vec {\nu }_j$

Let $D_s^+$ be a semi-disk of $D_s$ which contains $\delta (y_j(t)) \cap \mathring{D}_s$ in its interior. We will lift $\vec \nu _j$ to a rugose vector field $\vec \chi _j$ in $\ell _t^{-1}(\mathring{D}_s \cap \mathring{V}_j) \cap B_r$, which is zero on $\ell _t^{-1} \big ( \mathring{D}_s \cap \delta (y_j(t)) \big )$, tangent to the strata of $\mathcal {S}^{\prime }(t)$ and of $\ell _t^{-1}(\mathring{D}_s \cap \mathring{V}_j) \cap S_r$, where $S_r := \partial B_r$, in the following way:

• Recall that we can apply the induction hypothesis to the restriction:

  $$\begin{aligned} (\ell _t)_|: \ell _t^{-1}(D_s^+ \cap \mathring{D}_s \cap \mathring{V}_j) \cap \mathring{B}_r \rightarrow D_s^+ \cap \mathring{D}_s \cap \mathring{V}_j, \end{aligned}$$

  which has an isolated singularity at $x_j(t)$ in the stratified sense, since $\ell _t^{-1}(D_s^+ \cap \mathring{D}_s \cap \mathring{V}_j)$ has complex dimension $n-1$, where n is the dimension of X.

• Then we obtain a collapsing vector field $\vec \vartheta _+(j)$ and a collapsing cone $P_j^+$. Let:

  $$\begin{aligned} q_j: [0,+\infty [ \times \big ( \ell _t^{-1}(\mathring{D_s^+} \cap \mathring{V}_j) \cap S_r \big ) \rightarrow \ell _t^{-1}(\mathring{D_s^+} \cap \mathring{V}_j) \cap B_r \end{aligned}$$

  be the flow associated to $\vec \vartheta _+(j)$ and set:

  $$\begin{aligned} P_j(u) := q_j \big ( \{u\} \times \ell _t^{-1}(\mathring{D_s^+} \cap \mathring{V}_j) \cap S_r \big ), \end{aligned}$$

  where $u \ge 0$.

• The Whitney stratification $\mathcal {S}^{\prime }(t)$ induces a Whitney stratification of $P_j(u)$ (see Lemma 6 and notice that $P_j(0) = \ell _t^{-1}(\mathring{D_s^+} \cap \mathring{V}_j) \cap S_r$ is the intersection of $\ell ^{-1}(\mathring{D_s^+} \cap \mathring{V}_j)$ with the stratified space $X_t \cap S_r$). Moreover, the restriction of $\ell _t$ to each stratum has maximum rank. So by Proposition 16 we can lift the vector field $\vec \nu _j$ over $\mathring{D_s^+} \cap \mathring{V}_j$ to a rugose stratified vector field that is tangent to the strata of $P_j(u)$.

• On the other hand, for any point in $\ell _t^{-1}\big ( (\mathring{D}_s {\backslash } \mathring{D_s^+}) \cap \mathring{V}_j \big )$ we just ask the vector field $\vec \chi _j$ to be tangent to the strata of $\mathcal {S}^{\prime }(t)$ and to lift $\vec \nu _j$. This can be done locally and then $\vec \chi _j$ is obtained by a partition of unity.

Notice that at any point of $\mathring{B}_r \cap \ell _t^{-1} \big ( \mathring{D}_s^+ \cap \mathring{V}_j \big ) - \{x_j(t)\}$ and at any point of $(\mathring{B}_r {\backslash } B_{r^{\prime }}) \cap \ell _t^{-1} \big ( (\mathring{D}_s {\backslash } D_s^+) \cap \mathring{V}_j \big )$, for $r^{\prime }<r$ with $r-r^{\prime }\ll 1$, one can extend the vector field $\vec \vartheta _+(j)$ on a small open neighborhood. Now we construct $\vec \upsilon _{j,1}$ as follows:

• Over a small open neighborhood $U_{x_j(t)}$ of $x_j(t)$, consider the zero vector field.

• For any $z \in \mathring{B}_r \cap \ell _t^{-1}(D_s^+ \cap \mathring{D}_s \cap \mathring{V}_j) {\backslash } \{x_j(t)\}$, take an open neighborhood $U_z$ of z small enough such that it does not contain $x_j(t)$, it is contained in $\mathring{B}_r \cap \ell _t^{-1}(\mathring{D}_s \cap \mathring{V}_j)$ and $\vec \vartheta _+(j)$ is well defined on it. Then in $U_z$ we define the vector field:

  $$\begin{aligned} \vec \iota _z := \vec \vartheta _+(j)_{|U_z} + \vec \chi _j{_{|U_z}}, \end{aligned}$$

  where $\vec \vartheta _+(j)_{|U_z}$ and $\vec \chi _j{_{|U_z}}$ denote the restrictions of the vector fields $\vec \vartheta _+(j)$ and $\vec \chi _j$, respectively, to the neighborhood $U_z$. This vector field is rugose, tangent to the strata of $\mathcal {S}^{\prime }(t)$, non-zero outside the intersection of $U_z$ and $P_j$ and zero on $P_j \cap U_z$, where:

  $$\begin{aligned} P_j := P_j^+ \cap \ell _t^{-1}\big ( \delta (y_j(t)) \big ). \end{aligned}$$

• For any $z \in \mathring{B}_{r^{\prime }} \cap \ell _t^{-1} \big ( (\mathring{D}_s {\backslash } D_s^+) \cap V_j \big )$, take a small open neighborhood $U_z$ of z and set

  $$\begin{aligned} \vec \iota _z := \vec \chi _j{_{|U_z}} . \end{aligned}$$

• For any $z \in (\mathring{B}_r {\backslash } B_{r^{\prime }}) \cap \ell _t^{-1} \big ( (\mathring{D}_s {\backslash } D_s^+) \cap V_j \big )$, take a small open neighborhood $U_z$ of z contained in $(\mathring{B}_r {\backslash } B_{r^{\prime }}) \cap \ell _t^{-1} \big ( (\mathring{D}_s {\backslash } D_s^+) \cap V_j \big )$ and set:

  $$\begin{aligned} \vec \iota _z := \vec \vartheta _+(j)_{|U_z} + \vec \chi _j{_{|U_z}}. \end{aligned}$$

• Then considering a partition of unity $(\rho _z)$ associated to the covering $(U_z)$, we set the vector field:

  $$\begin{aligned} \vec \upsilon _{j,1} := \sum \rho _z \vec \iota _z \end{aligned}$$

  in $\ell _t^{-1}(\mathring{D}_s \cap \mathring{V}_j) \cap \mathring{B}_r$, which is continuous, rugose outside the point $x_j(t)$ (and therefore in $W_{j,1} {\backslash } P_j$), tangent to the strata of $\mathcal {S}^{\prime }(t)$, non-zero outside $P_j$ and zero on $P_j \cap \big ( \ell _t^{-1}(\mathring{D}_s \cap \mathring{V}_j) \big ) \cap \mathring{B}_r$.

• Notice that if $z \in \ell _t^{-1}(\mathring{D}_s {\backslash } D_s^+) \cap \mathring{B}_r$, its orbit by $\vec \upsilon _{j,1}$ has $\{x_j(t)\}$ as limit point, and the orbit by $\vec \upsilon _{j,1}$ of a point $z \in \ell _t^{-1}(D_s^+) \cap \mathring{B}_r$ has its limit point in $P_j$.

• 2. Construction of ${\vec \upsilon _{j,2}}$: Consider a smooth non-zero vector field $\vec \eta _j$ in $\mathring{V}_j \cap \mathring{D}_s$ as Fig. 19 and such that, for any $u \in \mathring{V}_j \cap \mathring{D}_s$ one has the following implication:

  $$\begin{aligned} \lambda \vec \nu _j(u) + \mu \vec \eta _j(u) =0, \ \ \lambda \ge 0, \ \ \mu \ge 0 \implies \lambda =\mu =0, \end{aligned}$$

  where $\vec \nu _j$ is the vector field defined above.

Fig. 19

The vector field $\vec {\eta }_j$

Then $\vec \upsilon _{j,2}$ is a lifting of $\vec \eta _j$ in $W_{j,2}$, which is rugose vector field, tangent to the strata of $\mathcal {S}^{\prime }(t)$ and of $\ell _t^{-1}(\mathring{D}_s) \cap S_r$.

• 3. Construction of ${\vec \upsilon _{j,3}}$: We set $\vec \upsilon _{j,3}$ to be the restriction of the vector field $\vec \Upsilon _j$ constructed above to $W_{j,3}$.

• 4. Construction of ${\vec \upsilon _{j,4}}$: Recall the vector field $\vec \vartheta _+(j)$ in $\ell _t^{-1}(D_s^+) \cap \mathring{B}_r$, obtained by the induction hypothesis, and restrict it to $\ell _t^{-1}(\partial D_s \cap \mathring{V}_j)$. Then transport it by the action of the vector field $\vec \Upsilon _j$. We obtain a vector field $\vec \sigma$ on $W_{j,4}^{\prime }$ that is rugose and tangent to the strata of $\mathcal {S}^{\prime }(t)$.

Over $W_{j,4} = W_{j,4}^{\prime } \cup W_{j,1}$, the vector fields $\vec \sigma$ and $\vec \upsilon _{j,1}$ glue in a vector field $\vec \upsilon _{j,4}$ that is continuous, rugose and non-zero on $W_{j,4} {\backslash } P_t$. The orbits of the points of $W_{j,4}$ by $\vec \upsilon _{j,4}$ has limit points in $P_t$.

7.5 Fifth step: gluing all the vector fields to obtain $\vec \vartheta _t$

Now, considering $\rho _2$, $\rho _3$ and $\rho _4$ a partition of unity associated to $W_{j,2}$, $W_{j,3}$ and $W_{j,4}$, we obtain the vector field:

$$\begin{aligned} \vec \upsilon _j := \rho _2 \vec \upsilon _{j,2} + \rho _3 \vec \upsilon _{j,3} + \rho _4 \vec \upsilon _{j,4} \end{aligned}$$

in $\ell _t^{-1}(\mathring{V}_j)$, which is continuous, rugose, non-zero on $\ell _t^{-1}(V_j) {\setminus } P_t$ and zero on $P_t$.

Gluing these vector fields $\vec \upsilon _j$, for $j=1, \ldots , k$, we get the vector field $\vec \upsilon$.

Finally, gluing the vector field $\vec \upsilon$ in $\ell _t^{-1}(V)$ and the vector field $\vec \tau$ in $\ell _t^{-1}(D_t {\backslash } V^{\prime })$ constructed in Section 6.1, we obtain a continuous vector field $\vec \vartheta _t$ in $X_t$ with the properties (i) to (v) of Proposition 35. We just have to check that the orbits of this vector field have a limit point when the parameter goes to infinity:

• (a) If $z \in \ell _t^{-1}(D_t {\backslash } V^{\prime })$, the orbit of z arrives to $W_{j,2} \cup W_{j,3} \cup W_{j,4}$ after a finite time.

• (b) If $z \in W_{j,2}$, the orbit of z arrives to $W_{j,3} \cup W_{j,4}$ after a finite time.

• (c) If $z \in W_{j,3} {\backslash } W_{j,4}$, it has a limit point on $\cup _{j=1}^k C_j$.

• (d) If $z \in W_{j,4} {\backslash } W_{j,3}$, it has a limit point on $P_t^{\prime }$.

• (e) If $z \in W_{j,3} \cap W_{j,4}$, we have that the orbit passing through z has a limit point that is the limit point by $\vec \Upsilon _j$ of the limit point of the orbit of z by $\vec \sigma$. Hence this limit point is on $P_j^{\prime } = P_t^{\prime } \cap \overline{C}_j$.