1 Introduction

As motivation, let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be a real analytic map, \(n\ge k\ge 2\), with \(0 \in {\mathbb {R}}^n\) an isolated critical point and dim(\(f^{-1}(0)) > 0\). By the implicit function theorem, for every sufficiently small sphere \(\mathbb {S}_\epsilon ^{n-1}\) around 0, one has the following transversality property: there exists \(\delta > 0\) such that for all \(t\in {\mathbb {R}}^k\) with \(||t|| \le \delta\) the fibre \(f^{-1}(t)\) meets \(\mathbb {S}_\epsilon ^{n-1}\) transversally or not at all. Hence, by the relative Ehresmann fibration theorem (see [16, p. 23]), one has a locally trivial fibration restricting f to the “tube”:

$$\begin{aligned} f:{\mathbb {B}}_{\epsilon }^n \cap f^{-1}(\mathbb {S}_\delta ^{k-1}) \longrightarrow \mathbb {S}_\delta ^{k-1} , \end{aligned}$$
(1)

where \({\mathbb {B}}_{\epsilon }^n\) is the closed ball in \({\mathbb {R}}^n\) of radius \({\epsilon }\) around the origin. In his now classical book [20] Milnor proves (see [20, Theorem 11.2] or [19, Theorem 2]) that (1) gives rise to a locally trivial fibration

$$\begin{aligned} \phi :\mathbb {S}^{n-1}_{\epsilon }{\setminus } f^{-1}(0)\rightarrow \mathbb {S}^{k-1} , \end{aligned}$$
(2)

where \(\mathbb {S}^{k-1}\) is the sphere of radius 1 around \(0 \in {\mathbb {R}}^k\). Milnor proves this constructing an integrable non-zero vector field on \({\mathbb {B}}_{\epsilon }{{\setminus }} f^{-1}(0)\) which is transverse to the fibres of f and to the spheres centered at 0 contained in \({\mathbb {B}}_{\epsilon }\). The integrable curves of this vector field carry diffeomorphically the tube to the complement of \(f^{-1}({\mathbb {B}}^{k}_\delta )\cap \mathbb {S}^{n-1}_{\epsilon }\) in \(\mathbb {S}^{n-1}_{\epsilon }\) keeping its boundary fixed.

Milnor points out in his book that this theorem has two main weaknesses: The condition for an analytic map \({\mathbb {R}}^n \rightarrow {\mathbb {R}}^k\) with \(n\ge k\ge 2\) to have an isolated critical point is very stringent, and even when this is satisfied, the projection map \(\phi\) may not always be taken as the natural map \(f/\Vert f\Vert\) (see [20, p. 99]).

The above discussion was extended in [23, Theorem 1.3] to functions f with an isolated critical value, a more general but still stringent setting. The authors provedFootnote 1 that if f has Thom’s \(a_f\) property, which implies the transversality property mentioned above, one also has fibration (1) on the tube, known as Milnor-Lê fibration, and using Milnor’s vector field one also has an equivalent fibration (2) on the sphere.

The question of whether we can take the projection \(\phi\) of (2) to be the natural one, \(\phi =f/\Vert f\Vert\), was answered in [7] for analytic functions with an isolated critical value, by introducing the concept of d-regularity (see also [4, 5, 9]): The map f is d-regular if and only if fibration (1) is equivalent to a fibration (2) where the projection is given by \(\phi =f/\Vert f\Vert\).

The d-regularity condition was introduced in [6] and is defined by means of a canonical pencil that somehow springs from [25, 26]. This is defined as follows: For every line \(0 \in \ell \subset {\mathbb {R}}^k\) consider the set

$$\begin{aligned} X_{\ell } = \{x \in {\mathbb {R}}^n \, \vert \, f(x) \in \ell \}. \end{aligned}$$

This is a pencil of real analytic varieties intersecting at \(f^{-1}(0)\) and smooth away from it. The map f is said to be d-regular at 0 if there exists \({\epsilon }_0>0\) such that every \(X_\ell {{\setminus }} V\) is transverse to every sphere centred at 0 and contained in \({\mathbb {B}}_{{\epsilon }_0}\), whenever the intersection is not empty.

Several works have been published in the last two decades studying Milnor fibrations for real analytic maps with either an isolated critical point or value, see for instance [1, 2, 7, 11, 13,14,15, 21,22,23,24,25,26] or the survey articles [3, 8, 9, 12, 27].

In this paper we envisage the general case of real analytic functions \(f :(U, 0) \rightarrow ({\mathbb {R}}^k, 0)\) with arbitrary critical set \(\Sigma _f\), where U is an open neighbourhood of the origin in \({\mathbb {R}}^n\) (cf. [10, 18]). If f has the transversality property then it admits a local Milnor-Lê fibration in tubes over the complement of its discriminant, in the following sense: For each \({\epsilon }>0\) small enough, there exists \(\delta _0>0\) such that for every \(0<\delta \le \delta _0\) the restriction

$$\begin{aligned} f:{\mathbb {B}}_{\epsilon }^n \cap f^{-1}( \mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon }) \longrightarrow \mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon }\end{aligned}$$
(3)

is the projection of a smooth locally trivial fibration, where \(\Delta _{\epsilon }:= f(\Sigma _f \cap {\mathbb {B}}_{\epsilon })\).

However, Milnor’s vector field on \({\mathbb {B}}_{\epsilon }^n{{\setminus }} f^{-1}(\Delta _{\epsilon })\) does not necessarily work to inflate the tube to the sphere, since in general \({\mathbb {B}}_{\epsilon }^n{{\setminus }} f^{-1}(\Delta _{\epsilon })\) is not invariant under its flow. This leads us to the following question: Is there a condition on f which ensures that there exists a fibration on the sphere?

The answer is d-regularity again. We show that if there exists a local homeomorphism \(({\mathbb {R}}^k,0) \buildrel {h} \over {\rightarrow } ({\mathbb {R}}^k,0)\) such that \(h^{-1}(\Delta _{\epsilon })\) is linear and the composition \(f_h=h^{-1}\circ f\) is d-regular in the ball \({\mathbb {B}}_{\epsilon }\), then the map

$$\begin{aligned} f_h/\Vert f_h\Vert :\mathbb {S}^{n-1}_{\epsilon }{{\setminus }} f^{-1}(\Delta _{\epsilon })\rightarrow \mathbb {S}^{k-1} \end{aligned}$$
(4)

is the projection of a smooth locally trivial fibration over its image (see Theorem 3.12 for the precise statement). In this case, we say that f is \(d_h\)-regular in \({\mathbb {B}}_{\epsilon }\). Every d-regular map is \(d_h\)-regular for h the identity map. Moreover, this fibration is equivalent to fibration (3).

Notice that the discriminant set of f can have real codimension 1 and therefore its complement may split into finitely many connected components, say \(S_1,\dots ,S_r\). The topology of the fibres \(f^{-1}(t)\) can change for values in different \(S_i\). This happens for instance for the map germs

$$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^2 , \sum _{i=1}^n b_i x_i^2 \right) , \end{aligned}$$
(5)

where \(a_i, b_i \in {\mathbb {R}}\) are constants in generic position, studied by López de Medrano in [17]. Notice that each sector determines an analytic family of smooth manifolds that degenerate to the singular fibre over 0. If we take an arc joining two adjacent sectors we get a family of smooth compact manifolds that transform into other smooth compact manifolds as we cross the discriminant. As it is pointed out in [17], it would be very interesting to determine how the topology changes as we move from one sector to another. In [17] partial results are presented for maps of the form (5).

Throughout this paper, we always assume (with no mention) that f is locally surjective, that is, the image by f of every neighborhood of the origin in \({\mathbb {R}}^n\) contains an open neighbourhood of the origin in \({\mathbb {R}}^p\). Nevertheless, it is easy to see that in the general case the same results hold if one intersects the bases of the locally trivial fibrations with their image. This choice is to avoid a heavy notation.

The results of this article extend to analytic map-germs \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) if we consider f in the class of nice analytic map-germs defined in [14, Definition 2.2] for which the germ of the discriminant depends only on the germ of f at \(0 \in {\mathbb {R}}^n\).

The article is organized as follows. In Sect. 2, we define when a real analytic map f as above has linear discriminant and we exhibit examples. Then we extend the concept of d-regularity and the corresponding fibration theorems to this case (Theorems 2.13 and 2.16). In Sect. 3, we define conic homeomorphisms on the target of f and \(d_h\)-regularity, and we give examples of both concepts. Then we discuss how using a conic homeomorphism one can linearize the discriminant of f. If the linearized map is d-regular we get the main fibration theorem (Theorem 3.12).

2 The d-regularity for analytic maps with linear discriminant

Let n and k be two integers with \(n>k\ge 2\). Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map with a critical point at 0. Assume that f is locally surjective (see the introduction).

Equip \({\mathbb {R}}^n\) with a Whitney stratification adapted to \(V=f^{-1}(0)\), and let \({\mathbb {B}}_{{\epsilon }_0}^n\) be a closed ball in \({\mathbb {R}}^n\), centred at 0, of sufficiently small radius \({\epsilon }_0>0\), so that every sphere in this ball, centred at 0, meets transversally every stratum of V. We call \({\epsilon }_0\) a Milnor radius for f at 0. In what follows, for \(0<{\epsilon }<{\epsilon }_0\) we shall consider the restriction \(f_{{\epsilon }}\) of f to the closed ball \({\mathbb {B}}_{{\epsilon }}^n\subset {\mathbb {B}}_{{\epsilon }_0}^n\).

Denote by \(\Sigma _{{\epsilon }}\) the intersection of the critical set of f with the ball \({\mathbb {B}}_{\epsilon }\) and set \(\Delta _{\epsilon }:= f_{\epsilon }(\Sigma _{{\epsilon }})\), which we call the discriminant of \(f_{{\epsilon }}\). The discriminant \(\Delta _{\epsilon }\) is a subanalytic set and it may depend on the choice of the radius \({\epsilon }\), as shown in [14].

Given \(\delta >0\) let \({\mathbb {B}}^k_\delta\) denote the closed ball around 0 in \({\mathbb {R}}^k\) with radius \(\delta\). We have:

Definition 2.1

We say that f has the transversality property if for every \({\epsilon }>0\) small enough there exists \(\delta = \delta ({\epsilon })>0\) such that for every \(y\in {\mathbb {B}}_\delta ^k{{\setminus }}\Delta _{\epsilon }\) either \(f^{-1}(y)\) does not intersect the sphere \(\mathbb {S}_{\epsilon }^{n-1}\) or \(f^{-1}(y)\) intersects \(\mathbb {S}_{\epsilon }^{n-1}\) transversally.

Remark 2.2

If f satisfies the Thom \(a_f\)-property then it has the transversality property (compare with [7, Remark 5.7]).

The transversality property gives a fibration on the “tube” (compare with [14, Lemma 3.3]):

Theorem 2.3

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\), with \(n>k\ge 2\), be an analytic map with a critical point at 0. If f has the transversality property then the map

$$\begin{aligned} f:{\mathbb {B}}_{\epsilon }^n \cap f^{-1}( \mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon }) \longrightarrow \mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon }\end{aligned}$$

is a locally trivial fibration.

Proof

It follows from the Relative Ehresmann Fibration Theorem. \(\square\)

Now we will extend the concept of d-regularity to real analytic maps with linear discriminant. This case was also studied in [24, Ch. 6] where it is called radial discriminant. The main idea is that d-regularity can be defined as transversality of preimages of lines, different from those lying in the discriminant, with sufficiently small spheres. Then we will show that in this context, d-regularity guarantees a fibration on the sphere.

First, let us make the definition of linear discriminant precise.

Definition 2.4

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map with a critical point at 0. Let \({\epsilon }_0\) be a Milnor radius for f and let \(0<{\epsilon }<{\epsilon }_0\). We say that f has linear discriminant in \(\mathbb {B}_{\epsilon }^n\) if \(\Delta _{\epsilon }\) is a union of line-segments with one endpoint at \(0 \in \mathbb {R}^k\). We say that \(\eta >0\) is a linearity radius for \(\Delta _{\epsilon }\) if each of these line-segments intersects \(\mathbb {S}_\eta ^{k-1}\), that is, if

$$\begin{aligned} \Delta _{\epsilon }\cap {\mathbb {B}}_\eta ^k = {{\,\textrm{Cone}\,}}\big ( \Delta _{\epsilon }\cap \mathbb {S}_\eta ^{k-1} \big ). \end{aligned}$$

Remark 2.5

The case when f has \(0\in {\mathbb {R}}^k\) as isolated critical value is considered to have linear discriminant with \(\Delta _{\epsilon }\cap \mathbb {S}_\eta ^{k-1}=\emptyset\).

Now suppose that f has linear discriminant in \(\mathbb {B}_{\epsilon }^n\) and consider a linearity radius \(\eta >0\) for f. Set:

$$\begin{aligned} \mathcal {A}_\eta := \Delta _{\epsilon }\cap \mathbb {S}_\eta ^{k-1}. \end{aligned}$$

Let \(\pi :\mathbb {S}_\eta ^{k-1}\rightarrow \mathbb {S}^{k-1}\) be the projection onto the unit sphere \(\mathbb {S}^{k-1}\) and set \(\mathcal {A}=\pi (\mathcal {A}_\eta )\). For each point \(\theta \in \mathbb {S}_\eta ^{k-1}\), let \(\mathcal {L}_\theta \subset {\mathbb {R}}^k\) be the open ray in \({\mathbb {R}}^k\) from the origin that contains the point \(\theta\). Set:

$$\begin{aligned} E_{\theta }:= f^{-1}(\mathcal {L}_\theta ). \end{aligned}$$

Notice that \(E_{\theta }\) is a smooth manifold for any \(\theta\) in \(\mathbb {S}_\eta ^{k-1} {\setminus } \mathcal {A}_\eta\).

We have:

Definition 2.6

We say that f is d-regular in \({\mathbb {B}}_{\epsilon }^n\) if \(E_{\theta }\) intersects the sphere \(\mathbb {S}_{{\epsilon }'}^{n-1}\) transversally in \({\mathbb {R}}^n\), for every \({\epsilon }'\) with \(0<{\epsilon }' \le {\epsilon }\) and for every \(\theta \in \mathbb {S}_\eta ^{k-1} {{\setminus }} \mathcal {A}_\eta\).

2.1 Some examples

The following two propositions help us to check that the examples are locally surjective or have the transversality property.

Proposition 2.7

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\), with \(n>k\ge 2\), be an analytic map with a critical point at 0. Suppose that for every sufficiently small \(\epsilon >0\) the vanishing set \(V(f)\cap {\mathbb {B}}_{\epsilon }^n\) is not contained in the critical set \(\Sigma _f\) of f, then f is locally surjective.

Proof

By hypothesis, for any \({\epsilon }>0\) there exists \(x \in V(f) \cap {\mathbb {B}}_{\epsilon }^n\) such that \(x \notin \Sigma _f\). So, up to a change of coordinates, f is a projection on an open neighborhood \(W_x\) of x in \({\mathbb {B}}_{\epsilon }^n\). Hence \(f(W_x)\) is an open neighborhood of 0 in \({\mathbb {R}}^k\). \(\square\)

Proposition 2.8

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\), with \(n>k\ge 2\), be an analytic map with a critical point at 0. If \(V(f) = \{0\}\) or if \(\Sigma _f(\mathring{{\mathbb {B}}}_{\epsilon })\cap V(f)= \{0\}\), then f has the transversality property.

Proof

Suppose that \(V(f) = \{0\}\) and fix \({\epsilon }>0\). By continuity of f there exists \(\delta >0\) sufficiently small such that \(f^{-1}(t)\) does not intersect the sphere \(\mathbb {S}_{{\epsilon }}^n\) for any \(t \in {\mathbb {B}}_\delta ^k\). So f has the transversality property by vacuity.

Now suppose that \(\dim V(f) >0\). Since f is real analytic, there exists \({\epsilon }_0>0\) sufficiently small such that V(f) intersects the sphere \(\mathbb {S}_{\epsilon }^{n-1}\) transversally in \({\mathbb {R}}^n\), for any \({\epsilon }\) with \(0<{\epsilon }\le {\epsilon }_0\).

Fix \({\epsilon }\) and consider the restriction \(f_|\) of f to \(\mathbb {S}_{\epsilon }^{n-1}\). Its critical points are the critical points of f that are in \(\mathbb {S}_{\epsilon }^{n-1}\) and the regular points x of f that are in \(\mathbb {S}_{\epsilon }^{n-1}\) such that \(f^{-1}(f(x))\) intersect \(\mathbb {S}_{\epsilon }^{n-1}\) not transversally at x in \({\mathbb {R}}^n\).

Since \(\Sigma _f(\mathring{{\mathbb {B}}}_{\epsilon }) \cap V(f) = \{0\}\), it follows that any \(x \in V(f) \cap \mathbb {S}_{\epsilon }^{n-1}\) is a regular point of f. Up to a change of coordinates, f is a projection on an open neighborhood of x in \({\mathbb {B}}_{\epsilon }^n\), therefore the fibres of f near V are transverse to \(\mathbb {S}_{\epsilon }^{n-1}\). So there exists an open neighborhood W of \(V(f) \cap \mathbb {S}_{\epsilon }^{n-1}\) in \(\mathbb {S}_{\epsilon }^{n-1}\) such that any \(y \in W\) is a regular point of \(f_|\). So we just have to take \(\delta = \delta ({\epsilon })\) sufficiently small such that \(f^{-1}({\mathbb {B}}_\delta ^k) \cap \mathbb {S}_{\epsilon }^{n-1}\) is contained in W. \(\square\)

Example 2.9

Consider the real analytic map \(f:{\mathbb {R}}^4 \rightarrow {\mathbb {R}}^3\) given by

$$\begin{aligned} f(x,y,z,w):= (x^2-y^2z, y, w). \end{aligned}$$

Its critical set is the plane \(\{x=y=0\}\) in \({\mathbb {R}}^4\), and its discriminant is the axis \(\{u_1 = u_2 =0\}\) in \({\mathbb {R}}^3\), which is linear. By Propositions 2.7 and 2.8 it is locally surjective and it has the transversality property, but one can check that f is not d-regular.

Example 2.10

López de Medrano [17] studied the topology of real analytic maps \((f,g): {\mathbb {R}}^n \rightarrow {\mathbb {R}}^2\) such that both f and g are homogeneous quadratic polynomials. For (f, g) satisfying some generic hypothesis, he completely described the topology of \(V(f,g) = f^{-1}(0) \cap g^{-1}(0)\) in terms of the coeficients of f and g.

Let us restrict to the diagonal case, that is, when (f, g) has the form:

$$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^2 , \sum _{i=1}^n b_i x_i^2 \right) , \end{aligned}$$

where \(a_i\) and \(b_i\) are real constants in generic position. This means that the origin is in the convex hull of the points \((a_i,b_i)\) (which guarantees that the link of V(f) is non-empty) and that no two of the directions \((a_i,b_i)\) are linearly dependent, that is, \(a_i b_j \ne a_j b_i\), for any \(i \ne j\) (Weak Hyperbolicity Hypothesis).

A simple calculation shows that the set \(\Sigma\) of critical points of (f, g) coincides with the coordinate axis of \({\mathbb {R}}^n\) and the discriminant \(\Delta (f,g)\) is the union of the n line-segments in \({\mathbb {R}}^2\) joining the origin and the points \(\lambda _i:= (a_i,b_i)\). Hence (f, g) has linear discriminant. One can check that the hypothesis of Propositions 2.7 and 2.8 are satisfied, and therefore (f, g) is locally surjective and it has the transversality property.

We are going to show that (f, g) is d-regular. Let (u, v) be a coordinate system in \({\mathbb {R}}^2\). First, notice that any ray \(\mathcal {L}_\theta\) is given by one of the following forms: \(\{u + {\alpha }v =0\} \cap \{v> 0\}, \{u + {\alpha }v =0\} \cap \{v < 0\}, \{v=0\} \cap \{u > 0\} \, \textrm{or}\) \(\{v=0\} \cap \{u < 0\}\) for some \({\alpha }\in {\mathbb {R}}\). Hence any \(E_{\theta }\) has one of the following forms: \(\{f + {\alpha }g =0\} \cap \{g> 0\}, \, \{f + {\alpha }g =0\} \cap \{g < 0\}, \, \{g=0\} \cap \{f > 0\} \, \textrm{or }\) \(\{g=0\} \cap \{f < 0\}\) for some \({\alpha }\in {\mathbb {R}}\).

In any case, all the equations and inequations are homogeneous, so it is easy to verify that \(E_{\theta }\) intersects any sphere centered at the origin transversally.

The following example generalizes López de Medrano’s maps:

Example 2.11

Let \({\mathbb {K}}\) be either \({\mathbb {R}}\) or \({\mathbb {C}}\). Let \((f,g): {\mathbb {K}}^n \rightarrow {\mathbb {K}}^2\) be a \({\mathbb {K}}\)-analytic map of the form:

$$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p , \sum _{i=1}^n b_i x_i^p \right) , \end{aligned}$$

where \(a_i, b_i \in {\mathbb {K}}\) are constants in generic position (as in Example 2.10) and \(p \ge 2\) is an integer.

As before, one has that the critical set \(\Sigma\) of (f, g) is given by the coordinate axis of \({\mathbb {K}}^n\) and the discriminant \(\Delta\) is linear. Moreover, the same argument of the example above shows that (f, g) is d-regular.

Let us show that the link Z of \(V=(f,g)^{-1}(0)\) is non-empty, as before. In order to deal simultaneously with both the real and the complex case, set \(c=1\) if \({\mathbb {K}}={\mathbb {R}}\) and \(c=2\) if \({\mathbb {K}}={\mathbb {C}}\).

If p is odd, consider the homeomorphism:

$$\begin{aligned} \begin{array}{cccc} \phi _1: &{} \! {\mathbb {K}}^n &{} \! \longrightarrow &{} \! {\mathbb {K}}^n \\ &{} \! (x_1, \dots , x_n) &{} \! \longmapsto &{} \! (x_1^p, \dots , x_n^p) \\ \end{array}. \end{aligned}$$

Then (f, g) equals the composition \((f^1,g^1) \circ \phi _1\), where \((f^1,g^1)\) is the submersion given by:

$$\begin{aligned} (f^1,g^1)(x):= \left( \sum _{i=1}^n a_i x_i, \sum _{i=1}^n b_i x_i \right) , \end{aligned}$$

Hence V is homeomorphic to a \((n-2)\)-\({\mathbb {K}}\)-dimensional plane in \({\mathbb {K}}^n\) and Z is homeomorphic to the sphere \(\mathbb {S}^{c(n-2)-1}\).

If \(p=2r\) for some odd integer \(r\ge 3\), consider the homeomorphism:

$$\begin{aligned} \begin{array}{cccc} \phi _2: \, &{} \! {\mathbb {K}}^n &{} \! \longrightarrow &{} \! {\mathbb {K}}^n \\ &{} \! (x_1, \dots , x_n) &{} \! \longmapsto &{} \! (x_1^r, \dots , x_n^r) \\ \end{array}. \end{aligned}$$

So \((f,g) = (f^2,g^2) \circ \phi _2\), where \((f^2,g^2):= \left( \sum _{i=1}^n a_i x_i^2, \sum _{i=1}^n b_i x_i^2 \right)\) is López de Medrano’s map as in Example 2.10. Therefore Z is homeomorphic to the link of \((f^2,g^2)\), which is non-empty.

Finally, if \(p=2r\) for some even integer \(r\ge 2\), consider the map:

$$\begin{aligned} \begin{array}{cccc} \phi _3: \, &{} \! {\mathbb {K}}^n &{} \! \longrightarrow &{} \! {\mathbb {K}}^n \\ &{} \! (x_1, \dots , x_n) &{} \! \longmapsto &{} \! (x_1^r, \dots , x_n^r) \\ \end{array}, \end{aligned}$$

which is not a homeomorphism. Once again we have that \((f,g) = (f^2,g^2) \circ \phi _3\). Hence \(V = (\phi _3)^{-1}({\hat{V}})\), where \({\hat{V}}:= (f^2,g^2)^{-1}(0)\) has dimension greater than zero. So the dimension of V is bigger than zero. Again, by Propositions 2.7 and 2.8 (f, g) is locally surjective and it has the transversality property.

2.2 Characterizations of d-regularity

Now we give a characterization of d-regularity for maps with linear discriminant, based on some definitions and preliminary results from [7, §3].

Let \(0<{\epsilon }<{\epsilon }_0\) be such that f has linear discriminant and it is d-regular in the ball \({\mathbb {B}}_{\epsilon }^n\). Set \(W:=f^{-1}(\Delta _{\epsilon })\) and consider the maps \(\Phi :{\mathbb {B}}_{\epsilon }^n{{\setminus }} W\rightarrow \mathbb {S}^{k-1} {{\setminus }} \mathcal {A}\) and \({\mathfrak {F}}:{\mathbb {B}}_{\epsilon }^n{{\setminus }} W\rightarrow {\mathbb {R}}^k{{\setminus }}\Delta _{\epsilon }\) given respectively by

$$\begin{aligned} \Phi (x)=\frac{f(x)}{\Vert f(x)\Vert },\quad \text {and}\quad {\mathfrak {F}}=\Vert x\Vert \Phi (x). \end{aligned}$$

Notice that given \(y\in \mathcal {L}_\theta\) with \(\theta \in \mathbb {S}^{k-1}_\eta {{\setminus }} \mathcal {A}_\eta\) the fibre \({\mathfrak {F}}^{-1}(y)\) is the intersection of \(E_\theta\) with the sphere of radius \(\Vert y\Vert\) centred at 0. Each \(E_\theta\) is a union of fibres of \({\mathfrak {F}}\), just as it is a union of fibres of f, so we have

$$\begin{aligned} E_\theta =f^{-1}(\mathcal {L}_\theta )={\mathfrak {F}}^{-1}(\mathcal {L}_\theta ). \end{aligned}$$
(6)

The map \({\mathfrak {F}}\) is called the spherefication of f.

The following proposition is a straightforward generalization of [7, Proposition 3.2].

Proposition 2.12

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map with linear discriminant. Let \({\epsilon }_0\) be a Milnor radius for f and let \(0<{\epsilon }<{\epsilon }_0\). The following conditions are equivalent

  1. (a)

    The map f is d-regular in \({\mathbb {B}}_{\epsilon }^n\).

  2. (b)

    For each sphere \(\mathbb {S}_{{\epsilon }'}^{n-1}\) in \({\mathbb {R}}^n\) centred at 0 of radius \({\epsilon }' < {\epsilon }\), the restriction map \({\mathfrak {F}}_{{\epsilon }'}:\mathbb {S}_{{\epsilon }'}^{n-1} {{\setminus }} W\rightarrow \mathbb {S}^{k-1}_{{\epsilon }'}{{\setminus }}\Delta _{\epsilon }\) of \({\mathfrak {F}}\) is a submersion.

  3. (c)

    The spherefication map \({\mathfrak {F}}\) is a submersion.

  4. (d)

    The map \(\phi =\frac{f}{\Vert f\Vert }:\mathbb {S}_{{\epsilon }'}^{n-1} {{\setminus }} W\longrightarrow \mathbb {S}^{k-1}{{\setminus }} \mathcal {A}\) is a submersion for every sphere \(\mathbb {S}_{{\epsilon }'}^{n-1}\) with \({\epsilon }'<{\epsilon }\).

We can now state the fibration theorem on the sphere (compare with [24, Thm. 6.1.6]):

Theorem 2.13

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) with \(n\ge k\ge 2\) be an analytic map with a critical point at 0. Let \({\epsilon }_0\) be a Milnor radius for f and let \(0<{\epsilon }<{\epsilon }_0\). Suppose f has linear discriminant and the transversality property in \(\mathbb {B}_{\epsilon }^n\). If f is d-regular in \(\mathbb {B}_{\epsilon }^n\), then the restriction of \(\Phi\) given by

$$\begin{aligned} \phi =\Phi |:\mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W \rightarrow \mathbb {S}^{k-1} {{\setminus }} \mathcal {A} \end{aligned}$$
(7)

is a smooth locally trivial fibration over its image, where \(W:= f^{-1}(\Delta _{\epsilon })\).

Proof

Suppose that f is locally surjective (the general case is analogous). Set

$$\begin{aligned} \mathcal {M}:= \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W . \end{aligned}$$

Notice that \(\mathcal {M}\) is an open submanifold of \(\mathbb {S}_{\epsilon }^{n-1}\) since \(\Delta _{\epsilon }\) is closed in \({\mathbb {R}}^k\). Consider the following decomposition

$$\begin{aligned} \mathcal {M} = \left( \mathcal {M} \cap f^{-1}({\mathbb {B}}_\delta ^k) \right) \cup \left( \mathcal {M} {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k) \right) , \end{aligned}$$

where \(\mathring{{\mathbb {B}}}_{\delta }^k\) is the interior of the closed ball \({\mathbb {B}}_{\delta }^k\). Both pieces are submanifolds-with-boundary of \(\mathcal {M}\) of dimension \(n-1\) and their intersection is the common boundary submanifold of dimension \(n-2\)

$$\begin{aligned} \mathbb {S}_{\epsilon }^{n-1} \cap f^{-1}(\mathbb {S}_\delta ^{k-1}{{\setminus }}\Delta _{\epsilon })= \left( \mathcal {M} \cap f^{-1}({\mathbb {B}}_\delta ^k) \right) \cap \left( \mathcal {M} {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k) \right) . \end{aligned}$$

We are going to show that the restriction of \(\phi\) to each of these components is a smooth fibre bundle, so they can be glued into a global fibre bundle.

The restriction of f given by \(f_1:\mathcal {M} \cap f^{-1}({\mathbb {B}}_\delta ^k)\rightarrow {\mathbb {B}}_{\delta }^k{{\setminus }} \Delta _{\epsilon }\) is proper since \(\mathbb {S}_{\epsilon }^{n-1}\cap f^{-1}({\mathbb {B}}_\delta ^k)\) is compact, and since f has the transversality property in the ball \(\mathbb {B}_{\epsilon }^n\) it is a submersion, and by Ehresmann fibration theorem it is a smooth fibre bundle. Now consider the radial projection \({\tilde{\pi }}:{\mathbb {B}}_{\delta }^k{{\setminus }} \Delta _{{\epsilon }} \rightarrow \mathbb {S}^{k-1}{{\setminus }}\mathcal {A}\) which is a (trivial and smooth) fibre bundle. The restriction

$$\begin{aligned} \phi _1:\left( \mathcal {M} \cap f^{-1}({\mathbb {B}}_\delta ^k) \right) \rightarrow \mathbb {S}^{k-1} {{\setminus }} \mathcal {A} \end{aligned}$$

of \(\phi\) is given by the composition \({\tilde{\pi }}\circ f_1\).

So now we just have to show that the restriction:

$$\begin{aligned} \phi _2: \mathcal {M} {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k) \rightarrow \mathbb {S}^{k-1} {{\setminus }} \mathcal {A} \end{aligned}$$

is a fibration. We have that \(\phi _2\) is proper since \(\mathbb {S}_{\epsilon }^{n-1} {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k)\) is compact.

Since f is d-regular, the map \(\phi :\mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\rightarrow \mathbb {S}^{k-1}{{\setminus }} \mathcal {A}\) has no critical points, by Proposition 2.12. So \(\phi _2\) is a submersion restricted to the interior \(\mathcal {M} {{\setminus }} f^{-1}({\mathbb {B}}_\delta ^k)\) of \(\mathcal {M} {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k)\). Since \(\phi _1\) and \(\phi _2\) coincide on the boundary \(\mathcal {M} \cap f^{-1}(\mathbb {S}_\delta ^{k-1})\) and we already know that \(\phi _1\) restricted to this boundary is a submersion, the result follows from the Ehresmann’s fibration lemma. \(\square\)

Remark 2.14

In the case of f with isolated critical value and transversality property, we have \(\Delta _{\epsilon }=\{0\}\) and \(\mathcal {A}=\emptyset\) and Theorem 2.13 in this case gives another proof of the existence of the fibration on the sphere \(\phi =\frac{f}{\Vert f\Vert }:\mathbb {S}_{\epsilon }^{n-1}{{\setminus }} f^{-1}(0)\rightarrow \mathbb {S}^{k-1}\) without “inflating” the fibration (3) on the tube (compare with [7, Theorem 5.3 (2)]).

2.3 Equivalence of fibrations

As before, let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map with a critical point at 0, linear discriminant and the transversality property in \(\mathbb {B}_{\epsilon }^n\).

Consider the locally trivial fibration (3). Composing it with the radial projection \({\tilde{\pi }}:{\mathbb {B}}_{\delta }^k{{\setminus }} \Delta _{{\epsilon }} \rightarrow \mathbb {S}^{k-1}{{\setminus }}\mathcal {A}\) we get a (equivalent) locally trivial fibration

$$\begin{aligned} {\tilde{f}}:={\tilde{\pi }}\circ f|:{\mathbb {B}}_{{\epsilon }}^n \cap f^{-1}(\mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{{\epsilon }}) \rightarrow \mathbb {S}^{k-1} {{\setminus }} \mathcal {A}. \end{aligned}$$
(8)

We want to show that this locally trivial fibration (restricted to the open ball) is equivalent to the locally trivial fibration (7), provided that f is d-regular. For this, we use the following generalization of a characterization of d-regularity given in [4, Theorem 3.7] to the case of f with linear discriminant.

Proposition 2.15

([7, Lemma 5.2]) The map f is d-regular in \({\mathbb {B}}_{\epsilon }^n\) if and only if there exists an analytic vector field \({\tilde{w}}\) on \(\mathring{{\mathbb {B}}}_{{\epsilon }} {{\setminus }} f^{-1}(\Delta _{\epsilon })\) which has the following properties:

  1. (1)

    It is radial, i.e., it is transverse to all spheres in \(\mathring{{\mathbb {B}}}_{{\epsilon }}\) centred at 0.

  2. (2)

    It is transverse to all the tubes \(f^{-1}(\mathbb {S}_\delta ^{k-1}{{\setminus }}\Delta _{\epsilon })\).

  3. (3)

    It is tangent to each \(E_\theta\) for \(\theta \notin \mathcal {A}_\eta\), whenever it is not empty.

Proof

Same as the proof of [5, Theorem 3.8] but replacing \(\mathring{{\mathbb {B}}}_{{\epsilon }}^n{\setminus } V\) by \(\mathring{{\mathbb {B}}}_{{\epsilon }}^n{\setminus } f^{-1}(\Delta _{\epsilon })\), see [5, Remark 3.10]. \(\square\)

Now we can use the vector field \({\tilde{w}}\) of Proposition 2.15 to give the equivalence of fibrations.

Theorem 2.16

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map with a critical point at 0 and linear discriminant. Suppose also that f has the transversality property and that is d-regular in \(\mathbb {B}_{\epsilon }^n\). Denote by \(\mathring{{\mathbb {B}}}_{\epsilon }^n\) the interior of the ball \({\mathbb {B}}_{\epsilon }^n\). The fibre bundles

$$\begin{aligned}&{\tilde{f}}:=\pi _\delta \circ f_{\epsilon }|:\mathring{{\mathbb {B}}}_{\epsilon }^n \cap f^{-1}(\mathbb {S}_\delta ^{k-1} {\setminus } \mathcal {A}_\delta ) \rightarrow \mathbb {S}^{k-1} {\setminus } \mathcal {A} \\&\quad \text {and}\quad \phi :\mathbb {S}_{\epsilon }^{n-1} {\setminus } f^{-1}(\Delta _{\epsilon }) \rightarrow \mathbb {S}^{k-1} {\setminus } \mathcal {A} \end{aligned}$$

are equivalent, where \(\pi _\delta :\mathbb {S}_{\delta }^{k-1} \rightarrow \mathbb {S}^{k-1}\) is the radial projection.

Proof

In the proof of Theorem 2.13 we saw that the restriction of \(\phi\) given by \(\phi |:\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k)\rightarrow \mathbb {S}^{k-1} {{\setminus }} \mathcal {A}\) is a fibre bundle. The flow associated to the vector field \({\tilde{w}}\) of Proposition 2.15 defines in the usual way a diffeomorphism \(\tau\) between \({\mathbb {B}}_{\epsilon }^n \cap f^{-1}(\mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon })\) and \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k)\): for a point \(x\in {\mathbb {B}}_{\epsilon }^n \cap f^{-1}(\mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon })\) follow the solution of \({\tilde{w}}\) that passes through x till it meets \(\mathbb {S}_{\epsilon }^{n-1}\) at some point \({\hat{x}}\). This point exists and is unique because \({\tilde{w}}\) satisfies conditions (1) and (2) in Proposition 2.15. Define \(\tau (x)={\hat{x}}\). By condition (3) in Proposition 2.15 the solutions of \({\tilde{w}}\) lie in an \(E_\theta\), then we have that \({\tilde{f}}(x):=\Phi (x)=\Phi ({\hat{x}})=:\phi ({\hat{x}})\). Therefore, the diffeomorphism \(\tau :{\mathbb {B}}_{\epsilon }^n \cap f^{-1}(\mathbb {S}_\delta ^{k-1} {{\setminus }} \Delta _{\epsilon })\rightarrow \left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k)\) gives an equivalence of fibre bundles

The fibres of fibration (8) are compact, while those of (7) are open manifolds. To have an actual isomorphism of the two fibrations one must restrict the fibration (8) to the open ball.

To complete the proof we must show that the fibration given by \(\phi\) on \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}({\mathbb {B}}_\delta ^k)\), whose fibres are open manifolds, is equivalent to fibration (7). One can see in the proof of Theorem 2.13 that the restriction of f given by \(f_1:\mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W \cap f^{-1}({\mathbb {B}}_\delta ^k)\rightarrow {\mathbb {B}}_{\delta }^k{{\setminus }} \Delta _{\epsilon }\) is a submersion. Hence we can lift the radial vector field \(u(z) = z\) on \({\mathbb {B}}_{\delta }^k{{\setminus }} \Delta _{\epsilon }\) to a smooth vector field on \(\mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W \cap f^{-1}({\mathbb {B}}_\delta ^k)\) whose flow preserves the fibres of \(\phi\) and is transverse to the intersection with \(\mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\) of all the Milnor tubes \(f^{-1}(\mathbb {S}_{\delta '}^{k-1} {{\setminus }} \Delta _{\epsilon })\) for all \(0 < \delta ' \le \delta\). The flow associated to this vector field carries diffeomorphically the intersection \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathbb {S}_{\delta '}^{k-1})\) to \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathbb {S}_\delta ^{k-1})\) and the total space \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_{\delta '}^k)\) to \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}(\mathring{{\mathbb {B}}}_\delta ^k)\) for all \(0 < \delta ' \le \delta\). Therefore, if we remove the boundary, the fibration \(\left( \mathbb {S}_{\epsilon }^{n-1} {{\setminus }} W\right) {{\setminus }} f^{-1}({\mathbb {B}}_\delta ^k)\rightarrow \mathbb {S}^{k-1}{{\setminus }} \mathcal {A}\) is equivalent to fibration (7). \(\square\)

Now we want to apply Theorem 2.16 in Example 2.11.

Corollary 2.17

Any map \((f,g):({\mathbb {K}}^n,0) \rightarrow ({\mathbb {K}}^2,0)\) as in Example 2.11 satisfies all the hypothesis of Theorem 2.16, hence it induces equivalent fibrations on the tube and the sphere.

3 A regularity condition for maps with arbitrary discriminant

In this section, we want to extend the concept of d-regularity for analytic maps with arbitrary discriminant.

3.1 Conical homeomorphisms

Recall that given \(\eta >0\), for each point \(\theta \in \mathbb {S}_\eta ^{k-1}\) the set \(\mathcal {L}_\theta \subset {\mathbb {R}}^k\) is the open segment of line that starts in the origin and ends at the point \(\theta\) (but not containing these two points).

Definition 3.1

Let \(h:({\mathbb {R}}^k,0) \longrightarrow ({\mathbb {R}}^k,0)\) be a homeomorphism. Suppose that there exists \(\eta >0\) sufficiently small, such that the restriction of h to the ball \({\mathbb {B}}_\eta ^k\)

$$\begin{aligned} h_\eta :{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k , \end{aligned}$$
(9)

with \(\mathcal {B}_\eta ^k:= h({\mathbb {B}}_\eta ^k)\), satisfies the following:

  1. (i)

    For each \(\theta \in \mathbb {S}_\eta ^{k-1}\) the image \(h_\eta (\mathcal {L}_\theta )\) is a smooth path in \({\mathbb {R}}^k\);

  2. (ii)

    The inverse map \(h^{-1}\) of h is smooth outside the origin;

  3. (iii)

    The map \(h^{-1}\) is a submersion outside the origin.

We say that the restriction (9) is a conic homeomorphism.

The identity map is obviously a conic homeomorphism. Given a conic homemorphism \(h_\eta :{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\), set:

$$\begin{aligned} \varphi _{h,\theta }:= h(\mathcal {L}_\theta ), \end{aligned}$$

for each \(\theta \in \mathbb {S}_\eta ^{k-1}\).

Example 3.2

Consider the map \(h(u,v)=(u,\root 3 \of {v})\), whose inverse is given by \(h^{-1}(u,v)=(u,v^3)\). Clearly, \(h^{-1}\) is a differentiable homeomorphism, although h is not differentiable at the u-axis. It is easy to check that h is a conic homeomorphism.

Example 3.3

Now consider the map \(h:({\mathbb {R}}^2,0) \rightarrow ({\mathbb {R}}^2,0)\) given by:

$$\begin{aligned} h(u,v)= {\left\{ \begin{array}{ll} \left( \sqrt{u}, \frac{v}{2}\right) , &{}\text {if}\; u \ge 0; \\ \left( -\sqrt{-u}, \frac{v}{2}\right) , &{}\text {if}\; u < 0. \end{array}\right. } \end{aligned}$$

whose inverse is given by

$$\begin{aligned} h^{-1}(u,v)= {\left\{ \begin{array}{ll} (u^2, 2v), &{}\text {if}\; u \ge 0; \\ (-u^2, 2v), &{}\text {if}\; u < 0. \end{array}\right. } \end{aligned}$$

One can check that h is a conic homeomorphism and the inverse map \(h^{-1}\) is not.

If \(h:{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\) is a conic homeomorphism, then each point \(x \in \mathcal {B}_\eta ^k\) belongs to some smooth path \(\varphi _{h,\theta (x)}\). Precisely, \(\theta (x) = \frac{h^{-1}(x)}{\Vert h^{-1}(x)\Vert }\).

So if we set

$$\begin{aligned} \mathcal {S}_\eta ^{k-1}:= h(\mathbb {S}_\eta ^{k-1}) , \end{aligned}$$

the conic homeomorphism h induces a continuous and surjective map

$$\begin{aligned} \begin{array}{cccc} \xi _h \, &{} \! \mathcal {B}_\eta ^k {\setminus } \{0\} &{} \! \longrightarrow &{} \! \mathcal {S}_\eta ^{k-1} \\ &{} \! x &{} \! \longmapsto &{} \! h \big ( \eta \frac{h^{-1}(x)}{\Vert h^{-1}(x)\Vert } \big ) \end{array} \,. \end{aligned}$$

That is, \(\xi _h(x)\) is the point where the smooth curve \(\varphi _{h,\theta (x)}\) that contains x intersects \(\mathcal {S}_\eta ^{k-1}\). In other words, \(\xi _h(x)\) sends each smooth curve \(\varphi _{h,\theta }\) to the point \(h(\theta ) \in \mathcal {S}_\eta ^{k-1}\).

So for each \(\theta\) in \(\mathbb {S}_{\eta }^{k-1}\) we have that

$$\begin{aligned} h(\mathcal {L}_\theta ):= \varphi _{h,\theta } = \xi _h^{-1}(h(\theta )) . \end{aligned}$$

3.2 The \(d_h\)-regularity

Given a map \(f:(U,0) \rightarrow ({\mathbb {R}}^k,0)\), U an open neighborhood of 0 in \({\mathbb {R}}^n\), and a conic homeomorphism \(h: ({\mathbb {R}}^k,0) \rightarrow ({\mathbb {R}}^k,0)\) we define

$$\begin{aligned} f_h:= h^{-1} \circ f , \end{aligned}$$

which we call a conic modification of f.

Once \(\eta >0\) is fixed, for each \(\theta\) in \(\mathbb {S}_{\eta }^{k-1}\) we set

$$\begin{aligned} E_{f,h,\theta }:= f^{-1}(\varphi _{h,\theta }) . \end{aligned}$$

Remark 3.4

For each \(\theta \in \mathbb {S}_{\eta }^{k-1}\) we have that

$$\begin{aligned} E_{f,h,\theta } = (f_h)^{-1}(\mathcal {L}_\theta ) = \left( \eta \frac{f_h}{\Vert f_h \Vert } \right) ^{-1}(\theta ) = E_{f_h,id,\theta } . \end{aligned}$$

One should also notice that \(E_{f,h,\theta }\) may be the empty set for some (or for any) \(\theta \in \mathbb {S}_{\eta }^{k-1}\), since f is not necessarily surjective.

Definition 3.5

We say that an analytic map \(f:(U,0) \rightarrow ({\mathbb {R}}^k,0)\) and a conic homeomorphism \(h: {\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\) are compatible in a subset \(\mathcal {C} \subset \mathbb {S}_{\eta }^{k-1}\) if \(E_{f,h,\theta }\) is a smooth manifold, for each \(\theta \in \mathcal {C}\). We say that f and h are compatible if f and h are compatible in \(\mathbb {S}_{\eta }^{k-1}\).

Remark 3.6

Note that if f has an isolated critical value, then any conic homeomorphism h is compatible with f, since each \(E_{f,h,\theta }\) is the inverse image by f of a smooth path \(\varphi _{h,\theta }\) that contains no critical value.

Now we can define d-regularity up to homemorphism:

Definition 3.7

(\(d_h\) -regularity) Let \(f:(U,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map. Let \({\epsilon }_0\) be a Milnor radius for f and let \(0<{\epsilon }<{\epsilon }_0\). Let \(h:{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\) be a conic homeomorphism compatible with f in \(\mathcal {C} \subset \mathbb {S}_{\eta }^{k-1}\) such that \(f({\mathbb {B}}_{{\epsilon }_0}^n) \subset \mathring{\mathcal {B}}_\eta ^k\). We say that f is \(d_h\)-regular relative to \(\mathcal {C}\) in \({\mathbb {B}}_{\epsilon }^n\) if for any \(0<{\epsilon }' \le {\epsilon }\) the sphere \(\mathbb {S}_{{\epsilon }'}^{n-1}\) intersects \(E_{f,h,\theta }\) transversally, whenever such intersection is not empty for each \(\theta \in \mathcal {C}\). If \(\mathcal {C} = \mathbb {S}_{\eta }^{k-1}\) we say that f is \(d_h\)-regular in \({\mathbb {B}}_{\epsilon }^n\).

Since by Remark 3.4 we have that \(E_{f,h,\theta } = E_{f_h,id,\theta }\) we get:

Proposition 3.8

Let \(f:(U,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map and let \(h:{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\) be a conic homeomorphism.

  1. (i)

    The map f is compatible with h in \(\mathcal {C}\) if and only if the conic modification \(f_h:= h^{-1} \circ f\) is compatible with the identity map in \(\mathcal {C}\).

  2. (ii)

    Suppose that f and h are compatible in \(\mathcal {C} \subset \mathbb {S}_{\eta }^{k-1}\). Then f is \(d_h\)-regular relative to \(\mathcal {C}\) in \({\mathbb {B}}_{\epsilon }^n\) if and only if \(f_h\) is d-regular relative to \(\mathcal {C}\) in \({\mathbb {B}}_{\epsilon }^n\).

Hence the definition of \(d_h\)-regularity above generalizes the definition of d-regularity, in the sense that an analytic map with linear discriminant (as in Sect. 2) is d-regular if and only if it is \(d_{id}\)-regular relative to \(\mathcal {C}:= \mathbb {S}_\eta ^{k-1} {{\setminus }} \mathcal {A}_\eta\).

Example 3.9

Let \(f:({\mathbb {R}}^3,0) \rightarrow ({\mathbb {R}}^2,0)\) be the analytic map given by

$$\begin{aligned} f(x,y,z):= (x^2z+y^3, x) . \end{aligned}$$

As pointed out in Example 3.9 of [9], the map f has an isolated critical value, but along the line \(L:= \{x-z=0 \, \ y=0 \}\), the spaces \(E_{f,id,\theta }\) are not transversal to the corresponding spheres. So f is not d-regular.

Now consider the conic homeomorphisms \(h(u,v) = (u,\root 3 \of {v})\) and its inverse \(h^{-1}(u,v) = (u,v^3)\) of Example 3.2. By Remark 3.6 we have that f is compatible with both h and \(h^{-1}\).

Let us show that f is \(d_{(h)}\)-regular:

$$\begin{aligned} f_{(h)}(x,y,z) = (x^2z+y^3, x^3) , \end{aligned}$$

it is an analytic map, whose critical set is the plane \(\{x=0\}\), and its discriminant is the line \(\{v=0\}\) in \({\mathbb {R}}^2\). Using [9, Proposition 3.8] one can check that \(f_{(h)}\) is d-regular and hence f is \(d_{(h)}\)-regular.

This leads us to the following:

Question 3.10

Is there always a conic modification of a real analytic map f with isolated critical value, which is d-regular?

Now we are going to study the possibility to modify the discriminant of a map by homeomorphism, making it linear. Then one can check if the new map is d-regular, and in that case we establish a fibration theorem. This process applies in particular to maps with isolated critical value and the transversality property.

Let \(f_{\epsilon }\) and \(\Delta _{\epsilon }\) be as before.

Definition 3.11

We say that a conic homeomorphism \(h:{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\) is a linearization for \(f_{\epsilon }\) if

$$\begin{aligned} h^{-1}(\Delta _{\epsilon }\cap \mathcal {B}_\eta ^k) = {{\,\textrm{Cone}\,}}(h^{-1}(\Delta _{\epsilon }\cap \partial \mathcal {B}_\eta ^k)) . \end{aligned}$$

Given a linearization h for an analytic map f, we have the smooth paths

$$\begin{aligned} \varphi _{h,\theta } = h(\mathcal {L}_\theta ) \end{aligned}$$

and the induced map

$$\begin{aligned} \xi _{h}:{\mathbb {B}}_{\eta }^k {\setminus } \{0\} \rightarrow \mathbb {S}_{\eta }^{k-1} , \end{aligned}$$

where \(\xi _{h}(x)\) is the point where the closure of the path \(\varphi _{h,\theta }\) that contains x intersects \(\mathbb {S}_{\eta }^{k-1}\). So \(\varphi _{h,\theta } = (\xi _h)^{-1}(\theta )\).

Also, given a linearization h for \(f_{\epsilon }\), we set

$$\begin{aligned} \mathcal {A}_{h,\eta }:= h^{-1}(\Delta _{\epsilon }\cap \partial \mathcal {B}_\eta ^k) = h^{-1}(\Delta _{\epsilon }) \cap \mathbb {S}_\eta ^{k-1} . \end{aligned}$$

Finally, for each \(\theta\) in \(\mathbb {S}_{\eta }^{k-1}\) we have

$$\begin{aligned} E_{f,h,\theta }:= f^{-1}(\varphi _{h,\theta }) , \end{aligned}$$

which is a smooth manifold if \(\theta \notin \mathcal {A}_{h,\eta }\). Recall that

$$\begin{aligned} E_{f,h,\theta } = (\xi _{h} \circ f)^{-1}(\theta ) = f^{-1}(h(\mathcal {L}_\theta )) . \end{aligned}$$

So any linearization h for an analytic map \(f_{\epsilon }\) is compatible with \(f_{\epsilon }\) in \(\mathbb {S}_{\eta }^{k-1} {\setminus } \mathcal {A}_{h,\eta }\).

Once again, let \(\pi :\mathbb {S}_\eta ^{k-1}\rightarrow \mathbb {S}^{k-1}\) be the projection onto the unit sphere \(\mathbb {S}^{k-1}\) and set \(\mathcal {A}_h=\pi (\mathcal {A}_{h,\eta })\). Recall the smooth map

$$\begin{aligned} \phi _{f,h}:\mathbb {S}_{\epsilon }^{n-1} {\setminus } K_f \rightarrow \mathbb {S}^{k-1} \end{aligned}$$

given by

$$\begin{aligned} (\phi _{f,h})(x) = \frac{f_h(x)}{\Vert f_h(x)\Vert } . \end{aligned}$$

As before, using Proposition 3.8 together with Theorems 2.13 and 2.16, we have:

Theorem 3.12

Let \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) be an analytic map with the transversality property in \({\mathbb {B}}_{\epsilon }^n\), and suppose it admits a linearization \(h:{\mathbb {B}}_\eta ^k \rightarrow \mathcal {B}_\eta ^k\) making f \(d_h\)-regular in \({\mathbb {B}}_{\epsilon }^n\). Then the restriction

$$\begin{aligned} (\phi _{f,h})|:\mathbb {S}_{\epsilon }^{n-1} {\setminus } f^{-1}(\Delta _{\epsilon }) \rightarrow \mathbb {S}^{k-1} {\setminus } \mathcal {A}_h \end{aligned}$$

is the projection of a smooth locally trivial fibration over its image, and this is equivalent to the fibration (3) above.

The following example generalizes Example 2.11:

Example 3.13

Let \((f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2\) be a real analytic map of the form

$$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p, \sum _{i=1}^n b_i x_i^q \right) \,, \end{aligned}$$

with \(p,q \ge 2\) integers, and \(a_i, b_i \in {\mathbb {R}}\) are non-zero constants such that the directions \(\lambda _i = (a_i,b_i)\) satisfy the Weak Hyperbolicity Hypothesis (that is, no two of them are linearly dependent and therefore \(a_ib_j\ne a_jb_i\) for all \(i\ne j\)). Example 2.11 concerns the case \(p=q\).

We begin computing the critical points. Let \(\textbf{x}=(x_1,\dots ,x_n)\in {\mathbb {R}}^n\). The Jacobian matrix at \(\textbf{x}\) is given by

$$\begin{aligned} D(f,g)_\textbf{x}=\left( \begin{array}{lll} pa_1x_1^{p-1} &{}\quad \cdots &{}\quad pa_nx_n^{p-1}\\ qb_1x_1^{q-1} &{}\quad \cdots &{}\quad qb_nx_n^{q-1} \end{array}\right) \end{aligned}$$

Hence, \(\textbf{x}\) is a critical point if all the minors

$$\begin{aligned} M_{ij}=\left| \begin{array}{ll} pa_ix_i^{p-1} &{}\quad pa_jx_j^{p-1}\\ qb_ix_i^{q-1} &{}\quad qb_jx_j^{q-1} \end{array}\right| =pq(a_ib_jx_i^{p-1}x_j^{q-1}-a_jb_ix_i^{q-1}x_j^{p-1}) \end{aligned}$$
(10)

are zero for any \(i\ne j\), with \(1\le i,j\le n\). The minor (10) is zero if \(x_i\) or \(x_j\) are zero, hence the origin is a critical point. If both \(x_i\) and \(x_j\) are non zero, (10) is zero if and only if

$$\begin{aligned} a_ib_jx_i^{p-1}x_j^{q-1}=a_jb_ix_i^{q-1}x_j^{p-1}, \end{aligned}$$

which is equivalent to have

$$\begin{aligned} x_j=\left( \frac{a_ib_j}{a_jb_i}\right) ^{\frac{1}{p-q}}x_i. \end{aligned}$$
(11)

For each \(\textbf{x}=(x_1,\dots ,x_n)\in {\mathbb {R}}^n\) define the element \(I_\textbf{x}=(\sigma _1,\dots ,\sigma _n)\in \{0,1\}^n\) by

$$\begin{aligned} \sigma _i={\left\{ \begin{array}{ll} 0 &{} \text {if}\; x_i=0,\\ 1 &{} \text {if}\; x_i\ne 0.\\ \end{array}\right. } \end{aligned}$$

We claim that for every element \(I\in \{0,1\}^n{\setminus }\{(0,\dots ,0)\}\) there is a line of critical points of the map (f, g).

Let \(I\in \{0,1\}^n{\setminus }\{(0,\dots ,0)\}\) and suppose \(\textbf{x}_I=(x_1,\dots ,x_n)\) is a critical point of (f, g) such that \(I_\mathbf {x_I}=I\). We want to find an explicit expression for \(\textbf{x}_I\). Let \(1\le k\le n\) be such that \(\sigma _k=1\) and \(\sigma _i=0\) for all \(i<k\), that is \(\sigma _k\) is the first non-zero entry of I, or equivalently, \(x_k\) is the first non-zero entry of \(\textbf{x}\). Set \(x_k=t\), by (11) we have that in order that the minors \(M_{kj}\) given by (10) vanish, for \(k<j\le n\), we need to have \(x_j=0\) or if \(x_j\ne 0\) then

$$\begin{aligned} x_j=\left( \frac{a_ib_j}{a_jb_i}\right) ^{\frac{1}{p-q}}x_k=\left( \frac{a_ib_j}{a_jb_i}\right) ^{\frac{1}{p-q}}t. \end{aligned}$$

So we get

$$\begin{aligned} \textbf{x}_I=(0,\dots ,t,\dots ,\sigma _{j}\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{1}{p-q}}t,\dots , \sigma _{n}\left( \frac{a_kb_{n}}{a_{n}b_k}\right) ^{\frac{1}{p-q}}t),\quad k<j\le n. \end{aligned}$$
(12)

To see that \(\textbf{x}_I\) is indeed a critical point of (f, g) we still need to check that all the minors \(M_{ij}\) with \(i\ne j\) and \(k<i,j\le n\) vanish. Suppose both \(x_i\) and \(x_j\) are non zero, then the minor \(M_{ij}\) is given by

$$\begin{aligned} M_{ij}=pq\left( a_ib_j\left( \frac{a_kb_{i}}{a_{i}b_k}\right) ^{\frac{p-1}{p-q}} \left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{q-1}{p-q}}-a_jb_i\left( \frac{a_kb_{i}}{a_{i}b_k}\right) ^{\frac{q-1}{p-q}}\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{p-1}{p-q}}\right) t^{p+q-2}. \end{aligned}$$

Dividing by \(pq\left( \frac{a_kb_{i}}{a_{i}b_k}\right) ^{\frac{q-1}{p-q}}\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{p-1}{p-q}}t^{p+q-2}\) we get

$$\begin{aligned} a_ib_j\left( \frac{a_kb_{i}}{a_{i}b_k}\right) ^{\frac{p-q}{p-q}}\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{q-p}{p-q}}-a_jb_i&=a_ib_j\left( \frac{a_kb_{i}}{a_{i}b_k}\right) \left( \frac{a_{j}b_k}{a_kb_{j}}\right) -a_jb_i\\&=b_ia_j-a_jb_i=0. \end{aligned}$$

Thus, given \(I\in \{0,1\}^n{\setminus }\{(0,\dots ,0)\}\) we have a line of critical points of (f, g) given by

$$\begin{aligned} t(0,\dots ,\underbrace{1}_k,\dots ,\sigma _{j}\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{1}{p-q}}, \dots ,\sigma _{n}\left( \frac{a_kb_{n}}{a_{n}b_k}\right) ^{\frac{1}{p-q}}),\quad k<j\le n,\quad t\in {\mathbb {R}}. \end{aligned}$$

Therefore, the set \(\Sigma\) of critical points consists of \(2^n-1\) lines in \({\mathbb {R}}^n\).

The critical value corresponding to \(\textbf{x}_I\) given in (12) is

$$\begin{aligned} \left( \left( a_k+\sum _{j=k+1}^n\sigma _ja_j\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{p}{p-q}}\right) t^p,\left( b_k+\sum _{j=k+1}^n\sigma _jb_j\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{q}{p-q}}\right) t^q\right) \end{aligned}$$
(13)

Set

$$\begin{aligned} A_I&=\left( a_k+\sum _{j=k+1}^n\sigma _ja_j\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{p}{p-q}}\right) ,\nonumber \\ B_I&=\left( b_k+\sum _{j=k+1}^n\sigma _jb_j\left( \frac{a_kb_{j}}{a_{j}b_k}\right) ^{\frac{q}{p-q}}\right) . \end{aligned}$$
(14)

Thus, by (13) the discriminant \(\Delta\) is the union of the parametrized curves \((A_I t^p, B_I t^q)\) in \({\mathbb {R}}^2\) (with \(t \in {\mathbb {R}}\)), for each \(I\in \{0,1\}^n{\setminus }\{(0,\dots ,0)\}\). It follows from Proposition 2.8 that (f, g) has the transversality property.

Moreover, if \(V(f,g):= (f,g)^{-1}(0)\) has dimension bigger than zero, it follows from Propositions 2.7 that (f, g) is locally surjective. If \(V(f,g) = \{0\}\), then (f, g) may not be locally surjective. Recall from Example 2.11 that if \(p=q\), then \(V(f,g) = \{0\}\) if and only if the origin is not in the convex hull of the points \(\lambda _i = (a_i,b_i)\).

We must consider four cases:

  1. (1)

    Suppose that both p and q are odd:

    Consider the germ of conic homeomorphism \(h(u,v) = (u^{1/q}, v^{1/p})\), whose inverse is given by \(h^{-1}(u,v) = (u^q, v^p)\). It is a linearization for the map (f, g), since for each \(i=1, \dots , n\) we have

    $$\begin{aligned} h^{-1}(A_I t^p,B_I t^q) = \big (A_I^q t^{pq},B_I^p t^{pq} \big ). \end{aligned}$$

    Let us show that (f, g) is \(d_h\)-regular. The conic modification \((f,g)_h:= h^{-1} \circ (f,g)\) is given by

    $$\begin{aligned} (f,g)_h = \left( \left( \sum _{i=1}^n a_i x_i^p \right) ^q, \left( \sum _{i=1}^n b_i x_i^q \right) ^p \right) . \end{aligned}$$

    So each \(E_{(f,g)_h, id, \theta }\) has one of the following forms:

    $$\begin{aligned} \{f^q + {\alpha }g^p =0\} \cap \{g > 0\} \end{aligned}$$

    or

    $$\begin{aligned} \{f^q + {\alpha }g^p =0\} \cap \{g < 0\} \end{aligned}$$

    or

    $$\begin{aligned} \{g=0\} \cap \{f > 0\} \end{aligned}$$

    or

    $$\begin{aligned} \{g=0\} \cap \{f < 0\} , \end{aligned}$$

    for some \({\alpha }\in {\mathbb {R}}\). The first one corresponds to \(\theta \in (0, \pi /2]\) if \({\alpha }\le 0\) and to \(\theta \in (\pi /2, \pi )\) if \({\alpha }>0\). The second one corresponds to \(\theta \in (\pi , 3\pi /2]\) if \({\alpha }\le 0\) and to \(\theta \in (3\pi /2, 2\pi )\) if \({\alpha }>0\). The third one corresponds to \(\theta = 0\) and the forth one corresponds to \(\theta = \pi\). Since all the equalities above deal with homogeneous polynomials, it is easy to see that each \(E_{(f,g)_h, id, \theta }\) intersects any sphere in \({\mathbb {R}}^n\) transversally.

  2. (2)

    Suppose that p is even and q is odd:

    Consider the germ of conic homeomorphism \(h: ({\mathbb {R}}^2,0) \rightarrow ({\mathbb {R}}^2,0)\) given by

    $$\begin{aligned} h(u,v)= {\left\{ \begin{array}{ll} (u^{1/q}, v^{1/p}), &{} \text {if}\; v \ge 0,\\ (u^{1/q}, -(-v)^{1/p}), &{}\text {if}\; v < 0,\\ \end{array}\right. } \end{aligned}$$

    whose inverse is given by:

    $$\begin{aligned} h^{-1}(u,v)= {\left\{ \begin{array}{ll} (u^q, v^p), &{} \text {if}\; v \ge 0,\\ (u^q, -v^p), &{} \text {if}\; v < 0. \\ \end{array}\right. } \end{aligned}$$

    It is a linearization for (f, g), since for each \(i=1, \dots , n\) we have:

    $$\begin{aligned} h^{-1}(a_i t^p, b_i t^q) = {\left\{ \begin{array}{ll} \big (A_I^q t^{pq}, B_I^p t^{pq} \big ), &{} \text {if}\; t \ge 0,\\ \big ( A_I^q t^{pq}, -B_I^p t^{pq} \big ), &{} \text {if}\; t < 0,\\ \end{array}\right. }\quad \text {if}\;B_I \ge 0, \end{aligned}$$

    and

    $$\begin{aligned} h^{-1}(a_i t^p, b_i t^q) = {\left\{ \begin{array}{ll} \big (A_I^q t^{pq}, -B_I^p t^{pq} \big ), &{} \text {if}\; t \ge 0,\\ \big (A_I^q t^{pq}, B_I^p t^{pq} \big ), &{} \text {if}\; t< 0.\\ \end{array}\right. }\quad \text {if}\; B_I < 0. \end{aligned}$$

    The conic modification \((f,g)_h\) is given by

    $$\begin{aligned} (f,g)_h (x) = {\left\{ \begin{array}{ll} \big ( f(x)^q, g(x)^p \big ), &{} \text {if}\; g(x) \ge 0,\\ \big ( f(x)^q, -g(x)^p \big ), &{} \text {if}\; g(x) < 0. \\ \end{array}\right. } \end{aligned}$$

    Proceeding as in Case (1), one can see that (f, g) is \(d_h\)-regular.

  3. (3)

    The case when p is odd and q is even is analogous, considering the linearization given by

    $$\begin{aligned} h(u,v)= {\left\{ \begin{array}{ll} (u^{1/q}, v^{1/p}), &{} \text {if}\; u \ge 0,\\ (-(-u)^{1/q}, v^{1/p}), &{} \text {if}\; u < 0. \\ \end{array}\right. } \end{aligned}$$
  4. (4)

    The case when both p and q are even is also analogous, considering the linearization given by

    $$\begin{aligned} h(u,v)= {\left\{ \begin{array}{ll} (u^{1/q}, v^{1/p}), &{} \text {if}\; u \ge 0 \; \text {and}\; v \ge 0,\\ (-(-u)^{1/q}, v^{1/p}), &{}\text {if}\; u< 0 \; \text {and}\; v \ge 0,\\ (u^{1/q}, -(-v)^{1/p}), &{}\text {if}\; u \ge 0 \;\text {and}\; v< 0, \\ (-(-u)^{1/q}, -(-v)^{1/p}), &{} \text {if}\; u< 0 \; \text {and}\; v < 0, \\ \end{array}\right. } \end{aligned}$$

So we have proved:

Theorem 3.14

Let \((f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2\) be a real analytic map of the form

$$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p , \sum _{i=1}^n b_i x_i^q \right) , \end{aligned}$$

where \(a_i, b_i \in {\mathbb {R}}\) are non-zero constants in generic position and \(p,q \ge 2\) are integers, as in Example 3.13. Then there exist \({\epsilon }\) and \(\delta\) sufficiently small, with \(0<\delta \ll {\epsilon }\), such that the restriction

$$\begin{aligned} (f,g)|:(f,g)^{-1}(\mathbb {D}_\delta ^2 {\setminus } \Delta ) \cap {\mathbb {B}}_{\epsilon }^n \rightarrow \mathbb {D}_\delta ^2 {\setminus } \Delta \end{aligned}$$

is a smooth locally trivial fibration over its image, where the discriminant \(\Delta\) of (f, g) is given by the parametrized curves \(t \mapsto (A_I t^p, B_I t^q)\) in \({\mathbb {R}}^2\), where \(A_I\) and \(B_I\) are given in (14) for each \(I\in \{0,1\}^n{\setminus }\{(0,\dots ,0)\}\).

Moreover, we have an equivalent smooth locally trivial fibration

$$\begin{aligned} \phi :\mathbb {S}_{\epsilon }^{n-1} {\setminus } \left( (f,g)^{-1}(\Delta ) \cap \mathbb {S}_{\epsilon }^{n-1} \right) \rightarrow (\mathbb {S}_{1}^1 {\setminus } \mathcal {A}_h) \cap {{\,\textrm{Im}\,}}(\phi ) \end{aligned}$$

where

$$\begin{aligned} \phi = \frac{h^{-1} \circ f}{\Vert h^{-1} \circ f\Vert } \end{aligned}$$

for the corresponding homeomorphism h as in Example 3.13, and \(\mathcal {A}_h:= h^{-1}(\Delta ) \cap \mathbb {S}_1^1\). If \(V(f,g) \ne \{0\}\), then the maps above are locally surjective.