RADIAL INDEX RELATED TO AN INTERSECTION INDEX

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INDEX

Julie Lapebie

To cite this version:

Julie Lapebie. RADIAL INDEX RELATED TO AN INTERSECTION INDEX. 2018. �hal-01829125�

JULIE LAPEBIE

Abstract. On a semialgebraic setX with an isolated singularity at 0, we are in- terested in relating the radial index of a vector eldV at the singularity to the one of−V. Then we write this radial index as an intersection index.

1. Introduction

To work on manifolds with singularities, we cannot work with the Poincaré-Hopf index, which is only dened on smooth manifolds, we need to work with another one, the radial index (or Schwartz index depend on the authors). This index was introduced by Marie-Hélène Schwartz ([11], [12]) but only concerned radial vector elds. Later, a generalization to other vector elds was made by Henry King and David trotman in [10] and independently by [1], [5] and [13]. We will use the denition of [6].

On a semialgebraic set X of dimension k in R^N with an isolated singularity at 0, we consider a vector eld V. We want to write the radial index of V at0 in terms of the zeros of V_|lk(X), the restriction of V to the link of the singularity (Proposition 2.5).

Then we relate, in Theorem 2.8, the radial index of V at 0to the one of −V at0:

• if k is even, then

ind_rad(−V,0) =ind_rad(V,0);

• if k is odd, then

ind_rad(V,0) +ind_rad(−V,0) = 2−χ(lk(X)).

It follows the corollary:

ind_rad(V,0) +ind_rad(−V,0)≡χ(lk(X)) mod 2.

In the case where we have an analytic map f : (Rⁿ,0) → (R^k,0) dened on a neighborhood of0and such that the ber f⁻¹(0)admits an isolated singularity, Giorgi Khimshiashvili ([9]) proved the following formula:

χ(f⁻¹(δ)∩B_εⁿ) = 1−sign(−δ)ⁿdeg₀∇f, where δ is a regular value of f, deg₀∇f the local degree of ∇f

k∇fk :S_εⁿ⁻¹ →Sⁿ⁻¹ and 0< δ ε1.

From Theorem 2.8, we obtain a singular version of Khimshiashvili's formula:

Let g : X → R be the restriction to X of a C²-semialgebraic function without any critical points on a neighborhood of0 and such that g(0) = 0. Then

2010 Mathematics Subject Classication. 14P10, 14P25, 58K05.

Key words and phrases. Radial index, semialgebraic set, Euler characteristic.

1

• if k is even,

χ(g⁻¹(δ)∩B_ε^N) = 1−ind_rad(∇g,0);

• if k is odd,

χ(g⁻¹(δ)∩B_ε^N) = χ(lk(X))−1 +ind_rad(∇g,0), where δ is a regular value of g.

This gives us a direct connection between the Euler characteristic of the Milnor ber of g and the radial index of∇g at0.

Finally, we remark that a coincidence point x∈lk(X) of V and −∇ρ, where ∇ρ is a radial vector eld, is an inward zero ofV|lk(X). Then, we would like to relate the radial index of V at 0 to an intersection index with the radial vector eld∇ρ. For this, we consider the normalized vector eldsVb and−∇ρbonlk(X)and their respective graphs GVb and G_−∇

ρb. In Proposition 3.3, we prove that:

I(G−∇ρˆ, G_Vˆ) = 1−ind_rad(V,0), where I(G−∇ρˆ, GVˆ)is the intersection index of the graphs.

The paper is organized as follows: in Section 2, we write the radial index ofV at0 as a sum involving only the zeros ofV|lk(X). This enables us to prove Theorem 2.8 and the two corollaries presented above. Section 3 relates the radial index of V at0to an intersection index of the graph of V with the graph of the opposite of a radial vector eld.

2. About the radial index

Let us consider a k-dimensional semialgebraic set X ⊂ R^N with an isolated singu- larity at 0.

We denote byB_ε^N the closed ball of dimensionN, radius ε and centered at0,S_ε^N−1 =

∂B_ε^N the sphere of dimension N −1and B˚_ε^N =B_ε^N \S_ε^N−1. We denote by lk(X) the link of X at 0, i.e. the intersection of X with S_ε^N−1 where ε >0 is such that 0 is the only singularity ofX in X∩B_ε^N and 0< ε1.

Before speaking about the radial index of a vector eld at a singularity, we need to dene what is a vector eld at a singularity and what is a radial vector eld.

Denition 2.1. [3] p.32.

Let V be a continuous vector eld on X ⊂ R^N. We say that V admits an isolated singularity at 0∈X if V is a continuous section ofTR^N|X tangent to X\ {0}and does not have any zeros on a small neighborhood of 0.

Denition 2.2. A continuous vector eld V_rad tangent to X is called radial (at 0) if it is transverse to the link of X. Moreover, we make the convention that it is pointing inward, i.e. it is not pointing toward0. A zero ofV_rad is called an inward zero (resp.

an outward zero) if V_rad is pointing inward (resp. outward) at that point.

Denition 2.3. [7] Denition 2.1, [5], [6] section 1 and [3] Chapter 2. Let us consider a continuous vector eld V on X with a singularity at 0.

Let ε >0be such that V does not have any zeros on X\ {0} ∩B_ε^N and let 0< ε⁰ < ε. Let us denote by X_εε⁰ = (X∩B_ε^N)\B˚_ε^N0 and by ∂X_εε⁰ =X_ε∪X_ε⁰ with X_ε=X∩S_ε^N−1 and X_ε⁰ =X∩S_ε^N0⁻¹.

Let Ve be a vector eld on X_εε⁰ such that:

• Ve|Xε =V;

• Ve_|X

ε0 =V_rad is a radial vector eld;

• Ve_|X˚

εε0 has a nite number of zeros (p_i)_1≤i≤s and they are all nondegenerate.

Then we dene the radial index of V at 0, which we write ind_rad(V,0), as:

ind_rad(V,0) := 1 +

s

X

i=1

ind_{P H}(V , pe _i).

For the next proposition, we will need to dene a correct zero of a vector eld.

Denition 2.4. We say that a zero p∈ lk(X) of V is correct if it is just a zero of V_|lk(X) and not of V on X.

Proposition 2.5. LetX be a semialgebraic closed set with 0as an isolated singularity and V a vector eld dened on a neighborhood U0 of 0 in X. Let U1 $ U0 be a neighborhood oflk(X). Let Ve be a C¹-perturbation ofV on U0 such thatVe|lk(X) admits a nite number of isolated nondegenerate zerosr1, ..., rtwhich are all correct; and such that Ve is equal to V on U₀\ U₁. Then we have the equality

indrad(V,0) = 1−

t

X

ri inward zeroi=1

indP H(Ve|lk(X), ri).

Remark 2.6. This proposition is a particular case of Denition 5.5 in [10].

Proof. Let0< ε⁰ < ε. Let us keep the notations of Denition 2.3. Let e

Ve a perturbation of Ve onU₀ such that:

• e

Ve_|X˚_εε0 admits a nite number of nondegenerate zeros p₁, ..., p_s;

• e

Ve|Xε =Ve|Xε =Ve|lk(X),

• e

Ve|X_ε0 is a C¹-perturbation of Vrad on Xε⁰ having only nondegenerate zeros q₁, ..., q_u which are all inward in X_εε⁰ (forV_rad is radial so is pointing inward at every point of X_ε⁰).

Applying Poincaré-Hopf theorem for manifold with boundary (see for example [2]

or [4]) to the vector eld e

Ve on the set X_εε⁰, we get the equality:

χ(X_εε⁰) =

s

X

i=1

ind_{P H}(e V , pe _i) +

u

X

qiinward ini=1 X_εε0

ind_{P H}(e

Ve|X_ε0, q_i)

+

t

X

riinward ini=1 X_εε0

ind_{P H}(Ve|Xε, r_i).

As e

Ve|X_ε0 only admits inward zeros on X_ε⁰, we have χ(X_ε⁰) =

u

X

i=1

ind_{P H}(e

Ve|_Xε0, q_i), and so

χ(X_εε⁰)−χ(X_ε⁰) =

s

X

i=1

ind_{P H}(e V , pe _i) +

t

X

riinward ini=1 X_εε0

ind_{P H}(Ve|_Xε, r_i).

But X_ε⁰ is a retract by deformation of X_εε⁰ soχ(X_εε⁰) = χ(X_ε⁰). Then we nd

s

X

i=1

ind_{P H}(e

V , pe _i) = −

t

X

riinward ini=1 X_εε0

ind_{P H}(Ve|_Xε, r_i),

and therefore, 1 +

s

X

i=1

ind_{P H}(e V , pe _i)

| {z }

=ind_rad(V,0)

= 1−

t

X

riinward ini=1 X_εε0

ind_{P H}(Ve|_Xε, r_i),

which gives us the required result.

Remark 2.7. The proof of the next theorem is similar to the one of Poincaré-Hopf theorem for manifolds with boundary.

Theorem 2.8. LetX be a closed semialgebraic set of dimensionk with0as an isolated singularity and let V be a vector eld on X with a singularity at 0. Then we have the following equalities:

• if k is even, then

ind_rad(−V,0) =ind_rad(V,0);

• if k is odd, then

ind_rad(V,0) +ind_rad(−V,0) = 2−χ(lk(X)).

Proof. Let Ve be a C¹-perturbation of V satisfying conditions of Proposition 2.5. We denote by (q_i)1≤i≤t the inward zeros ofVe and (p_i)1≤i≤s the outward zeros. By Propo- sition 2.5, we can write

ind_rad(−V,0) = 1− X

rinward zero of−eV|_lk(X)

ind_{P H}(−Ve|_lk(X), r),

and

X

rinward zero of−Ve|_lk(X)

ind_{P H}(−Ve|_lk(X), r) =

s

X

i=1

ind_{P H}(−Ve|_lk(X), p_i).

As X has dimension k, the linklk(X) has dimension k−1 and so ind_{P H}(−Ve|_lk(X), p_i) = (−1)^k−1ind_{P H}(Ve|_lk(X), p_i).

Moreover,

ind_rad(V,0) = 1− X

rinward zero ofVe|_lk(X)

ind_{P H}(Ve|_lk(X), r)

= 1−

t

X

i=1

ind_{P H}(Ve|_lk(X), q_i).

Thus,

S := indrad(V,0) +indrad(−V,0)

= 2−

t

X

i=1

ind_{P H}(eV|_lk(X), q_i)−(−1)^k−1

s

X

i=1

ind_{P H}(eV|_lk(X), p_i).

We also have the equality

χ(lk(X)) =

t

X

i=1

ind_{P H}(Ve|_lk(X), q_i) +

s

X

i=1

ind_{P H}(Ve|_lk(X), p_i).

Let us study the result according to the parity of k.

• If k is even, then

S = 2−

t

X

i=1

ind_{P H}(Ve|lk(X), q_i) +

s

X

i=1

ind_{P H}(Ve|lk(X), p_i)

= 2−

t

X

i=1

ind_{P H}(Ve|lk(X), q_i) +χ(lk(X))−

t

X

i=1

ind_{P H}(Ve|lk(X), q_i)

= 2−2

t

X

i=1

ind_{P H}(Ve_|lk(X), q_i) +χ(lk(X))

= 2 ind_rad(V,0) +χ(lk(X)).

As dimension of lk(X) is odd (equal to k−1), its Euler characteristic van- ishes. Therefore

ind_rad(−V,0) =ind_rad(V,0).

• If k is odd, then

S = 2−

t

X

i=1

indP H(eV|lk(X), qi)−

s

X

i=1

indP H(Ve|lk(X), pi)

= 2−χ(lk(X)).

Once we have proved this theorem, the following result is straightforward:

Corollary 2.9. Let X be a closed semialgebraic set with an isolated singularity at 0 and let V be a vector eld on X with a singularity at 0. Then

ind_rad(V,0) +ind_rad(−V,0)≡χ(lk(X)) mod 2.

Proof. In the previous theorem, we have seen that ifk is even, then S :=ind_rad(V,0) +ind_rad(−V,0) = 2 ind_rad(V,0)≡0 mod 2, and in that case χ(lk(X)) = 0, so

S ≡χ(lk(X)) mod 2.

Ifk is odd, then

S= 2−χ(lk(X)), and hence

S ≡χ(lk(X)) mod 2.

Finally, we have the following corollary, which is a singular version of Khimshi- ashvili's formula:

Corollary 2.10. Let X ⊂ R^N a closed semialgebraic set of dimension k with an isolated singularity at 0and let us consider a functiong :X →R restriction toX of a C²-semialgebraic function. Let us suppose that g(0) = 0 and that g does not have any critical points on a neighborhood of 0. Let δ be a regular value of g and ε > 0 such that 0< δ ε 1. Then we have the following equalities:

• if k is even,

χ(g⁻¹(δ)∩B_ε^N) = 1−indrad(∇g,0);

• if k is odd,

χ(g⁻¹(δ)∩B_ε^N) = χ(lk(X))−1 +ind_rad(∇g,0).

Proof. From [7] Example 2.6, we know that for 0< δ ε1, we have χ(g⁻¹(−δ)∩B_ε^N) = 1−ind_rad(∇g,0).

So

χ(g⁻¹(δ)∩B_ε^N) = 1−ind_rad(−∇g,0).

Thus, from the previous theorem,

• if k is even, then

ind_rad(−∇g,0) =ind_rad(∇g,0), and so

χ(g⁻¹(δ)∩B_ε^N) = 1−ind_rad(∇g,0);

• if k is odd, then

ind_rad(−∇g,0) = 2−χ(lk(X))−ind_rad(∇g,0), and therefore

χ(g⁻¹(δ)∩B_ε^N) = 1−ind_rad(−∇g,0)

= χ(lk(X))−1 +ind_rad(∇g,0).

3. A link with an intersection index

First, we recall the denition of the intersection index of two manifolds [8].

Denition 3.1. Let X, Y and Z be three oriented manifolds without boundary such that X, Z ⊂Y. We suppose X compact, Z closed and dimX+ dimZ = dimY. If X and Z are transverse, then we dene the intersection index I(X, Z) of X and Z counting the points xof the intersection X∩Z and associating to each a plus sign + or a minus sign−. More precisely, if forx∈X∩Z the given orientation ofT_xX×T_xZ (where we denoted by T_xX the tangent plane of X at x) is the same as the one given by the tangent plane T_xY, then we associate a plus sign + to x. We set or(x) = 1. Otherwise, we associate a minus sign − and we set or(x) = −1. Let C(X, Z) be the set of intersection points of X and Z. Then we have the equality

I(X, Z) := X

x∈C(X,Z)

or(x).

Remark 3.2. When X and Z are not transverse, we perturb X and Z in X˜ and Z˜ respectively so thatX˜ andZ˜ are. Then the intersection indexI( ˜X,Z˜)exists according to the previous denition. Moreover, as this index is invariant under perturbations, we dene I(X, Z) as I(X, Z) =I( ˜X,Z)˜ .

LetX ⊂R^N be a semialgebraic variety of dimensionk+1with an isolated singularity at 0. We suppose that X \ {0} is oriented. For example, we can consider a set of the form X = {f₁ =...=f_l = 0} such that rank(∇f₁(x), ...,∇f_l(x)) = 0 for all x∈X\ {0}.

We consider a C²-tangent vector eld V on X \ {0} and ∇ρ a C²-radial vector eld on X. Then we consider the link of X at 0, which we denote by lk(X), i.e. the

Figure 1. Example in dimension 2.

intersection ofX with S_ε^N−1 whereε >0is such that 0is the only singularity of X in X∩B_ε^N and 0< ε1.

The vector eld V does not admit any zero on lk(X). For a care of simplicity, we set Y :=lk(X). We can remark that Y is an oriented submanifold of X of dimension k. Indeed, we know that there exists a (k+ 1)-dimensional manifold with boundary Z such that the boundary ∂Z of Z is equal to Y. Here, Z = X∩B_ε^N \ {0}. We recall that the orientation ofY is given by the one ofZ. Let(v₁, ..., v_k)be a basis ofY =∂Z and let x ∈ Y. The space T_xY has codimension 1 in T_xZ so there exist exactly two unit vectors~v and−~v inT_xZ which are perpendicular toT_xY,~v pointing outward and

−~v inward. We suppose that (v₁, ..., v_k, ~v) is a positively oriented basis of T_xZ. Then the basis (v1, ..., vk) is positively oriented for TxY and so is for Y.

For all x∈Y, we introduce these two vector elds:

Vb(x) = V(x)

kV(x)k and − ∇ρ(x) =b − ∇ρ(x) k∇ρ(x)k.

We say that x is a coincidence point of Vb and −∇ρbif Vb(x) = −∇ρ(x)b . We want to relate these coincidence points to the inward zeros of the vector eld V|Y. Indeed,

Vb(x) = −∇ρ(x)b ⇐⇒ V(x) =− kV(x)k

k∇ρ(x)k∇ρ(x)

⇐⇒ x is an inward zero ofV|Y. Let us introduce the following set:

T U Y =n

(x, ~v) | x∈Y, ~v ∈T_xX and k~vk= 1o , which is included in the unit tangent bundle of X, that is

T U X =n

(x, ~v) | x∈X, ~v ∈T_xX and k~vk= 1o .

The bundleT U X is locally the product of an open set in R^k with the k-sphereS^k so has dimension dimX+ dimS^k =k+ 1 +k = 2k+ 1.

The setT U Y is an oriented manifold of dimension2k. Indeed, T U Y = T U X ∩ {ρ(x) = ε} × R^k+1

. The tangent bundle T X of X is oriented so is T U X. Actually, T U X is the transverse intersection of T X with Rⁿ× {~v | k~vk= 1}, that is also an oriented manifold. As the two manifoldsT U X and

{ρ(x) =ε} ×R^k+1 are oriented and that their intersection is transverse, T U Y is also oriented. Moreover, its dimension is equal to2k for it is equal todimT U X−1.

We considerG_Vb and G−∇ρb the respective graphs ofVb and −∇ρb: G_Vb =n

(x,Vb(x))| x∈Yo

and G−∇ρb=n

(x,−∇ρ(x))b | x∈Yo .

These graphs are oriented subsets ofT U Y. A positively oriented basis ofT_(x,Vb(x))G_Vb is of the formD

(ε₁, DVb(x)(ε₁)), ...,(ε_k, DVb(x)(ε_k))E

for(ε₁, ..., ε_k)is a positively oriented basis of Y.

Then we prove the following result:

Proposition 3.3. Let X ⊂ R^N be a semialgebraic set of dimension k + 1 with an isolated singularity at 0 and such that X\ {0} is oriented. Let V be a C²-vector eld and ∇ρ a C²-radial vector eld on X, and let us consider their respective normalized vector eld Vb and ∇ρbdened on the link of X at the singularity. Then we have the equality:

I(G−∇bρ, G_Vb) = 1−ind_rad(V,0).

Proof. The graphsG

Vb andG−∇ρbofVb and−∇ρbrespectively are oriented compact sub- manifolds ofT U Y, each of dimensionk. Then their intersection index I(G−∇ρb, G

Vb)is well dened.

Let us consider a coincidence point x ∈ Y (= lk(X)) of Vb and −∇ρb. The point (x,Vb(x))is in that case an intersection point of the two graphsG_Vb andG−∇bρ. We are going to compute the value of or((x,Vb(x))), which we denote for simplicity by or(x), to relate it with ind_{P H}(V|Y, x), the Poincaré-Hopf index of V|Y atx.

To compute or(x), we must nd a positively oriented basis of the tangent plane to T U Y at (x,Vb(x)) and one for the two tangent planes T_(x,

Vb(x))G

Vb and T_(x,bV(x))G−∇ρb. Let us begin with T_(x,Vb(x))T U Y.

AsX\ {0}is a manifold of dimensionk+ 1, we know that there exists a neighborhood U₁ of x ∈Y in X, and a dieomorphismφ₁ such that φ₁(U₁) is an open set of R^k+1. Moreover, Y is a submanifold of X of dimension k so U₁∩Y is a neighborhood of x in Y and φ₁(U₁∩Y) =φ₁(U₁)∩R^k is an open set of R^k, where we identify R^k with R^k× {0} ⊂R^k+1. We will write R^k× {0}={x_k+1= 0}. So

φ₁(U₁∩Y) =φ₁(U₁)∩ {x_k+1 = 0}.

Moreover, as Z is a manifold with boundary included in X of dimension k + 1 and such that∂Z =Y, the open setU1∩Z is a neighborhood ofx in Z and

φ₁(U₁∩Z) = φ₁(U₁)∩ {x_k+1 ≥0},

which is an open set of R^k×R+ (where we wrote {x_k+1 ≥0}=R^k×R+).

Locally, if x ∈ U₁, the bundle T X is dieomorphic to φ₁(U₁)×R^k+1 and T U X to φ₁(U₁)×S^k. Thus, T U Y is dieomorphic to φ₁(U₁∩Y)×S^k.

Now, to lighten notations, let us suppose that T X = φ₁(U₁) × R^k+1 and T Y =

φ₁(U₁∩Y)×R^k+1. We also know that

T_(x,Vb(x))G−∇ρb=G−D∇ρ(x)b =n

(u,−D∇ρ(x)(u))b | u∈T_xYo , and

T_(x,bV(x))G_Vb =G_DVb(x) =n

(u, DVb(x)(u))| u∈T_xYo . These sets are both of dimension k.

As the vector eld ∇ρbis unit and radial on Y and so that −∇ρbpoints in each point of φ₁(U₁)∩Y toward φ₁(Z), we can write in local coordinates that −∇ρbis constant equal to (0, ...,0,1)∈R^k+1 pointing toward U₁ ∩ {x_k+1 ≥0}.

Then we nd a basis ofT_(x,Vb(x))G−∇ρb: D

(e₁,−D∇ρ(x)(eb ₁)), ...,(e_k,−D∇ρ(x)(eb _k))E . But D∇ρ(x)(eb _i) = 0 for all i∈ {1, ..., k} so the basis is nally

D

(e₁,0), ...,(e_k,0)E

, with (e_i,0)∈R^k× {0}^k.

Moreover, the sphereS^kis locally dieomorphic to an open set ofR^k(by the projection (x₁, ..., x_k+1)∈S^k7→(x₁, ..., x_k) at(0, ...,0,1)) and thus,

T_(x,bV(x))T U Y ∼=R^k×R^k, where ∼=means "dieomorphic to".

Then we choose a basis for T_(x,Vb(x))T U Y: B=D

(e₁,0), ...,(e_k,0),(0, e₁), ...,(0, e_k)E ,

where(e₁, ..., e_k)is the canonical basis positively oriented ofR^kand for alli∈ {1, ..., k}, (e_i,0)∈R^k× {0}^k.

Regarding T_(x,Vb(x))G_Vb, we have

DVb(x) :T_xY −→R^k,

so the image of DVb(x) is also in R^k. In the same way as previously, a basis of T_(x,bV(x))G

Vb is of the form D

(e1, DVb(x)(e1)), ...,(ek, DVb(x)(ek)) E

.

Thus we obtained a basis of T_(x,Vb(x))G−∇ρb×T_(x,bV(x))G_Vb: B⁰ =D

(e₁,0), ...,(e_k,0),(e₁, DVb(x)(e₁)), ...,(e_k, DVb(x)(e_k))E .

Let us compute DVb(x)(e_i) for all i∈ {1, ..., k}.

We know that Vb(x) = (0, ...,0,1)∈S^k so in local coordinates, we have Vb(x) =

Vb₁(x), ...,Vb_k(x)

=

V₁(x)

kV(x)k, ..., V_k(x) kV(x)k

. Then for all i, j ∈ {1, ..., k}, we have

∂Vb_i

∂x_j(x) =

∂V_i

∂x_j(x)kV(x)k − ∂kVk

∂x_j (x)Vi(x) kV(x)k² . But V_i(x) = 0 for all i∈ {1, ..., k}, thus

∂Vb_i

∂x_j(x) = 1 kV(x)k

∂V_i

∂x_j(x).

We nally have

DVb(x)(e_i) = 1 kV(x)k

k

X

j=1

∂V1

∂x_j(x)e^j_i, ...,

k

X

j=1

∂Vk

∂x_j(x)e^j_i

! ,

where we wrote e_i = (e¹_i, ..., e^k_i) the k components ofe_i. We also know that e^j_i = 0 for alli6=j and eⁱ_i = 1.

We set A: =

DVb(x)(e₁) · · · DVb(x)(e_k)

= 1

kV(x)k













∂V₁

∂x1

(x) · · · ∂V₁

∂xk

(x)

... ...

∂Vk

∂x₁(x) · · · ∂Vk

∂x_k(x)













×





e¹₁ · · · e¹_k ... ...

e^k₁ · · · e^k_k





| {z }

:=IdRk

= 1

kV(x)kDV|Y(x).

Now we can write the change of basis P fromB⁰ to B: P =

Id_R^k Id_R^k

(0) A

, then

detP = detA= 1

kV(x)kdetDV|Y(x).

Even if we consider a perturbation Ve of V which only admits nondegenerate zeros on Y, we suppose, to simplify, that it is already the case for V. Thus, as x is a nondegenerate zero of V, we have detDV_|Y(x)6= 0. We nally have

sign(detP) = sign(detDV|Y(x)).

Thus we have proved that for each coincidence pointx of −∇ρband Vb, which gives us an intersection point (x,Vb(x))∈T U Y of G−∇ρband G

Vb, we have sign(detDV|Y(x)) = sign(detP),

where sign(detP) is justor(x). Thus,

X

x∈C(−∇bρ,bV)

or(x) = X

xinward zero ofV|Y

sign(detDV_|Y(x)).

But X

xinward zero ofV_|Y

sign(detDV|Y(x)) = X

xinward zero ofV_|Y

ind_{P H}(V|Y, x), and we proved that

X

xinward zero ofV|Y

ind_{P H}(V|Y, x) = 1−ind_rad(V,0).

Last we have, by denition,

I(G−∇ρb, G_Vb) = X

x∈C(−∇ρ,bVb)

or(x).

This nally gives us

I(G−∇bρ, G_Vb) = 1−indrad(V,0).

References

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Uspekhi Mat. Nauk, 34:338, 1979.

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[9] G.M. Khimshiashvili. On the local degree of a smooth map. Soobshch. Akad. Nauk Gruz., 85:309 311, 1977.

[10] H. King and D. Trotman. Poincaré-hopf theorems on singular spaces. Proc. Lond. Math. Soc., 108:682703, 2014.

[11] M.-H. Schwartz. Classes caractéristiques dénies par une stratication d'une variété analytique complexe. C.R. Acad. Sci. Paris Sér. I Math., 260:32623264,35353537, 1965.

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Aix-Marseille Université, CNRS, Centre Gilles-Gaton Granger, Aix-en-Provence, France

E-mail address: julie.lapebie@univ-amu.fr