Bahri invariants for fractional Nirenberg-type flows

This paper is concerned with prescribing the fractional Q-curvature on the unit sphere Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{n}$$\end{document} endowed with its standard conformal structure g0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_0$$\end{document}, n≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 4$$\end{document}. Since the associated variational problem is noncompact, we approach this issue with techniques passed by Abbas Bahri, as the well known theory of critical points at infinity, as well as some lesser known topological invariants that appear here as criteria for existence results.


Introduction
The armor is not for protecting the soul from spoilage. 1 In conformal geometry the Nirenberg problem is the well-known problem of prescribing the scalar (resp. Gauss) curvature on the sphere S n for n ≥ 3 (resp. n = 2). Given a prescribed function K : S n −→ R this problem amounts to solve the following equations − g 0 u + 1 = K e u on S 2 , and P g 0 where P g 0 1 = − g 0 + n−2 4(n−1) R g 0 is the conformal Laplacian and R g 0 the scalar curvature associated to g 0 . The operator P g 0 1 is the first of a sequence of conformally covariant elliptic operators. The second operator was introduced by Paneitz in [29] (see also [19]) and is P g 0 2 = (− g 0 ) 2 − div g 0 (a n R g 0 g 0 + b n Ric g 0 )d + where Ric g 0 and Q g 0 n are respectively the Ricci and the standard Q-curvature of g 0 , a n and b n are constants depending on n. In [21] Graham, Jenne, Masion and Sparling gave the first construction of a sequence of conformally covariant elliptic operators (P g 0 k ) k , which exist for all k ∈ N when the dimension n is odd, but only for k ∈ {1, 2, . . . , n 2 } when n is even. In [30], Peterson constructed an intrinsically defined, arbitrary real number order, conformally covariant pseudo-differential operator. In [22], Graham and Zworski showed that the operators P g 0 k can be realized as the residues at s = k of a meromorphic family of scattering operators. Using this point of view, a family of conformally covariant pseudo-differential operators (P g 0 s ) for noninteger s was given.
In recent years we have seen several works on the properties of the fractional Laplacians as nonlocal operators and their applications to various problems (see e.g. [9][10][11][12][13]). It is a known fact that (− g 0 ) s on R n with s ∈ (0, 1) is a nonlocal operator. In [10], Caffarelli and Sylvestre expressed (− g 0 ) s as a generalized Dirichlet-Neumann map for an elliptic boundary value problem with local differential operators defined on R n+1 + . This work was extended in [14] where Chang and Gonzalez characterized (P g 0 s ) as such a Dirichlet-to-Neumann operator on a conformally compact Einstein manifold.
Observe now that when γ ∈ (0, n 4 ), under the conformal change of metric g = u 4 n−2s g 0 , the new operator P g s associated to the new metric g is connected with the initial operator P  (1).
We will in the sequel omit the superscript g 0 in P g 0 s and put P g 0 s = P s . We recall that P s is the 2s order conformal Laplacian on S n , which is an intertwining operator having the expression where is the Gamma function and In this paper we focus on the fractional Nirenberg problem on the standard sphere S n , n ≥ 4, that is the prescribing fractional Q-curvature problem: find a conformal change of metric g = u 4 n−2s g 0 such that Q g s = K where K : S n −→ R is a prescribed function. This problem amounts to solve the following system Several authors studied problem (1.1) on S n , see e.g. [15,16,23,24,26], but their methods differ from the our. Note however that in [1] and [17] the authors used, as we do, the theory of critical points at infinity, but they conclude in different ways from us. More particularely in [17] Chen, Liu and Zheng addressed the problem (1.1) using the same framework that ours; they established an Euler-Hopf-type formula, and obtained some existence results under Bahri-Coron-type assumptions. However our results are original since they are obtained in a completely different way: by using some topological invariants which are associated to variational flows. These invariants are actually intersection numbers who have been introduced the first time by Bahri in [2], where in order to treat the scalar curvature problem on high dimensional spheres in the Riemannian framework, the author has taken the basic elements of the variational theory, that is to say the deformation along the variational flows and the analysis of changes of topology which they induce, even if they drive to infinity. As we shall see, these intersection numbers are proving to be criteria of existence of solutions. Our paper will be organized as follows: In Sect. 2 we introduce some notations and constructions and we give our main results. In Sect. 3 we recall the variational framework. Section 4 and 5 are devoted to prove our results.

Notations, assumptions and main results
In this first section we will introduce three invariants which will be denoted by γ , τ and μ, and we will state our existence results. For this purpose, we need to introduce several notations, definitions and constructions.
We assume that K : S n −→ R is a C 2 function having a finite set of h +1 nondegenerate critical points y 0 , . . . , y , y +1 . . . , y h , ordered such that and For 0 i h, we denote the Morse index of K at y i by and we assume that y 0 is an absolute maximum of K . For a chosen integer k ≥ 1, we introduce the set Let Z be a pseudogradient for K such that (generically) the intersections of the unstable and the stable manifolds of the various critical points of K are transverse. We assume that where W s (y) (resp. W u (y)) is the stable (resp. unstable) manifold of y with respect to Z . For a subset I of I + k which will be specified in due time, we define X is then a k-dimensional compact manifold without boundary. For an integer p 2, we define where δ a i is the Dirac measure at a i . Now, for λ large enough, we define the map where + is the positive unit half sphere of our functional space H s (S n ) and ϕ (a i ,λ) given by (3.1) is a solution of Eq. (3.2), (see Sect. 3). Note that f λ (B p (X )) is a manifold of dimension p(k + 1) − 1, (its singularities arise in dimension p(k + 1) − 3 and lower). We let in the sequel and m = dim N = p(k + 1) − 1. In Sect. 4 we show that: More precisely, in terms of homology, the boundary of W u (N ) is the m-singular chain: Here we denote where

11)
A ∞ = φ, critical point at infinity of J in + , (2.12) ind(J, φ) is the Morse index of J at φ, and N .φ is the intersection number N .W s (φ) (see [28] for the whole definition). This intersection number is well defined because W is transverse to N = f λ (B p (X )) outside f λ (B p−1 (X )), which cannot dominate critical points of index m (for dimensional reasons). Furthermore, W s (φ) adds to W s (φ) only stable manifolds of critical points of index larger than or equal to m + 1, and, since N is of dimension m, these manifolds can be assumed to avoid it. Notice that here, ∂ denotes the boundary operator of the Floer-Milnor Homology associated to J , as it was defined in [32] (see also [20]). We recall that in this homology, singular chains are generated by the unstable manifolds of the critical points of J . Observe that since ∂ • ∂ = 0 and since N is a cycle in dimension m, that is ∂ N = 0, it follows that: From this lemma we derive the following lemma which proof is in Sect. 4: We continue our constructions. Let ν −→ X be a tubular neighborhood of X in S n . We denote ν(z), for z ∈ X , the fiber at z of this tubular neighborhood. The fibers ν(z) has been built such that K (z) = max ν(z) K and z is the unique maximum of K on the fiber ν(z). We now introduce for > 0 and z 1 , . . . , z p ∈ X such that − K (z i ) > 0 (1 i p), the set where (V 0 ) designates the orthogonality relations (with respect to the inner product given by (3.7)): For δ > 0, let us define the set In Sect. 4 we sketch the proof of this lemma.

The intersection numbers γ and τ
In [2] Bahri introduced the intersection number γ (see also [32]) which can be defined by the formula: (2.14) In Sect. 4 below, we will sketch the proof of the following Proposition: Observe that + is not a manifold, but there exists a neighborhood V of + , which is open in H s (S n ), and such that + is a retract of V. We are thinking of intersection theory in that sense. On the other hand, another invariant was introduced by Bahri in [2]. This is the quantity τ , which is the intersection number We introduce the following assumption (H 2 ) There are no critical points at infinity of index m, which are dominated by N .
Our first existence result is based on a comparison between γ and τ : Observe that the value of the invariant γ can be computed. In Sect. 4 we prove that: Thus, Theorem 2.9 becomes A simple application of the preceding corollary is given by the following result:

A min-max characterization
From another side, one can provide a min-max characterization of some critical points of J . Indeed, let

The intersection number μ
In contrast to the precedent Theorems, we have the following results based on another intersection number, also introduced in [2] by Bahri, and which is denoted by μ. The advantage with this local invariant is that one can dispense with the hypothesis (H 2 ). We will consider two cases.

First case
In this case, using (2.5) with I = {y i 0 }, we take C y 0 (X ) is a manifold in dimension k + 1. We define, for λ large enough, the intersection number This intersection number is well defined by the same arguments as above. We then have has a solution of index k or k + 1.

Second case
In this case, we consider a more general situation. Using (2.5) with where we denoted by d the cardinality of I + k . X is a k-dimensional compact manifold without boundary. For each j = 1, . . . , d, we define, for λ large, where W s (y 0 , y i j ) ∞ is the stable manifold of the critical point at infinity (y 0 , y i j ) ∞ for the pseudogradient W . By the same arguments as above, this intersection number is well defined, and we have

Variational framework
Denote by H s (S n ) the completion of C ∞ (S n ) with respect of the norm where dv g 0 is the volume element of g 0 . Let in the sequel The Euler functional associated to problem (1.1), denoted by J , is then defined on H s (S n ) by 2 v is a solution for (1.1), and therefore the new metric g = u 4 n−2s g 0 has its fractional Q-curvature Q g s = K . Problem (1.1) therefore amounts to finding the critical points of the functional J . However, this problem is known to be delicate because it is part of the class of noncompact problems (e.g. Yang-Mill's equations, nonlinear wave equations under nonrationally depending limit conditions, periodic orbits problem for contact flows on contact manifolds...). In addition, there are also topological obstructions of Kazden-Warner condition type to solve (1.1) (see e.g. [25]).

Defect of compactness
The functional J does not satisfy the Palais-Smale condition [denoted (P.S.) for short], and there exist noncompact orbits for the decreasing gradient flow of J , that is to say there exist critical points at infinity (asymptotes) for the functional J .
Let us introduce for a ∈ S n and λ > 0 the function where d = d g 0 is the geodesic distance on S n and c n > 0 is chosen such that the functions ϕ (a,λ) are solutions of Notice that, under the stereographic projection, Eq. (3.2) is transformed to equation whose solutions are functions given for a ∈ R n and λ > 0 by Using Lemma A1 of [17], we introduce the sharp constant S for the continuous-but noncompact-Sobolev embedding H s (S n ) → L 2n n−2s (S n ), which is reached by the functions ϕ (a,λ) , namely: We introduce the set E = (0, +∞) p × (S n ) p × (0, +∞) p which is the space of the variables (α, a, λ) =  (α 1 , . . . , α p , a 1 , . . . , a p , λ 1 , . . . , λ p ) and we consider the following subset of E: where Now, we define the set of potential critical points at infinity of J in + to be: As we said above the functional J fails to satisfy (PS) condition (this failure was analyzed in several earlier works see e.g. [8,27,31]), which leads to the existence of noncompact sequences along which the functional is bounded and its gradient goes to zero. The characterization of the sequences failing the (P.S) condition for J in + was achieved in [17] (lemma 2.2), such sequences are described by the following result: Notice that any function u ∈ V ( p, ) possesses an optimal representation. The following proposition follows from the corresponding results in [4,5,17]: has a unique solution (α, a, λ) in B up to permutation.
Thus, for > 0 sufficiently small, any function u ∈ V ( p, ) can be uniquely written as: where (3.6) designates the following orthogonality relations with respect to the inner product associated to the norm : Otherwise, from Lemma 3.5 of [17], one has Lemma 3.3 There exists a C 1 -mapv : B → H (a, λ), (α, a, λ) →v(α, a, λ), such that for any (α, a, λ) ∈ B ,v =v(α, a, λ) is the unique minimum of the functional J on H (a, λ): and we have the estimate For such av satisfying (3.8) we set in the sequel  H (a, λ). Then we have the following expansion of J : where Q is a quadratic form,v is the minimizer given by (3.8), c 0 , c 1 and c 2 are nonnegative constants. H (a, λ), furthermore it is bounded.

Lemma 3.5 Assume the ε i j are small enough. Then there exists
For a proof, one can operate as in the proof of Proposition 3.1 in [3].

Critical points at infinity
Definition 3.6 A critical point at infinity of J in + is a limit of a flow line u(t) of the equation ( p, (t)) for all t t 0 , where (t) → 0 when t → +∞, and where u 0 is an initial condition.

By Proposition 3.2, u(t) can be written as u(t)
and a i = lim t→∞ a i (t), then such a critical point at infinity is denoted by +∞) or (a 1 , . . . , a p ) ∞ . Let us introduce Recall that A ∞ is the set of critical points at infinity of J in + . It was shown in section 5 of [17], (or in [33]), that A ∞ is in bijective correspondence with F + ∞ . Proposition 3.7 φ ∈ A ∞ iff there exists p ∈ {1, . . . , + 1} and (y i 1 , . . . , y i p ) ∈ F + p , such that φ = (y i 1 , . . . , y i p ) ∞ .

Proposition 3.8 The Morse index of J at the critical point at infinity
The next lemma provides a vector field said pseudogradient of the Morse Lemma at infinity, which has a very nice behavior near critical points at infinity (see Lemma 4.3 in [17]): Lemma 3.9 There exists on V ( p, ) a vector field W for which there exists constants c, c , C > 0 independent of i α i ϕ i in V ( p, ) such that (v resp.ū is given by (3.8) resp. (3.10)): (a 1 , . . . , a p ) is close to some (y i 1 , . . . , y i p ) ∈ F + p , and the λ i are comparable.

The only region where the λ i are not bounded along the decreasing flow lines of W , is where
In particular such a pseudogradient of the Morse Lemma at infinity satisfies (PS) condition outside the critical points at infinity. Given that within our framework critical points at infinity may come into play, we need such appropriate type of pseudogradients. Following [2], we introduce the notion of a Morse-Smale pseudogradient for J and the more general notion of flow belonging to P: Definition 3.10 (Morse-Smale pseudogradient for a functional) We say that a vector field or a flow V on + is a Morse-Smale pseudogradient for J , if V has the expected critical points at infinity of J and it behaves around them as the pseudogradient W of the Morse Lemma at infinity (given by Lemma 3.9 above), and if its flow lines are transverse to N , where N is defined by (2.7).
Of course such Morse-Smale pseudogradients for J exist as soon as the pseudogradient W of the Morse Lemma at infinity exists. For example we could assume that the Morse-Smale pseudogradients for J are equal to W in an arbitrary neighborhood of the critical points at infinity.

Stable and unstable manifolds of a critical point at infinity
Denote by φ the critical point at infinity (y i 1 , . . . , y i p ) ∞ . We can define its stable manifold W s (φ) quite simply as the set of all points of + which are attracted by the asymptote φ through the decreasing flow lines of a Morse-Smale pseudogradient W for J . Before talking about the unstable manifold, which is a more subtle notion that the stable manifold, note that near the critical point at infinity φ, for a function u = V ( p, ), the expansion of the functional J after further computations and suitable change of variables becomes: and then we see that the flow W splits the variable λ from the other variables (α, a).
The unstable manifold is more delicate to define because it exists only as the limit of W u (u λ ) the unstable manifold of the critical point u λ of the reduced problem in a section relative to λ. Since there is a Morse lemma in this section and since the flow W splits the variable λ from the other variables near φ, one can therefore think of W u (φ) to be represented by some W u (u λ ) for λ very large. Ultimately, when λ j = +∞ for j = 1, . . . p, the unstable manifold of φ can be defined as: where W u (ᾱ, y) is the unstable manifold of the critical point (ᾱ, y) of the functional ).
(For anymore details we send back the curious reader at [17] or [33]).

Proof of Proposition 2.2
We begin by noting that the condition W u (N ) ∩ N = N can be satisfied. Just assume that 2m + 1 < n, then we can assume generically on −∇ J (the opposite of the gradient of J ) or any Morse-Smale pseudogradient W that W u (N ) ∩ N = N . Denoting by η t the one parameter group generated by the flow W , one has One can easily see that the boundary of

Proof of Proposition 2.3
From the algebraic topological point of view W u (N ) is a chain of dimension m + 1 and its boundary ∂ W u (N ) is a chain of dimension m. Observe that the upper boundary {0} × N is represented by the chain N . As to the lower boundary {+∞}× N , we know by using Proposition 7.24 and Theorem 8.2 of [6], that the corresponding chain is made of unstable manifolds of critical points or critical points at infinity of J which are dominated by N and whose Morse indexes are equal to m, that is the chain φ∈A 0 We recall that N .φ is the intersection number N .W s (φ). Hence we have:

Proof of Lemma 2.5
+ is contractible, then in particular the homology group H m ( + ) = 0, and so there exists a (m + 1)-chain σ , such that and thus, W u (N ) − σ is a (m + 1)-cycle of + , modulo N , i.e: Observe that the boundary ∂σ splits as it follows: ∂σ = ς 0 + ς ∞ , where ς 0 (resp. ς ∞ ) is the contribution of the critical points (resp. of the critical points at infinity) of J of Morse index m, into the boundary ∂σ .

Proof of Lemma 2.6
The proof of Lemma 2.6 is very technical, we will recall its key steps. For anymore details we send back to [2] or [7] (in particular see the Remark p. 394 in [7]). Let Considering the expansion of J (u) given by Proposition 3.4 and observing that α i is equal to its optimal value (see Section 5 in [17]) , then, after further calculation using in particular the estimates given by (3.9) and Lemma 3.5 we obtain the following expansion of the functional: wherec, c i j are positive constants, Q is a definite positive quadratic form on v −v,v is a uniquely determined function satisfying (V 0 ) and We now use the fact that K (z i ) is the maximum of the function K on the fiber ν(z i ) to observe that c ∞ (z 1 + h 1 , . . . , 2s n , and it is close to it as we will by making small enough. Then using the Morse Lemma for the function K we derive that where α 0 is a positive constant. Observe that the boundary of (z 1 , . . . , z p ) is defined by: therefore we can choose δ = δ( ) small enough such that the boundary of (z 1 , . . . , z p ) does not intersect the level J −1 {c ∞ + δ}, and then D = (z 1 , . . . , z p ) ∩ J −1 {c ∞ + δ}, is a Fredholm manifold of codimension p(k + 1) = m + 1.

Proof of Proposition 2.7
The proof of the fact that the invariant γ is independent of the variational flow was first performed by Bahri in [2] (lemma 7), in the case where = 1 and p = 2. The proof was extended to ≥ 1 and p ≥ 2 in [7] (proposition 3.1). But this proof could induce the misconception that this outcome is only valid for scalar curvature type flows, whereas it depends on a more general variational principle, which was mentioned in [32]. Let us sketch the main idea of this proof.
Taking two flows V 1 and V 2 in the set P, (see Definition 3.12), we let for i = 1, 2, γ ( Note that + is contractible which implies in particular that H m+1 ( + ) = 0 and thus there exists a chain C ∈ H m+2 ( + ) such that ∂C = c. Observe now that since D is of codimension (m + 1) and without boundary, we may assume up to perturbation, that generically C ∩ D is a one dimensional chain and thus with zero boundary (modolo 2). We derive that

Proof of Proposition 2.10
We know by Proposition 2.7 that the invariant γ does not depend on the variational flow of our functional J , but it relies on a more general variational principle. To compute the value of γ , one can use the variational flow on + of the functional denoted J (here the prime is not the derivative): (4.2) Problem (4.2) is a compact variational problem, and J satisfies (P.S) condition. We denote by W u (N ) the unstable manifold of N with respect of a pseudogradient W for J , (for example W = −∇ J is here an appropriate pseudogradient for J ). We claim that: Proof of (i). The Euler-Lagrange equation of the functional J is which is transformed under the stereographic projection to the equation There is a unique solution to Eq. (4.4) on R n , which is positive an spherically symmetric (see e.g. [18]), and which by the stereographic projection, gives rise to a unique minimumû of the functional J in + . The Morse index of J atû is then zero, and claim (i) is proved.
Proof of (ii). Note that claim (i) implies that N is a set of regular points of the functional J , and thus J is an admissible functional (see Remark 3.14).
Proof of (iii). Observe that since W u (N ) is compact, using Proposition 7.24 of [6] we have that: Then, taking and δ small enough in the construction of D, we can make W u (N ) avoiding D, that is W u (N ).D = 0. Hence claim (iii), and Lemma 4.1 is proved. Denote by ς 0 and ς ∞ the chains defined as in (4.1), but with respect to the pseudogradient W (see Remark 3.13). Since there are no critical points of J of index m and since there are no critical points at infinity of any index (this problem being compact), we derive that ς 0 = ς ∞ = 0. Considering definition (2.14) and Proposition 2.7, we can compute the value of the invariant γ by using the variational flow of J , and thus we have: where σ is the (m + 1)-chain such that ∂σ = ς 0 + ς ∞ . Since ∂σ = 0, the chain σ can be taken to be zero so one gets γ (W ) = W u (N ).D = 0.
We can now use Proposition 2.7, with V 1 = W and V 2 = W , to conclude that since the pseudogradients W and W clearly belong to P. Hence γ = γ (N , D) = 0 which completes the proof of Proposition 2.10.

Proof of Theorem 2.9
Proof Notice that assumption (H 2 ) is written ∀φ ∈ A ∞ m , W u (N ) ∩ W s (φ) = ∅. Using (H 2 ), we derive that the intersection number N .φ = N .W s (φ) = 0, and then On the other hand, assuming that there is no solution for (1.1) we have and then ∂σ = 0. Thus, σ can be taken to be zero, and so ∂ W u (N ) = N . Using (2.14) and (2.15) we obtain γ = τ . A contradiction! Thus, ς 0 = 0, i.e., ∂ W u (N ) contains some unstable manifold W u (w) of some solution w of (1.1), which is dominated by N , that is of Morse index equal to dim N = m.

Proof of Corollary 2.12
Proof The most massive critical point at infinity is φ = (y 0 , y 1 , . . . , y ) ∞ , and its Morse index (see Proposition 3.8) is precisely m ∞ = + j= j=1 k j , (k 0 = 0). Since any other critical point at infinity φ = (y i 1 , y i 2 , . . . , y i ) ∞ , with , has a Morse index smaller than the one of φ , and since m ∞ < m, we derive that A ∞ m = ∅, so condition (H 2 ) is satisfied, and then, since τ = 0 = γ , the result follows by Theorem 2.9.

Proof of Theorem 2.14
For c ∈ R we let J c = {u ∈ + , J (u) c}. First, using the characterization of d, we know that we can find a cycle homologous to N in W u (N ) ∩ J d+ε , for ε > 0 sufficiently small. Next, we can write the exact homology sequence of the pair (W u (N ) ∩ J d+ε , W u (N ) ∩ J d−ε ). Arguing by contradiction, we assume that there is no critical point of index m at the level d. Observe that since + is contractible, we have in particular that H m+1 ( + ) = H m ( + ) = 0. Hence, under the assumption (H 2 ), using Proposition 7.24 of [6], we derive that Therefore the homomorphism H m (W u (N ) ∩ J d+ε ) −→ H m (W u (N ) ∩ J d−ε ) is onto, and so there is a cycle homologous to N in W u (N ) ∩ J d−ε . This contradicts the definition of d. The result follows.
The flow of W acts essentially on α. The action of bringing α to zero or to one depends on whether α < 1 2 , in which case u goes to W u (y i 0 ) ∞ = X , or α > 1 2 , in which case u goes to W u (y 0 ) ∞ = {y 0 }. On the other hand, we have another action on x ∈ X = W s (y i 0 ), when α = 1 − α = 1 2 . Observe that we then have u = 1 2 ϕ (y 0 ,λ) + 1 2 ϕ (x,λ) .
Since only x can move, then y 0 remains one of the concentration points of u, and x goes to W s (y j ), where y j is a critical point of K dominated by y i 0 . Thus, u goes to W u (y 0 , is a manifold in dimension at most k−1. Therefore, X ∪W u (y 0 , y i 0 )∪Z contains a strong retract of f λ (C y 0 (X )).
Since μ(y 0 ) = 0, one can be more precise. This strong retract does not intersect W u (y 0 , y i 0 ), and thus it is contained in X ∪ Z . It can be written as X ∪ S, where S ⊂ Z is a stratified set (see Proposition 7.24 in [6]), and thus is of dimension at most k − 1. Therefore H q (X ∪ S) = 0 for all q ∈ N * , since f λ (C y 0 (X )) is a contractible set. Using the exact homology sequence of the pair (X ∪ S, X ), we have Since H q (X ∪ S) = 0 for each q ∈ N * , then we derive that H k (X ) = H k+1 (X ∪ S, X ). Observe that (X ∪ S, X ) is a stratified set of dimension at most k, so we have H k+1 (X ∪ S, X ) = 0, and therefore, H k (X ) = 0 which is a contradiction. Whence the result.

Proof of Theorem 2.16
Proof By the same argument as above, is a strong retract of f λ (C y 0 (X )), where S ⊂ Z is a stratified set, and is of dimension at most k. Now, since μ j (y 0 ) = 0 for each j = 1, . . . , d, then we derive that X ∪ S is a strong retract of f λ (C y 0 (X )), and, because f λ (C y 0 (X )) is a contractible set, then H q (X ∪ S) = 0 for all q ∈ N * . Using the exact homology sequence of the pair (X ∪ S, X ), we derive that H k+1 (X ∪ S, X ) = H k (X ) = 0, a contradiction. The result then follows.
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