Morse theory for $S$-balanced configurations in the Newtonian $n$-body problem

For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the $n$-body is a constant up to rotations and scalings named \textit{central configuration}. For $d\leq 3$, the only possible homographic motions are those given by central configurations. For $d \geq 4$ instead, new possibilities arise due to the higher complexity of the orthogonal group $O(d)$, as observed by Albouy and Chenciner. For instance, in $\mathbb R^4$ it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $d\geq 4$ there is a wider class of $S$-\textit{balanced configurations} (containing the central ones) providing simple solutions of the $n$-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $\mathbb R^d$, for arbitrary $d\geq 4$, and establish a version of the $45^\circ$-theorem for balanced configurations, thus answering some questions raised by Moeckel. Also, a careful study of the asymptotics of the coefficients of the Poincar\'e polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of $S$-balanced configurations. In the last part of the paper, we focus on the case $d=4$ and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational $n$-body problem which improves a previous celebrated result of McCord.


Introduction
The Newtonian n-body problem concerns the motion of n point particles with masses m j ∈ R + and position q j ∈ R d , where j = 1, . . . , n and d ≥ 2, interacting each other according to Newton's law of inverses squares. The equations of motion read m jqj = ∂U ∂q j , where U (q 1 , . . . , q n ) := i<j m i m j |q i − q j | .
As the center of mass has an inertial motion, there is no loss in generalities in assuming that the center of mass lies at the origin. The configuration space with center of mass at the origin is defined as The potential U is not defined when (at least) two particles collide, meaning that they have the same position. Therefore, we define the space of collision free configurations as X := { q = (q 1 , . . . , q n ) ∈ X | q i = q j for i = j } = X \ ∆, where ∆ := q = (q 1 , . . . , q n ) ∈ R dn q i = q j for i = j and let ·, · M := M ·, · and | · | M := M ·, · 1/2 be respectively the mass scalar product and the mass norm, where ·, · denotes the Euclidean product in R nd . In this notation, the equations of motion can be written as (1.1)q = M −1 ∇U (q).
Among all possible configurations of the system, a crucial role in penetrating the intricate dynamics of this problem, is played by the so-called central configurations (CC in shorthand notation), namely configurations in which M −1 ∇U (q) is parallel to q: (1.2) M −1 ∇U (q) + λq = 0 In other words, the acceleration vector of each particle is pointing towards the origin with magnitude proportional to the distance to the origin. As a straightforward consequence of the homogeneity of the potential we obtain that the proportionality constant λ is actually equal to −U (q)/|q| 2 M . Equation (1.2) is a nonlinear algebraic equation which turns out to be extremely hard to solve. Despite substantial progresses (starting from the work of -among others -Smale, Conley, Albouy, Chenciner, McCord, Moeckel, Pacella) have been made in the last decades, many basic questions about CC still remain unsolved. Nevertheless, there are several reasons why CC are of interest in the n-body problem and more generally in Celestial Mechanics: -Every CC defines a homothetic solution of (1.1), namely a solution which preserves its shape for all time while receding from or collapsing into the center of mass.
-Planar CC give rise to a family of periodic motions of (1.1), the so-called relative equilibria, in which each of the bodies moves on a circular Kepler orbit. In other words, the configuration rigidly rotates at a constant angular speed about the center of mass. These motions are true equilibrium solutions in a uniformly rotating coordinate system and correspond to elliptical orbits of the Kepler problem of eccentricity e = 0. It is worth observing that relative equilibria are the simplest periodic solutions of the n-body problem having an explicit formula.
-Planar CC more generally give rise to a family of homographic solutions of (1.1) in which each particle traverses an elliptical orbit with eccentricity e ∈ (0, 1). In this case, both the radius r(t) of the orbit and the angular speedθ(t) are time-dependent solutions of the Kepler problem (in polar coordinates). Moreover, the configuration remains similar to the initial configuration throughout the motion, varying only in size.
-CC control the qualitative behavior of an important class of colliding solutions (and parabolic motions) of the n-body problem. More precisely, if an orbit approaches or recedes from total collapse, it does so approximating homothetic orbits. So, orbits passing arbitrarily close to a total collision do it by first approaching one homothetic motion and then receding from the collision following a (possibly different) homothetic motion.
In other words, for the n-body problem in R d , d ≤ 3, any CC generates a homothetic ejection or collapse. Morevorer, any planar CC generates planar Keplerian homographic motions, whereas spatial non-planar CC can only produce homothetic motions. Configurations which are not central cannot produce homographic motions at all. If we instead allow dimensions d ≥ 4, then -as observed by Albouy and Chenciner in [AC98] (cfr. also [Moe14]) -there is a wider class of the so-called "S-balanced configurations" which produces relative equilibria of the n-body problem. These new phenomena are due to the higher complexity of the group of rotations in higher dimensions, which allows, for example, to rotate in two mutually orthogonal planes with different angular velocities. This leads to new ways of balancing the gravitational forces with centrifugal forces in order to get new relative equilibria of the n-body problem. We shall notice that, in contrast with the case d = 2, such relative equilibria need not be periodic in time. Indeed, if the angular velocities are rationally independent, then the resulting motions will be only quasi-periodic.
In order to formally define such a class of configurations, we consider positive real numbers s 1 ≥ . . . ≥ s d > 0 (possibly not all different) and set S := diag(S, ..., S n-times ) ∈ R nd×nd , where S = diag(s 1 , ..., s d ). An S-balanced configuration, (SBC in shorthand notation), is an arrangement of the masses whose associated configuration vector q ∈ X satisfies (1.3) ∇U (q) + λ SM q = 0 for some real (positive) constant λ. We observe that for n = 3 a planar non-equilateral and non-collinear isosceles triangle is a SBC (but not a CC) as soon as the two symmetric masses are equal. Also, the class of SBC strictly includes CC (corresponding, in fact, to S = I), and Equation (1.3) is in general only invariant under the (diagonal) action of a subgroup of SO(d), which is proper as soon as S = I. In the extremal case in which the (diagonal) entries of S are pairwise distinct, the symmetry group of Equation (1.3) is trivial.
Remark 1.1. The usual definition of S-balanced configurations (cfr. e.g. [Moe14]) requires S to be a symmetric positive definite matrix. However, since the problem is invariant by unitary transformation, there is no loss of generality in assuming that S be in diagonal form, i.e. that the spectral basis coincide with the canonical basis of R d . Also, one requires S to be minus the square of a skew-symmetric matrix (e.g. a complex structure), and hence, in particular, that all eigenvalues of S have even multiplicity. As we shall see later, it will be very convenient to allow S to have eigenvalues of odd multiplicity (in particular simple).
It is worth pointing out that, in even dimension, if S = −A 2 where A is an antisymmetric matrix, every SBC gives rise to a uniformly rotating relative equilibrium solution of Equation (1.1). To get more general homographic solutions it is still necessary to have a CC in which the bodies run on planar Keplerian ellipses. However, ellipses corresponding to different bodies may lie in different planes.
In order to find solutions of (1.3), we will exploit the fact that SBC (as well as CC) admit a variational characterization as critical points of the restriction of U to the collision free configuration sphere is the configuration sphere and I S (q) if the S-weighted moment of inertia, namely the norm squared associated to the scalar product induced by SM . The proof of Shub's lemma carries over word by word to this more general situation, thus showing that SBC cannot accumulate on the collision set ∆. In particular, we can find a neighborhood of ∆ in S which contains no SBC. This opens up the possibility of using Morse theoretical methods to obtain lower bounds on the number of solutions to (1.3). However, in doing this, one has to keep in mind that: • Equation (1.3) has a non-trivial symmetry group whenever at least two entries of the matrix S are equal, and hence critical points are not isolated unless all entries of S are distinct. Also, such an action is not free (actually, non even locally free) as soon as d ≥ 3. Therefore, the quotient space is in general not even an orbifold but rather an Alexandrov space, and hence a delicate stratified Morse theory is needed when working on the quotient space. Methods such as Morse-Bott theory, which allow to deal with functions having (Morse-Bott) non-degenerate critical manifolds, do not take into account the symmetries of the problem and hence necessarily lead to weaker results.
• Even if one is able to give lower bounds on the number of SBC, for instance via equivariant Morse theory or Morse-Bott theory, one has additionally to show that the SBC that one finds are actually not CC. This is the same problem that one has to face when trying to exclude that spatial CC are actually not planar: despite the lower bounds provided by Pacella [Pac86], it is so far not known whether there is more than one spatial CC which is not planar.
• All previously mentioned methods are essentially based on equivariant homology and intersection homology theories, respectively, which are computationally unaccessible in full generality.
We will circumvent all these difficulties at once by restricting our attention to a subspace of R d in which the eigenvalues of S are pairwise distinct (in what follows this will be called the "reduction to Assumption (H1)" argument). Assumption (H1) will namely kill all symmetries of Equation (1.3), thus allowing us to apply standard Morse theory arguments. Also, the fact that all entries of S are distinct implies that solutions of (1.3) which are not collinear cannot be CC. More about Assumption (H1) will be said in Section 2. Therefore, applying classical Morse theory to U := U | S under Assumption (H1), we derive lower bounds on the number of SBC assuming non-degeneracy. These lower bounds will depend on suitable spectral gap conditions involving the eigenvalues of the matrix S (cfr. Theorems 4.2, 4.4, and 4.6 for the precise statements). A first key step in this direction is provided by a careful investigation of collinear SBC (in shorthand notation CSBC) and their inertia indices (cfr. Lemma 2.13 for further details). The Poincaré polynomial of S is the same as in the case of CC, however establishing precise growth estimates on its coefficients (cfr. Proposition 3.2) we are able to derive non-trivial and rather unexpected qualitative consequences on the count of SBC: roughly speaking, CSBC corresponding to smaller eigenvalues of S will in many cases contribute more than CSBC corresponding to the largest eigenvalue of S (this should be compared with [Moe14, Pag. 151]). We stress on the fact that all lower bounds obtained via the reduction to (H1) argument provide genuine SBC.
In the last section we prove a sharper lower bound on the number of non-degenerate SBC in dimension d = 4 for S = diag(s, s, 1, 1). Even if the proof is again based on Morse theory, the argument is completely different than the one used in the proof of Theorem 4.2. After restricting our attention to the {0} × R 3 ⊂ R 4 , we quotient out S by the SO(2)-action given by rotations in the {0} × R 2 -plane after removing the manifold Y of collinear configurations contained in R× {0}. This is done by first establishing a so-called 45 • Theorem for SBC (cfr. Theorem 5.2). Therefore, the quotient is a manifold. We then compute in Theorem 6.2 the homology of S by using Alexander duality and the Gysin long exact sequence, and finally obtain the desired lower bound by explicitly computing the sum of the Betti numbers of S. This has an important consequence on the lower bound of periodic orbits of the gravitational n-body problem. In fact, assuming non-degeneracy, we have at least relative equilibria in R 4 for fixed s > 1. Such a number is roughly twice as large as the lower bound proved by McCord in [McC96]: among all of these relative equilibria, at least n! 2 h(n) + 1) are produced by CC, namely n!h(n)/2 corresponding to planar non-collinear CC (McCord's estimate) and n!/2 corresponding to collinear CC (CCC in shorthand notation). By subtracting the latter integer to the former one, we get at least n! h(n) 2 + 1 n relative equilibria which do not come from McCord's estimate, but could anyway come from CC (unlike in Theorem 4.2, this cannot be excluded a priori). Finally, we observe that the relative equilibria coming from genuine SBC are periodic in time for s ∈ Q, and quasi-periodic otherwise. We finish this introduction by summarizing, for the reader's sake, some of the notation that we shall use henceforth throughout the paper.
• M is the nd-dimensional mass matrix; I d is the d-dimensional identity matrix; S is a d-dimensional diagonal matrix, S := diag(S, . . . , S).
• ·, · M , | · | M denote respectively the mass scalar product and the mass norm.
• I(q), I S (q) denote the moment of inertia and the S-weighted moment of inertia (w.r.t. origin).
• X is the configuration space with center of mass at the origin; ∆ is the collision set and X := X \ ∆ is the collision free configuration space; S is the configuration sphere and S := S \ ∆ is the collision free configuration sphere.
• U := U | S is the restriction of U to S.
• CC, SBC, CCC, CSBC denote respectively the set of central configurations, S-balanced configurations, collinear CC and finally collinear SBC.
Acknowledgments. Luca Asselle is partially supported by the DFG-grant 380257369 "Morse theoretical methods in Hamiltonian dynamics".

S-balanced configurations: definition and basic properties
This section is devoted to recall the definition of S-balanced configurations and to collect some of their properties. Our basic reference, for this section, is [Moe14]. For n ≥ 2, let m 1 , . . . , m n be positive real numbers (which can be thought of as the masses of the n points) and let M be the diagonal (block) (nd × nd)-matrix defined as where I d denotes the d-dimensional identity matrix, d ≥ 1. Denoting by ·, · the Euclidean product in R nd , we denote by ·, · M := M ·, · and | · | M := M ·, · 1/2 respectively the mass scalar product and the mass norm. As the center of mass has an inertial motion, it is not restrictive to prescribe its position at the origin. Therefore, we define the configuration space with center of mass at the origin as It is readily seen that X is an N -dimensional (real) vector space where N := d(n − 1). We define the space of collision free configurations as where ∆ := q = (q 1 , . . . , q n ) ∈ R dn q i = q j for i = j denotes the collision set. The Newtonian potential U : X → R at the configuration vector q is given by and so, Newton's equations of motion read as follows where ∇U denotes the nd-dimensional gradient. Given positive real numbers s 1 . . . s d > 0 (possibly not all different), we let S be the diagonal (d × d)-matrix defined by S = diag(s 1 , ..., s d ), and Definition 2.1. An S-balanced configuration, (SBC in shorthand notation), is an arrangement of the masses whose associated configuration vector q ∈ X satisfies (2.1) ∇U (q) + λ SM q = 0 for some real (positive) constant λ.
Remark 2.2. In [Moe14] the matrix S is assumed to be symmetric and positive definite. Nevertheless, we are not leading in generalities assuming here that S is in diagonal form. Indeed, a symmetric matrix can always be diagonalized by choosing a spectral basis. In other words, here we are simply assuming that the spectral basis coincides with the standard basis of R d . This is possible since the formulation of the problem doesn't depend upon this choice. Also, one usually assumes that S be minus the square of a skew-symmetric matrix, hence in particular that each eigenvalue of S have even multiplicity. As we shall see later, it will be very convenient to allow S to have eigenvalues of odd multiplicity, in particular simple eigenvalues.
The two extreme cases correspond to s 1 = . . . = s d = 1, respectively to s i = s j for all i = j. In the former case we get the well-known notion of central configuration (CC for short) for which a rich literature is nowadays available, whereas the latter case will be the main object of interest of the present paper. where | · | 2 S denotes the norm squared induced by the scalar product ·, · S := SM ·, · . Remark 2.4. Taking the scalar product of Equation (2.1) with q and using Euler's theorem for positively homogeneous functions, we get that the (positive) constant λ appearing in Equation (2.1) is given by In particular, for any SBC q, we get a continuous family of SBC by scaling.
Definition 2.5. Under the previous notation, we define the configuration sphere and the collision free configuration sphere as follows S := { q ∈ X | I S (q) = 1 } and S := S \ ∆ We refer to its elements as (normalized) configuration vectors.
It is immediate to check that S is a smooth compact manifold diffeomorphic to a (N − 1)-dimensional sphere. As a direct consequence of the scaling property of Equation (2.1), with any SBC we can associate a unique normalized SBC. Hereafter, we will refer to normalized SBC simply as SBC.
The next result can be obtained by a straightforward modification of the proof of the variational characterization of CC and provides a variational characterization of SBC as critical points of the restriction of U to the collision free configuration sphere.
Lemma 2.6. A (normalized) configuration vector q is an SBC if and only if it is a critical point of U := U | S .
In the next result, we provide a representation of the Hessian quadratic form at any SBC w.r.t. to the the Euclidean as well as w.r.t. the mass scalar product.
Lemma 2.7. The Hessian of U : S → R at a critical point q is the quadratic form on T q S that can be represented w.r.t. the Euclidean and mass scalar product respectively by the dn-dimensional matrices Remark 2.8. The latter representation is the natural choice if we are looking at the Hessian at an SBC as the linearization of the gradient flow D ∇ U (q), where ∇ is the gradient on S with respect to the Riemannian metric induced by the weighted mass-scalar product of the ambient manifold.
By setting a direct computation shows that the (i, j)-entry of the block symmetric matrix D 2 U (q) is given by The group SO(d) acts diagonally on S. Since the potential is rotationally invariant, we get that In particular, the Hessian of a CC is SO(d) invariant, whereas for general SBC the Hessian is invariant by a proper subgroup of SO(d), which is the trivial subgroup in case s i = s j for all i = j.

CSBC and Assumption (H1)
In this subsection we provide a useful representation of the Hessian at a collinear SBC (CSBC in shorthand notation) and we discuss Assumption (H1) that will be used throughout the paper. As a preliminary observation, we see from an easy inspection of Equation (2.1) that the line containing each body q k of a CSBC must lie in an eigenspace of the matrix S. In particular, if the s j 's are pairwise distinct, then the only lines containing CSBC are the coordinate axis.
Notation 2.9. We can assume without loss of generality that each particle of the CSBC q is in the "j-th eigenspace" e j of S, where as usual e j denotes the j-th vector of the canonical basis. Up to relabeling the e j 's if necessary, we can assume that each body q k of the CSBC q lies on R × (0) d−1 ⊂ R d , and so the unit vectors u ij are all multiples of e 1 .
By a straightforward computation we get that each block D ij has the following representation After rearranging the dn coordinates of the configuration vector q by collecting the components into groups of n variables with all of the e 1 components first, the e 2 components second and so on, the Hessian at the CSBC is of the form where B(q) is the n × n matrix whose (i, j)-entry is given by Remark 2.10. We recall that the dimension of the configuration sphere is N − 1 = d(n − 1) − 1 = dim T q S and so, for d = 2 (planar case) we get 2n − 3, whereas for d = 3 (spatial case), we get 3n − 4. In the particular case of CCC (corresponding to s 1 = .... = s d = 1), the (n × n)-block matrix M −1 B(q) has Morse index (n − 1) (this result can be traced back to Conley) with largest eigenvalue −U (q), and nullity 1 with corresponding eigenvector (1, . . . , 1) (being transverse to the tangent space at q to S) which does not contribute to the inertia indices. Similarly, the block −2M −1 B(q) has Morse coindex (n − 1) and nullity 1, which once again does not contribute to the inertia indices. In particular, the inertia indices of the CCC q are Starting from Remark 2.10, we provide some interesting information about the inertia indices of a CSBC. Hereafter, until the end of Section 4, we will always implicitly assume that Such an assumption might appear at a first glance rather restrictive. However, it has several noteworthy advantages that we will exploit throughout the paper and that we now thoroughly discuss. Before commenting further on Assumption (H1), we shall notice that there is no loss of generality in assuming that the s i 's be strictly decreasing and that s d = 1. This follows from the fact that we can always rearrange the vectors of the standard basis of R d and that the problem is invariant under scaling.
• Assumption (H1) "kills" all symmetries of Equation (2.1), thus allowing to apply standard Morse theory arguments (rather than more complicated theories, such as e.g. equivariant Morse theory or Morse-Bott theory) when looking for lower bounds on the number of SBC.
• The lower bounds on the number of SBC that one obtains assuming (H1) still allow to give (possibly not optimal but still remarkable) lower bounds on the number of SBC in the general case via a suitable "reduction to (H1) argument" as we now explain. Thus, suppose that for d 2 positive real numbers s 1 > ... > s d = 1 and natural numbers µ 1 , . . . , µ d ∈ N are given. On R D , D := µ 1 + . . . + µ d d, we consider the S-balanced configurations problem with ).
Notice that, under the assumptions above, Equation (2.1) is invariant by the diagonal action on the configuration space of the subgroup of SO(D) given by the product group SO(µ 1 ) × ... × SO(µ d ). For each i = 1, ..., d we fix a line L i on the µ i -dimensional subspace of R D relative to s i , and restrict our attention to SBC which are contained in Ignoring all vanishing components of configuration vectors which lie in Y , we obtain that SBC which are contained in Y satisfy Equation (2.1) on R d with which is nothing else but (H1) on a proper subspace of R D . We call this procedure the reduction to (H1) argument.
• Applying equivariant theories for the S-balanced configurations problem in R D seems hopeless in full generality due -among other facts -to the amount of cases that one should treat (corresponding to different choices of the µ i 's) and to the fact that, for d 3, the diagonal action of the symmetry group SO(µ 1 ) × ... × SO(µ d ) of Equation (2.1) on the configuration space is not free, thus yielding a quotient which is not a manifold (actually, not even an orbifold).
• Theories like Morse-Bott theory, which allow to deal with functions having "non-degenerate" critical manifolds rather than critical points, do not take into account the symmetries of the problem and thus necessarily lead to weaker results. For instance, it is possible to show that for the S-balanced configurations problem on R 4 with S = diag(s 1 , s 1 , 1, 1), the lower bounds on the number of SBC that one obtains via the reduction to (H1) argument are precisely twice as much as the lower bounds that one obtains by Morse-Bott theory.
• All methods involving equivariant or stratified Morse theories are essentially based on equivariant homology and intersection homology theories, which are computationally unaccessilbe in full generality.
• If the singular set consists only of collinear configurations, under suitable assumptions, one could try to overcome these difficulties by quotienting out the configuration space by the group action after removing the singular set. We will say more about this in Section 6, where we will compare the lower bounds on the number of SBC in R 4 with S = diag (s 1 , s 1 , 1, 1) obtained implementing such a strategy with the ones obtained via the reduction to (H1) argument (thus, by considering the S-balanced configurations problem in R 2 with matrix diag(s 1 , 1)).
• Even if one is able to give lower bounds on the number of SBC not assuming (H1), for instance via equivariant Morse theory or Morse-Bott theory, one has additionally to show that the SBC that one finds are actually not CC. Notice indeed that, for SBC which are contained in a subspace of R D corresponding to some s i , Equation (2.1) reduces to the central configurations equation. This is the same problem that one has to face when trying to exclude that spatial CC are actually not planar: despite the lower bounds provided by Pacella [Pac86], it is so far not known whether there is more than one spatial CC which is not planar.
Assuming (H1), instead, we easily overcome this problem: indeed, under Assumption (H1), every solution of (2.1) which is not collinear cannot be a CC.
• Under Assumption (H1), solutions of Equation (2.1) yield relative equilibria for the n-body problem in R 2d via "complexification", which can be thought of as the inverse procedure to the reduction to (H1)

CSBC and corresponding CCC: spectra and inertia indices
In what follows, we refer to a CSBC in which every particle q k belongs to e j (the eigenspace corresponding to the eigenvalue s j ) just as s j − CSBC. In this subsection, after describing the relation between s j − CSBC and the corresponding CCC, we provide an explicit description of their inertia indices.
Lemma 2.11. Let q be a s-CSBC, where s = s j for some j = 1, ..., d. Then: Proof. By assumption, the CSBC q lies in the eigenspace of S corresponding to the eigenvalue s and | √ s q| M = 1. Thus, by a direct computation, we get Setting q := √ s · q, we obtain ∇U ( q) = −U ( q)M q which implies that q is a CCC. The homogeneity of U implies that and hence the eigenvalues of B(q) are easily obtained by multiplying each eigenvalue of B( q) by s √ s.
Remark 2.12. As we shall see below, from Lemma 2.11 it follows that the nullity of q is generically zero, but might be non-zero for certain choices of the s i 's. Moreover, the coindex of a CSBC can possibly be greater than the coindex of the corresponding CCC, as it could increase (but not decrease) for varying s. Analogously, the index of a CSBC can possibly be lower than the index of the corresponding CCC, as it could decrease (but not increase) for varying s. Loosely speaking, since the eigenvalues of S are assumed to be greater than 1, we get that adding to the block matrix M −1 B(q) a block diagonal matrix of the form s i U (q)I n , we can move part of the spectrum located on the negative real line to the positive thus changing the inertia indices.
Another important property which is pointed out by Lemma 2.13 is the different role (with respect to the inertia indices of a CSBC) played by the different eigenspaces. In fact, for an s − CSBC the index is maximal, and the coindex is minimal, for s = s 1 .
Lemma 2.13. Let q be a CSBC. The following facts hold.
1. The index of q is at most (d − 1)(n − 1) and the coindex of q is at least (n − 2) 2. If q is an s 1 − CSBC, then the Morse index is precisely (d − 1)(n − 1) and the Morse coindex is precisely n − 2. Furthermore, if q is an s j − CSBC and j < d, then q cannot be a local minimum of U .
Proof. We only prove the second statement being the first a straightforward consequence of the first one. We assume that q is a s j -CSBC for some j ∈ {1, . . . , d}. By using Lemma 2.11, for i = j we get that where q is the c corresponding to q. Depending on i and j, the quantity can be positive or negative. For j = 1 (corresponding to one of the two extremal cases), this term is always negative (i.e. independent of i). Thus, also the nullity of the central configuration q contributes to the Morse index of the s 1 -CSBC (in fact, the contribution is 1 for each of the (d − 1) blocks). The contribution to the coindex coming from the first block of the Hessian is n − 2. Arguing analogously, we see that for j > 1 the Morse index is strictly less than (d − 1)(n − 1), whereas the coindex is strictly greater than n − 2. This follows from the fact that, for i < j, the above quantity is strictly positive and hence the nullity of the CC q contributes for such i's to the Morse coindex. Finally, for j < d, the above quantity for i = d is strictly negative, thus the nullity of q contributes to the Morse index.
The next result describes the properties of a CSBC in terms of the spectra of the corresponding CCC (or s d − CSBC which is the same) as well as of the eigenvalues of S. Then: • q is a local minimum of U provided that η + s d−1 U ( q) > 0 • q has non-vanishing Morse index if and only if η + s d−1 U ( q) < 0.
• If η + s 1 U ( q) < 0, then for every j = 1, ..., d, then the s j -CSBC whose ordering of the masses coincides with the one of q has non-vanishing Morse index for the restriction of U to any possible subspace of R d of the form span {e j , e i1 , ..., e i h }, h 1.
Proof. For i < d, the smallest eigenvalue of the i-th block of the Hessian The first two claims readily follows. As far as the last claim is concerned, we observe that for the s j − CSBC q whose ordering of the masses coincides with the one of q and for every i ∈ {1, . . . , d} we have and hence in particular for the smallest eigenvalue (up to the positive factor s j √ thus showing that the Morse index is at least one. Clearly, the value of U ( q), as well as the matrix B( q) and its eigenvalues, depend on the choice of the CCC (i.e. on the choice of the ordering of the masses), unless all masses are equal. Therefore, the conditions above are a priori different for different ordering of the masses, and there is in general no direct way of comparing the inertia indices of CSBC with different ordering of the masses. Nevertheless, we can still use this information to deduce existence of non-collinear SBC as we now show.
Corollary 2.15. Under the notation above, if all masses are equal and then there are at least d(d−1) 2 planar non-collinear SBC. In particular, the n-body problem in R 2d ∼ = C d has at least d(d−1) 2 continua of relative equilibria in which periodic motions form a dense subset. If the masses are not all equal, then the same conclusion holds provided one replaces the spectral condition above with Proof. For 1 ≤ i < j ≤ d fixed, we consider the (e i , e j )-plane in R d . If all masses are equal, then Lemma 2.14 implies that the global minimum of U restricted to span {e i , e j } cannot be attained at a CSBC, and hence must be attained at a planar non-collinear SBC. The first part of the claim follows, since the cardinality of the set . We now set The stability property of minima implies that the maps yield the desired continua of relative equilibria. Such relative equilibria will be periodic motions whenever s 1 , ..., s d−1 , 1 are rationally dependent, and quasi-periodic motions otherwise. This completes the proof in the case all masses are equal. The proof is analogous in the case of masses which are not all equal. Indeed, the new spectral gap condition implies that none of the CSBC can be a local minimum for the restriction of U to span {e i , e j } as well. The details are left to the reader. Actually, by using similar arguments, it is possible to prove the existence of others continua of relative equilibria coming from minmax points (that must exist for topological reasons). Remark 2.17. To any planar non-collinear CC we can associate two integers, the so-called planar index and normal index. As shown by Moeckel in [Moe94, Pag. 69] (by using an argument originally due to Pacella) estimating from below the trace of −M −1 B(q) at such a planar non-collinear CC, it is possible to provide a lower bound on the normal index of a central configuration. A direct consequence of this estimate is that the global minimum of U doesn't occur at a planar CC, and so proving the existence of at least one non-planar CC. It would be interesting to understand if an analogous argument can be carried over to the case of planar non-collinear SBC.
We conclude this section by determining the inertia indices of CSBC in case d = 2. Thus, let S = diag (s 1 , 1) with s 1 > 1. By Lemma 2.13 we get that the inertia indices of a s 1 − CSBC, denoted q, are whereas for a s 2 − CSBC (which are nothing else but CCC, thus denoted q) we have In the next result we provide a complete characterization of the inertia indices of s 2 − CSBC depending on s 1 and on the spectrum of M −1 B at q. Thus, let q be a s 2 − CSBC (or, equivalently, a CCC), and denote by η k ( q) < ... < η 1 ( q) < −U ( q) < 0 the eigenvalues of the matrix M −1 B( q). Even though the η i 's depend on the choice of q, to ease the notation we hereafter drop the dependence on q. It is well known that the eigenvalues −U ( q) and 0 are simple, but any other igenvalue might have multiplicity. For j = 1, . . . , k we denote by α j the multiplicity of η j as an eigenvalue of M −1 B( q). We also set η 0 := −U ( q) and α 0 := 1.
Proposition 2.18. Let q be an s 2 − CSBC. Then, the following hold.

If s
In particular, in this case, q is a local minimum of U .
Proof. In order to prove the first claim we start observing that by adding the term s 1 U ( q)I n to the matrix M −1 B( q) part of the spectrum becomes positive. Since the assumption is equivalent to η j+1 < −s 1 U ( q) < η j , we get η j+1 + s 1 U ( q) < 0 < η j + s 1 U ( q).
So, to the Morse coindex of q we add the contribution provided by the first (j + 1) eigenvalues of M −1 B( q) which become positive by adding s 1 U ( q). Clearly, the same contribution has to be subtracted from the Morse index. The proof of the second item is completely analogous. We just need to observe that the (j + 1)-th eigenvalue η j of M −1 B( q) gives a non-trivial contribution to the nullity. The third item follows straightforwardly since, under the assumption s 1 U ( q) + η k > 0, the whole spectrum of M −1 B( q) moves into the positive real axis.
Remark 2.19. As a direct consequence of Proposition 2.18, we get that there are only finitely many values of s 1 for which s 2 − CSBC can be degenerate as critical points of U . In other words, s 2 − CSBC are generically non-degenerate. We shall however remark that the values of s 1 that make an s 2 − CSBC degenerate depend on the s 2 − CSBC, unless the masses are all equal. In a similar vein, where and how big the jumps on the Morse index and coindex are depend on the s 2 − CSBC.
A result analogous to Proposition 2.18, which will be omitted to keep the exposition as simple as possible, can be proved also for d > 2, thus showing as a corollary that CSBC are generically non-degenerate.

Estimates on the coefficients of the Poincaré polynomial
In this section we provide some elementary but rather not trivial (asymptotic) estimates on the coefficients of the Poincaré polynomial of the collision free configuration sphere. The interest in such estimates relies mainly on the following facts: • They yield non-trivial lower bounds for the number of SBC, since they show that also s j − CSBC, j ≥ 2, can contribute to the count of critical points.
• They have highly non-trivial and rather unexpected qualitative consequences on the count of critical points. Indeed, in "some cases" s j − CSBC, j ≥ 2, will contribute more than s 1 -collinear SBC (this should be compared with [Moe14, Pag. 151]).
• They might have applications in the study of limiting problems (i.e. with very large number of bodies).
We start by describing the Poincaré polynomial of the collision free configuration sphere S. This result is actually well-known, however for the reader's convenience we provide a sketch of the proof.
Lemma 3.1. The Poincaré polynomial of S is given by Proof. We start observing that the collision free configuration sphere S is homotopy equivalent to R dn \ ∆, as By Künneth formula, the Poincaré polynomial of a product space is the product of the Poincaré polynomials; so, in particular, the Poincaré polynomial of R dn \ ∆ is equal to the Poincaré polynomial of S. Thus, it remains to determine the Poincaré polynomial of R dn \ ∆. We argue by induction over n.
For n = 2, we get that R d \ (0) is a strong deformation retract of R 2d \ ∆; so, Since, for d ≥ 2, the (d − 1)-dimensional sphere has non-vanishing Betti number only in dimension 0 and d − 1, we get that the Poincaré polynomial of R 2d \ ∆ is For n > 2 the map π : R nd \ ∆ → R d(n−1) \ ∆, π(q 1 , . . . , q n ) := (q 1 , . . . , q n−1 ), is a fiber bundle projection known as the Fadell-Neuwirth projection map. Although the fiber bundle is non-trivial, it admits a smooth global cross-sectionand so the Poincaré polynomial of the total space is the product of the Poincaré polynomial of the base and the one of the fiber (cfr. e.g. [FN62,FH01,Coh10] for further details). The claim follows by the inductive hypothesis.
We are now ready to prove some results about the growth rate for the coefficient c It is worth observing that the Poincaré polynomial P (t) of S is nothing else but p n (t d−1 ). For notational convenience we hereafter set c (n) j = 0 for every j ≥ n. To our best knowledge, such a representation has never appeared in the literature. Notice also that Part 1 immediately implies that all but at most one coefficients c (n) j are smaller than n!/2. We observe that the inequality in Part 2 is strict as soon as n ≥ 4.
Remark 3.4. Alternatively, one can show that for j ∈ N fixed, the coefficient c (n) j has a polynomial growth in n.

The first claim is trivial. As far as c
and the claim follows. Now, for fixed j ≥ 2 we compute ...
Remark 3.5. For j ∈ N sufficiently large the estimate provided by the last item in Proposition 3.2, is very far from being optimal. In fact, as we shall see below, a much stronger statement holds. More precisely, for every sequence (j n ) n ⊂ N we have The last two items of Proposition 3.2 confirm the validity of (3.1) when (n − j n ) n and (j n ) n are uniformly bounded. In the case of positively divergent sequences j n , the behavior is described by Proposition 3.6.
Proposition 3.6. Let (j n ) n ⊂ N be a positively divergent sequence. Then, the following hold.
2. If j n log log n, then 3. If j n log n, then for every k ∈ N fixed Remark 3.7. Proposition 3.6 states that every sequence (c (n) kn ) n of coefficients of the polynomials p n (z) grows slower than n!. Moreover, if the sequence (k n ) n does not grow "too fast", then we get even a slower growth; in fact, c (n) kn grows slower than (n − 1)! if k n n − log log n and slower than (n − k)! for every k ∈ N provided k n n − log n. Notice that for (k n ) n bounded we have polynomial growth, and that (n − 1)! is precisely the coefficient of p n (z) of degree (n − 1).
Before proving the proposition we observe that for every n ∈ N and every j ≤ n − 1: The key step is the following result, whose proof is postponed to Appendix A.
Lemma 3.8. For every n, j ∈ N we have Proof of Proposition 3.6. and we readily see that the exponent goes to −∞ for n → +∞, as ǫ log n dominates log log n and j n log j n dominates the remaining terms.
2. Arguing in a similarly way, we finally get c (n) n−jn−1 (n − 1)! log jn (n) j n ! ∼ e log jn+log log n− 1 2 log(2πjn)−jn log jn+jn → 0, as j n log j n is the leading term for j n log log n.
3. For fixed k ≥ 2, we compute ∼ e (k−1) log n+log jn+log log n− 1 2 log(2πjn)−jn log jn+jn → 0, as j n log j n is the leading term provided j n log n.

Lower bounds on the number of SBC
This section is devoted to prove lower bounds on the number of SBC under the additional assumption where R(t) is a polynomial with non-negative integer coefficients. So, as a direct consequence of the positivity of the coefficients of R(t) one obtains that β k is a lower bound for γ k .
Remark 4.1. Even if the collision free configuration sphere S is not compact, we are still allowed to use the Morse inequalities to derive lower bounds on the number of SBC. Indeed, as it is well-known, U (q) → +∞ as q approaches the singular set ∆.
In Theorem 4.2 below we provide a lower bound on the number of planar SBC when all masses are equal. This lower bound depends on a spectral gap condition at any CCC, which is independent of the chosen CCC. Moreover, as a direct consequence of the Assumption (H1), non-collinear SBC cannot be CC.
are not collinear.
Proof. All statements are consequences of the estimates for the coefficients of the Poincaré polynomial proved in Section 3 and the Morse inequalities, which may be written using Equation (4.1) as for some polynomial R(z) having non-negative integer coefficients r i , where γ i denotes the number of critical points of U having Morse index i. Recall that the Morse index of s 1 − CSBC is n − 1, as it follows from Lemma 2.11, independently of s 1 .
2. Under the given assumption, the Morse index j of a s 2 − CSBC is strictly smaller than n − 2 (in fact, at least η 1 moves into the positive line by adding s 1 U ( q)). Applying again the full Morse inequalities together with Proposition 3.2 we obtain γ n−1 = (n − 1)! + r n−2 + r n−1 ≥ n! ⇒ r n−2 + r n−1 ≥ n! − (n − 1)!
3. In this case s 2 − CSBC are local minima of U and hence which yields the desired lower bound.
In the next result we apply the asymptotic estimates for the coefficients of the Poincaré polynomial proved in Section 3 to improve the lower bounds on the number of SBC for large values of n in cases all masses are equal. The outcome will be that s j − CSBC, j 2, asymptotically will -in many cases -give more contribution than the s 1 − CSBC to the count of SBC. As already explained, the importance of such a result relies more in its qualitative rather than in its quantitative aspect, since such improvements will only be marginal. We recall that α j denotes the multiplicity of the eigenvalue η j of M −1 B( q), where q is any CCC of n equal masses (cfr. Subsection 2.2 for further details). Proof. Under the assumptions, the Morse index j of a s 2 − CSBC will be smaller than n − log n − 1. Using Item 3 of Proposition 3.6, we find an h ≥ 1 such that and this yields by the Morse inequalities The claim follows by arguing as in the proof of Theorem 4.2.
We now drop the condition that all masses be equal and derive lower bounds on the number of SBC assuming (H1) and (H2). As already observed, in this case the jumps on the Morse index of s 2 − CSBC might occur at different values of s 1 for different orderings of the masses. Hence, a precise statement as in Theorem 4.2 is unfortunately no longer available. Imposing further conditions, one could still apply the asymptotic estimates proved in Section 3, thus obtaining better lower bounds. However, we refrain to do this in order to keep the exposition as reader friendly as possible. Notice also that, by the pidgeonhole principle, at least (n − 1)! of the s 2 − CSBC must have the same Morse index; these will then contribute to the count of critical points provided the corresponding coefficient of the Poincaré polynomial be smaller than (n − 1)! (which is, in virtue of the discussion in Section 3, asymptotically very likely to be the case).
Proof. Under the given spectral gap assumption, we readily see that every s 2 − CSBC is a local minimum of U . The claim follows as in Part 3 of Theorem 4.2. The other claim is obtained using the Morse inequalities only taking into account the contribution given by s 1 − CSBC. The details are left to the reader.
For d > 2 the Poincaré polynomial is given by 3. Suppose that, for every j ≥ 1, the Morse indices of s j − CSBC and s j+1 − CSBC differ at least by two, and that for every j > 1 the Morse index is not an integer multiple of (d − 1). Then, there are at least (d + 1)n! − 2(n − 1)! non collinear SBC. Proof.
1. In this case, the Morse indices of CSBC are precisely so that the corresponding coefficients of the Poincaré polynomial vanishes only for the smallest resp. largest Morse index. The full Morse inequalities and Proposition 3.2 yield now for the coefficients of the remainder R(z): The asymptotic estimate is obtained using again Item 3 of Proposition 3.2.
2. Denoting by µ j the Morse index of s j − CSBC for j > 1, from we obtain j γ j ≥ n! + 2(n! − (n − 1)!) + (d − 1)n! = (d + 2)n! − 2(n − 1)! 3. In this case we have r µj −1 + r µj ≥ n! − c (n) µj = n! ∀j > 1 and hence j γ j ≥ n!+2(n!−(n−1)!)+2(d−1)n! = (2d+1)n!−2(n−1)! In the proposition above we applied Morse theory on the whole collision free configuration sphere in R nd , and as we can see, in two of the three cases we obtain lower bounds independent of the dimension d. Implementing a strategy similar to the one used in Corollary 2.15, we can instead give a general lower bound (i.e. independent of the Morse indices of the CSBC as well as of the fact that the masses be equal or not) on the number of non-collinear SBC which grows quadratically in d.
Before doing that in the next theorem, we shall notice that Part 1 of the proposition above holds also for non-equal masses provided all s j are sufficiently close to one. Indeed, also in this case all s j − CSBC have the same Morse index and the difference of Morse indices between s j − CSBC and s j+1 − CSBC is always equal one. This shows that, at least for s j close to one, all CSBC contribute to the count of SBC, even if the masses are not all equal. Notice also that, ignoring s j − CSBC for j 2 (as done in Theorem 4.2) does not provide any non-trivial lower bound for d > 2. Proof. For all 1 i < j d, the manifold P i,j of configurations which are contained in the (e i , e j )-plane is invariant under the gradient flow of U . Therefore, we can do Morse theory on P i,j , which in turn is nothing else but the collision free configuration space in the plane. Notice indeed, that Assumptions (H1) and (H2) hold also for the restriction to P i,j . Therefore, using Theorem 4.4, we obtain that there are at least n! − 2(n − 1)! non-collinear SBC in the (e i , e j )-plane. Furthermore, Assumption (H1) implies that the symmetry group of Equation (2.1) is trivial, and hence planar SBC contained in different planes can be counted separately. This together with the fact the number of different pairs 1 i < j d is precisely d(d− 1)/2 implies the claim.
Also in this case, we could obtain better lower bounds by a case distinction and implementing the asymptotic growth estimates for the coefficients of the Poincaré polynomial. Again, we refrain to do this to obtain better readability. We finish this section noticing that information on the inertia indices of planar SBC might help to improve the above result by e.g. showing the existence of non-planar SBC. This is indeed object of ongoing research.

The 4• theorem for balanced configurations
The well-known 45 • -Theorem for collinear CC is a "global version" of the fact that collinear CC are an attractor for the projectivized gradient flow of U . This theorem, indeed, consists in providing an explicit neighborhood of the manifold of collinear configurations such that orbits of the gradient flow of U starting off from such a neighborhood become more and more collinear. In this section we will extend the validity of the 45 • -Theorem to s 1 − CSBC in R 3 . More precisely, we will show that there is a neighborhood of the manifold of collinear configurations along the e 1 -axis such that orbits of the gradient flow of U starting off from such a neighborhood become more and more collinear along the e 1 -axis.
Such a 45 • -Theorem for s 1 − CSBC holds for any S = diag (s 1 , s 2 , 1), and actually even for every d ≥ 3. However, to keep the discussion as elementary as possible we assume hereafter that S = diag(s, 1, 1) for some s > 1, which is actually the case we will be interested in in Section 6. We set The gradient flow equation of U is given bẏ Definition 5.1. The collinearity angle of the configuration vector q ∈ S w.r.t. the e 1 -direction is Theorem 5.2. The function pointwise defined in Equation (5.1) is a Lyapunov function on for ∇ U . In particular there are no non-collinear SBC for ϑ(q(t)) ∈ (0, 45 • ].
Remark 5.3. To our best knowledge it is not clear whether a word by word generalization of the CC 45 • -Theorem hold for SBC. In fact, there is a priori no reason why collinear configurations along some line which is not one of the main axes should be mapped into collinear configurations by the gradient flow.
Proof. Choose (i, j) such that ϑ(q) := ∢(q i − q j , e 1 ). Sincė we get that By using the SO(2)-symmetry provided by rotations in the yOz-plane, we can w.l.o.g. suppose that q i − q j belongs to the xOz-plane. Set α(t) := ∢(q i (t) − q j (t), e 1 ). Up to interchanging i and j, we have where e ⊥ 1 denotes the orthogonal projection of e 1 on the plane perpendicular to q i − q j . We observe that e ⊥ 1 = sin α and e ⊥ 1 = sin α(sin α, − cos α) By using Equations (5.2) and (5.3) and by taking into account that (q i − q j ) · e 1 = 0, we get where we set q = (x, y, z) T . By arguing precisely as in [Moe94, P. 62-63], we get that that the first summand in the (RHS) of Equation (5.4) is non-negative (actually positive provided q is not collinear along some line). The last term in Equation (5.4) can be rewritten as follows where F ik is the force exerted by the particle q k on the particle q i . It is readily seen that the (RHS) in Equation (5.5) is positive. So,α < 0 and this concludes the proof.
A direct consequence of the 45 • -theorem for s 1 − CSBC is the following.
Corollary 5.4. The set is an attractor for the gradient flow of U . The set V is invariant under the gradient flow and the function t → ϑ q(t) is strictly decreasing along gradient flow lines, provided ϑ q(0) ∈ (0, 45 • ]. Remark 5.5. Corollary 5.4 allows us to remove the set Y from S before quotienting out by the (diagonal) SO(2)action on S induced by rotations in the yOz-plane, which we recall is the symmetry group of Equation (2.1) in case S = diag (s, 1, 1). Since Y is precisely the singular set of the SO(2)-action, we obtain that is a manifold. Such an argument is actually very similar to the one used by Merkel in [Mer08] in the case of spatial CC (the main difference being that we remove only collinear configuration in the e 1 -direction instead of the whole manifold of collinear configurations), unlike in Merkel's case however the lower bound that we will obtain is much larger than McCord's lower bound on the number of planar CC (cfr. Section 6).
The goal of the next section will be to compute the Poincaré polynomial of S. This will give us -via the "complexification argument" discussed in Section 2 -a lower bound on the number of SBC in R 4 with matrix diag (s, s, 1, 1), that will be then compared with the lower bounds provided by Theorem 4.2 via the reduction to (H1) argument.
We finish this section stating the general 45 • -theorem for s 1 − CSBC. The proof can be obtained from the one of Theorem 5.2 with some minor modifications and will be omitted. Then, the set q ∈ S q is collinear along some line in R µ1 × {0} ⊂ R D is an attractor for the gradient flow of U . More precisely, the collinearity function where L runs all over the lines through the origin in R µ1 × (0), is a Lyapounov function for the gradient vector field of U on the set {q | θ(q) ∈ (0, 45 • ]}. Remark 5.7. The singular set of the action of the symmetry group of Equation 2.1 does not consist of only collinear configurations as soon as d ≥ 4. This implies that the quotient of S by the group action will not be a manifold even if the 45 • -theorem is available in any dimension. Indeed, the whole singular set cannot be removed, in general. This is already the case for planar CC, which are in general not an attractor for the gradient flow. Besides that, the computations of the homology of the (singular) quotient are unaccessible for large values of d, and even if in the particular case considered in this section and in Section 6, we succeed to obtain a quotient manifold and to compute (though with some effort) its homology, in the count of SBC we still have to keep in mind that planar SBC which are contained in the yOz-plane are actually CC, whereas the non-collinear SBC provided by Theorem 4.2 cannot be CC. All these facts put on evidence, once again, the importance of the reduction to (H1) argument.

A lower bound on the number of SBC in R 4á la McCord
In this last section we compute the homology of the manifold S defined in (5.6) and, by means of this information, give lower bounds on the number of SBC in R 4 with matrix diag (s, s, 1, 1) that will be then compared with the ones obtained in Theorem 4.2. The two major ingredients for such a computation are: 1. Corollary 5.4: The manifold Y is an attractor for the gradient flow of U , and the neighborhood V of Y does not contain SBC other than the s 1 -collinear SBC.
2. Shub's lemma for SBC * : SBC cannot accumulate on the singular set ∆. In particular, there exists an invariant neighborhood of ∆ containing no SBC.
We set Y := Y \ ∆ and assume hereafter that n ≥ 4, as for n = 3 some arguments need some slight modification. We start by collecting some well-known facts. The first two are direct consequence of the fact that both S and Y are homological spheres: It is also readily seen that ∆ is a codimensional-3-submanifold of S. In particular, the set where c j is the j-th coefficient of the polynomial p(t) = (1 + t) · . . . · 1 + (n − 1)t and H * (resp. H * ) denotes the reduced homology (resp. cohomology). Therefore, by Alexander duality, H * (∆) ∼ = H 3n−5− * ( S) = R cj if * = 3n − 5 − 2j, 1 ≤ j ≤ n − 1, 0 otherwise.

Homology of some intermediate manifolds
The computation of the homology of S = S \ (∆ ∪ Y) /SO(2) is divided in the following intermediate steps: ( Step 1) We compute the homology of Y ∪ ∆. ( Step 2) We compute the reduced homology of S \ (Y ∪ ∆) using Alexander duality. * The proof of the original Shub's lemma carries over word by word to SBC.

(
Step 3) We conclude using the Gysin long exact sequence of the fibration In this subsection we will discuss Steps 1 and 2, leaving Step 3 for the next subsection. Thus, we start computing the homology of Y ∪ ∆ by using the Mayer-Vietoris sequence and the homological properties of Y, Y ∩ ∆, and ∆. Since denoting by ∆ a fattened open neighborhood of ∆ in Y, we get that Each connected component of the intersection Y ∩ (Y ∩ ∆) is homotopy equivalent to S n−3 (recall that the connected components of Y are homotopically (n − 2)-dimensional disks). Since we have n! connected components in total, we obtain implies that Proof. The first identity follows evaluating the polynomial at t = 1. We now prove the second identity by induction over n. A straightforward computation shows that for n = 4 both RHS and LHS are equal to 7. Suppose now that the claim be true for n. We want to show that (6.2) n−2 j=0 ξ (n+1) j (n − 1 − j) = (n + 1)! 2 h(n + 1).
As one readily sees, we have ξ This completes the proof.
For notational convenience, we hereafter drop the superscript from ξ (n) j . By the very definition of the coefficients c j , we see that c j = ξ j + ξ j−1 , ∀j = 0, ..., n − 1, where as above ξ −1 = ξ n−1 := 0. Therefore, using Lemma 6.3 we can rewrite (6.1) as (n − 1)! + This gives a lower bound on the number of critical points of U on S assuming non-degeneracy. However, such a set of critical points contains also the CC in the yOz-plane, which are in virtue of McCord's result (taking into account also collinear CC) at least n! 2 h(n) + 1).