Daciberg Lima

Gonc ¸alves

John Guaschi
email: john.guaschi@unicaen.fr

Vinicius Casteluber Laass
email: vinicius.laass@ufba.br

Free cyclic actions on surfaces and the Borsuk-Ulam theorem

Keywords: Borsuk-Ulam theorem, cyclic groups, braid groups, surfaces

Let M and N be topological spaces, let G be a group, and let τ : G×M -→ M be a proper free action of G. In this paper, we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pair (G, τ ) that generalises the classical antipodal Borsuk-Ulam theorem of maps from the n-sphere S n to R n . In the cases where M is a finite pathwise-connected CW-complex, G is a finite, non-trivial Abelian group, τ is a proper free cellular action, and N is either R 2 or a compact surface without boundary different of S 2 and RP 2 , we give an algebraic criterion involving braid groups to decide whether a free homotopy class β ∈ [M, N ] has the Borsuk-Ulam property. As an application of this criterion, we consider the case where M is a compact surface without boundary equipped with a free action τ of the finite cyclic group Z n . In terms of the orientability of the orbit space M τ of M by the action τ , the value of n modulo 4 and a certain algebraic condition involving the first homology group of M τ , we are able to determine if the single homotopy class of maps from M to R 2 possesses the Borsuk-Ulam property with respect to (Z n , τ ). Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is R 2 .

Introduction

The famous Borsuk-Ulam theorem states that every continuous map from the m-dimensional sphere to the m-dimensional Euclidean space collapses a pair of antipodal points [START_REF] Borsuk | Drei Sätze über die n-dimensionale Euklidische Sphäre[END_REF]Satz II]. We can state this result in terms of a free action of the cyclic group of order 2 in the following way: let Z 2 = {0, 1} be the cyclic group of order 2, and let τ : Z 2 × S m -→ S m be the free action defined by τ (i, x) = (-1) i x for all i ∈ {0, 1} and x ∈ M . If f : S m -→ R m is a continuous map, then there exists x ∈ S m such that f (τ (0, x)) = f (τ (1, x)). A very natural generalisation of this theorem is to replace S m and R m by manifolds M and N , and replace the antipodal map by a free involution on M . One may then ask if any continuous map from M to N collapses an orbit of this Z 2 -action. We will refer to this question as the Borsuk-Ulam problem for M and N . We focus our discussion on the case where M and N are surfaces, for which much progress has been made over the past fifteen years with respect to this problem. In 2005, classical techniques of algebraic topology were used to obtain a complete answer to the problem for every compact surface M without boundary equipped with a free involution and N = R 2 [START_REF] Gonc ¸alves | The Borsuk-Ulam theorem for surfaces[END_REF]. A few years later, the problem in the case where M and N are compact surfaces without boundary was studied, and using the theory of surface braid groups, a complete answer to the problem was given [START_REF] Gonc ¸alves | The Borsuk-Ulam theorem for maps into a surface[END_REF]. To our knowledge, this was the first time in the literature that these groups were used to study Borsuk-Ulam-type problems. In a similar vein, the Borsuk-Ulam problem was solved in the case where the domain is the product of two compact surfaces with boundary equipped with the diagonal involution, and the target is R n [START_REF] Gonc ¸alves | Diagonal involutions and the Borsuk-Ulam property for product of surfaces[END_REF] or a compact surface without boundary [START_REF] Gonc ¸alves | The Borsuk-Ulam property for maps from product of two surfaces into a surface[END_REF], braid theory being used in the latter case. More recently, another natural and more refined problem that generalises the Borsuk-Ulam theorem arose, namely the question of the classification of the free homotopy classes of maps between M and N , where M is equipped with a free Z 2 -action for which every representative map of a given homotopy class collapses an orbit of the action. This problem was solved by the authors in three recent papers in the cases where M and N are compact surfaces without boundary of Euler characteristic zero [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero[END_REF][START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle[END_REF][START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle -part 2[END_REF]. The results depend on the choice of free involution on M , and once more, braid theory plays an important rôle in the solution of the problem.

In this paper, our aim is to initiate the study of another generalisation of the Borsuk-Ulam theorem by considering free actions of groups other than Z 2 on spaces as follows. Let M and N be topological spaces, and let G be a non-trivial group for which there exists a free action τ : G × M -→ M of G on M . Following [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero[END_REF][START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle[END_REF][START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle -part 2[END_REF], we say that a free homotopy class β ∈ [M, N ] has the Borsuk-Ulam property with respect to the pair (G, τ ) if for every representative map f : M -→ N of β, there exist g 1 , g 2 ∈ G, where g 1 ̸ = g 2 , and x ∈ M such that f (τ (g 1 , x)) = f (τ (g 2 , x)), and that the quadruple (M, G, τ ; N ) satisfies the Borsuk-Ulam property if every homotopy class of maps between M and N has the Borsuk-Ulam property with respect to (G, τ ). For technical reasons, we also define the Borsuk-Ulam property for pointed homotopy classes. More precisely, a pointed homotopy class α ∈ [M, m 1 ; N, n 1 ] has the Borsuk-Ulam property with respect to the pair (G, τ ) if for every representative map f : (M, m 1 ) -→ (N, n 1 ) of α there exist g 1 , g 2 ∈ G, where g 1 ̸ = g 2 , and

x ∈ M such that f (τ (g 1 , x)) = f (τ (g 2 , x)).
The first case that we study is that of free actions of a finite cyclic group on a compact surface M without boundary, and where N is the Euclidean plane R 2 . Note that in this situation, there is just one homotopy class, and in this sense, our results generalise those of [START_REF] Gonc ¸alves | The Borsuk-Ulam theorem for surfaces[END_REF]. We underline the important rôle of braid theory in our work.

To state Theorem 1.1, which is the main result of this paper, we first recall some facts and notation. Given a pathwise-connected CW-complex M and a proper free cellular action τ : G×M -→ M , where G is a non-trivial discrete group, we say that two points x 1 , x 2 ∈ M are equivalent if there exists g ∈ G such that τ (g, x 1 ) = x 2 . Let M τ denote the quotient space (or orbit space) of M by this action, and let p τ : M -→ M τ denote the natural projection of M onto M τ . Then p τ is a regular covering map, and the associated group of deck transformations is G. Thus the space M τ is also a pathwise-connected CW-complex whose dimension is that of M , and we have the following short exact sequence (we omit the basepoints):

{1} / / π 1 (M ) (pτ ) # / / π 1 (M τ ) θτ / / π 1 (M τ ) (p τ ) # (π 1 (M )) ∼ = G / / {1}, (1)
where θ τ is the natural projection. Suppose now that G is an Abelian group. Then the homomorphism θ τ of (1) factors through the Abelianisation (π 1 (M τ )) Ab of π 1 (M τ ) that is isomorphic to the first homology group H 1 (M τ , Z) of M τ with integral coefficients. We denote this factorisation by (θ τ ) Ab : H 1 (M τ ) -→ G. If M τ is a compact, non-orientable surface without boundary, let δ denote the unique element of H 1 (M τ ) of order two. The main goal of this paper is to prove the following theorem. (2) M τ is non-orientable, and (θ τ ) Ab (δ) is non trivial.

Notice that for M = S 2 and n = 2, Theorem 1.1 is a special case of the classical Borsuk-Ulam theorem.

Following Theorem 1.1, it is natural to ask what happens for actions of finite non-cyclic groups. Our techniques may be can be applied in part to this more general case, and in Section 5, we illustrate their use by giving a solution to the Borsuk-Ulam problem in certain cases where G is the symmetric group Σ n , where n ≥ 2. More precisely, in Proposition 5.1, we show that for all n ≥ 2, there exist actions of Σ n on two compact surfaces M 1 and M 2 without boundary. In the case of M 1 , the Borsuk-Ulam property does not hold for maps into R 2 , while in the case of M 2 , we do not know the answer in general.

Apart from this Introduction, this paper has four sections. In Theorem 2.4 of Section 2, we give an algebraic criterion involving configuration spaces and braid groups that enables us to decide whether a given homotopy class of maps from M to N has the Borsuk-Ulam property, where M is a connected manifold equipped with a free action of an Abelian group G, and N is either the Euclidean plane R 2 or a compact surface without boundary different from the 2-sphere S 2 and the real projective plane RP 2 . One implication of the condition of Theorem 2.4 is valid in general, while for the other implication, at present, we only know that it holds in the case where G is Abelian. This is the reason for which we do not know presently if the surface M 2 of the previous paragraph satisfies the Borsuk-Ulam property. In Section 3, in Theorem 3.1 we exhibit a presentation for the braid group that arises in the statement of Theorem 2.4 in the case where G is cyclic of order n and N = R 2 . Section 4 is devoted to the proof of Theorem 1.1. Finally in Section 5, we discuss the above-mentioned examples of two surfaces on which Σ n acts freely with regard to the Borsuk-Ulam property.

Preliminaries and Generalities

We start by stating and proving the following result that is an algebraic and homotopy theoretical characterisation of the set of pointed homotopy classes of maps between pairs of CW -complexes. It generalises [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero[END_REF]Theorem 4] in some sense, and it will be used in the proof of Lemma 2.2.

Theorem 2.1. Let (M, m 1 ) and (N, n 1 ) be pointed pathwise-connected CW -complexes such that π i (N, n 1 ) is trivial for

2 ≤ i ≤ dim(M ). Then the map Γ : [M, m 1 ; N, n 1 ] -→ Hom(π 1 (M, m 1 ), π 1 (N, n 1 )) defined by Γ ([f ]) = f # is a bijection.
Proof. Let N = K(π 1 (N ), 1) be the first stage of the Postnikov tower of N , and let h : N -→ N be a map that induces an isomorphism on the level of fundamental groups. By the definition of h and the hypothesis, the homomorphism h # :

π i (N, n 1 ) -→ π i ( N , h(n 1 )) is an isomorphism if 1 ≤ i ≤ dim(M )
, and it is surjective if i = dim(M ) + 1. From these observations, standard arguments of homotopy theory and [17, Chapter VI, Theorems 6.12 and 6.13], the map Γ [START_REF] Whitehead | Elements of homotopy theory[END_REF]Chapter V,Theorem 4.3]. Now let Γ 3 : Hom(π 1 (M, m 1 ), π 1 ( N , h(n 1 ))) -→ Hom(π 1 (M, m 1 ), π 1 (N, n 1 )) be the map defined by Γ 3 (φ) = (h # ) -1 • φ. Then Γ 3 is a bijection, and the result follows by taking

1 : [M, m 1 ; N, n 1 ] -→ [M, m 1 ; N , h(n 1 )] defined by Γ 1 ([f ]) = [h•f ] is a bijection. Further, the map Γ 2 : [M, m 1 ; N , h(n 1 )] -→ Hom(π 1 (M, m 1 ), π 1 ( N , h(n 1 ))) given by Γ 2 ([g]) = g # is a bijection by
Γ = Γ 3 • Γ 2 • Γ 1 .
Let M and N be pathwise-connected CW -complexes, and let G be a non-trivial discrete group for which there exist proper free cellular actions τ : G × M -→ M and τ 1 :

G × N -→ N . A map f : M -→ N is said to be (τ, τ 1 )-equivariant if f (τ (g, x)) = τ 1 (g, f (x)) for all g ∈ G and all x ∈ M . Given a pointed homotopy class α = [f ] ∈ [M, m 1 ; N, n 1 ], let α # = f # : π 1 (M, m 1 ) -→ π 1 (N, n 1 ).
Note that the homomorphism α # is independent of the choice of the representative map f of α. The following result provides an algebraic criterion to decide whether a pointed homotopy class has an equivariant representative map, and it generalises [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero[END_REF]Lemma 5].

Lemma 2.2. Let (M, m 1 ) and (N, n 1 ) be pointed pathwise-connected CW -complexes (resp. compact surfaces without boundary) such that π i (N, n 1 ) is trivial for 2 ≤ i ≤ dim(M ), let G be a non-trivial group for which there exist proper free cellular actions τ : G × M -→ M and τ 1 : G × N -→ N , and let α ∈ [M, m 1 ; N, n 1 ] be a pointed homotopy class. Then the following conditions are equivalent: (a) there exists a representative map (resp. homeomorphism) f : (M, m 1 ) -→ (N, n 1 ) of α that is (τ, τ 1 )-equivariant. (b) there exists a homomorphism (resp. isomorphism) ψ : π 1 (M τ , p τ (m 1 )) -→ π 1 (N τ 1 , p τ 1 (n 1 )) such that the following diagram is commutative:

π 1 (M, m 1 ) α # / / (pτ ) # π 1 (N, n 1 ) (pτ 1 ) # π 1 (M τ , p τ (m 1 )) ψ / / θτ & & π 1 (N τ 1 , p τ 1 (n 1 )) θτ 1 w w G. (2) Proof. We first prove that condition (a) implies condition (b). Let f : (M, m 1 ) -→ (N, n 1 ) be a (τ, τ 1 )-equivariant representative map of α. Then the map f : (M τ , p τ (m 1 )) -→ (N τ 1 , p τ 1 (n 1 )) between the corresponding orbit spaces defined by f (y) = p τ 1 • f (x) for all y ∈ M τ , where x ∈ p -1
τ (y), is well defined, and the following diagram is commutative:

(M, m 1 ) f / / pτ (N, n 1 ) pτ 1 (M τ , p τ (m 1 )) f / / (N τ 1 , p τ 1 (n 1 )).
(

) Let ψ : π 1 (M τ , p τ (m 1 )) -→ π 1 (N τ 1 , p τ 1 (n 1 ) 3
) denote the homomorphism induced by f . The diagram (3) then gives rise to the upper square of the diagram [START_REF] Borsuk | Drei Sätze über die n-dimensionale Euklidische Sphäre[END_REF]. It remains to show that

θ τ = θ τ 1 • ψ. To see this, let a ∈ π 1 (M τ , p τ (m 1 )), let α be a loop in M τ based at p τ (m 1 ) such that a = [α], let α ′ be the loop in M τ based at p τ (m 1 ) given by α ′ = f • α, let g = θ τ (a), and let ξ : [0, 1] -→ M be the lift of α to M for which ξ(0) = m 1 . By commutativity of (3), α ′ = f • p τ • ξ = p τ 1 (f • ξ), and since f (ξ(0)) = f (m 1 ) = n 1 , f • ξ : [0, 1] -→ N is the lift to N of α ′ whose initial point is n 1 . Using the construction of the short exact sequence (1), we have ξ(1) = τ (g, m 1 ). It follows from the fact that f is (τ, τ 1 )-equivariant that f • ξ(1) = f (τ (g, ξ(1))) = τ 1 (g, f • ξ(1)
), and so

θ τ 1 ([α ′ ]) = g, from which we see that θ τ 1 • ψ([α]) = θ τ 1 ([α ′ ]) = g = θ τ (g) as required. Further, if f is a homeomorphism, then f is too, and ψ = f # is an isomorphism. We now prove that condition (b) implies condition (a). Since p τ 1 : N -→ N τ 1 is a covering map, the triviality of π i (N, n 1 ) for 2 ≤ i ≤ dim(M ) implies that of π i (N τ 1 , p τ 1 (n 1 )) for all 2 ≤ i ≤ dim(M ).
It follows from Theorem 2.1 that there exists a map f : (M τ , p τ (m 1 )) -→ (N τ 1 , p τ 1 (n 1 )) such that f # = ψ, and so by [START_REF] Borsuk | Drei Sätze über die n-dimensionale Euklidische Sphäre[END_REF], there exists a lift f : (M, m 1 ) -→ (N, n 1 ) of the map f • p τ for the covering p τ 1 , from which we obtain the commutative diagram [START_REF] Fadell | Configuration Spaces[END_REF]. The injectivity of (p τ 1 ) # implies that α # = f # , and so f is a representative map of α by Theorem 2.1. Let us show that f is (τ, τ 1 )-equivariant. We will do this in two steps. We first claim that f (τ (g,

m 1 )) = τ 1 (g, f (m 1 )) for all g ∈ G. To see this, let γ : [0, 1] -→ M be a path such that γ(0) = m 1 and γ(1) = τ (g, m 1 ). Then α = p τ • γ is a loop in M τ based at p τ (m 1 ) such that θ τ ([α]) = g. The commutative diagrams (2) and (3) imply that f • α is a loop in N τ 1 based at p τ 1 (n 1 )
, and that:

θ τ 1 ([f • α]) = (θ τ 1 • f # )([α]) = θ τ ([α]) = g. So if γ 1 : [0, 1] -→ N is the lift of f • α by the covering p τ 1 such that γ 1 (0) = n 1 , then γ 1 (1) = τ 1 (g, n 1 ). Now (f • γ)(0) = f (m 1 ) = n 1 and: p τ 1 • (f • γ) = f • p τ • γ = f • α = p τ 1 • γ 1 .
It follows from the path lifting property that f • γ = γ 1 , and hence:

f (τ (g, m 1 )) = (f • γ)(1) = γ 1 (1) = τ 1 (g, n 1 ) = τ 1 (g, f (m 1 )), (4)
which proves the claim. Now let x ∈ M , let g ∈ G, let ξ : [0, 1] -→ M be a path from m 1 to x, and let ξ 1 , ξ 2 : [0, 1] -→ N be the paths defined by ξ

1 (t) = f (τ (g, ξ(t))) and ξ 2 (t) = τ 1 (g, f • ξ(t)
). Then:

ξ 1 (0) = f (τ (g, ξ(0))) = f (τ (g, m 1 )) (4) = τ 1 (g, f (m 1 )) = τ 1 (g, f • ξ(0)) = ξ 2 (0). (5)
Using ( 3) and the definition of p τ and p τ 1 , for all t ∈ [0, 1], we have:

p τ 1 • ξ 1 (t) = p τ 1 • f (τ (g, ξ(t))) = f • p τ (τ (g, ξ(t))) = f • p τ (ξ(t)) = p τ 1 (f • ξ(t)) = p τ 1 (τ 1 (g, f • ξ(t))) = p τ 1 • ξ 2 (t).
Thus ξ 1 and ξ 2 are lifts of the path p τ 1 • ξ 1 by the covering p τ 1 that have the same initial point by ( 5), and therefore ξ 1 = ξ 2 by the path lifting property. From this, we conclude that:

f (τ (g, x)) = f (τ (g, ξ(1))) = ξ 1 (1) = ξ 2 (1) = τ 1 (g, f • ξ(1)) = τ 1 (g, f (x)),
and f is indeed (τ, τ 1 )-equivariant. Finally, if M and N are compact surfaces without boundary and ψ is an isomorphism, it follows from the classification theorem for surfaces and [18, Theorem 5.6.2] that ψ is induced by a homeomorphism. So we may take f to be a homeomorphism, in which case f is also a homeomorphism.

Let S be either the Euclidean plane or a compact surface without boundary. Recall that

F n (S) = {(x 1 , . . . , x n ) ∈ S n | x i ̸ = x j if i ̸ = j} is the n th ordered configuration space of S.
It is a pathwiseconnected 2n-dimensional manifold, and if S is different from the 2-sphere S 2 or the projective plane RP 2 , then by [3, Corollary 2.2], π i (F n (S)) is trivial for all i ≥ 2. The fundamental group π 1 (F n (S)), denoted by P n (S), is the n-string pure braid group of S. Let Σ n be the symmetric group on the set {1, . . . , n}. We adopt the convention that a product of permutations is read from left to right. Note that the map

τ 1 : Σ n × F n (S) -→ F n (S) defined by τ 1 (σ, (x 1 , . . . , x n )) = (x σ -1 (1) , . . . , x σ -1 (n)
) is a free action. Using the notation and the constructions given in the Introduction, F n (S) τ 1 is the orbit space corresponding to this action, and the group π 1 (F n (S) τ 1 ), which we will denote by B n (S), is the n-string (full) braid group of the surface S. Let

ι 1 = (p τ 1 ) # : P n (S) = π 1 (F n (S)) -→ π 1 (F n (S) τ 1 ) = B n (S) and let π 1 = θ τ 1 : B n (S) = π 1 (F n (S) τ 1 ) -→ π 1 (F n (S) τ 1 )/ Im ((p τ 1 ) # ) ∼ = Σ n . Now
let G be a finite group of order n, and let i : G -→ Σ n be a monomorphism. Let τ 2 : G × F n (S) -→ F n (S) be the free involution defined by τ 2 (g, x) = τ 1 (i(g), x). We may regard G as a subgroup of Σ n , and τ 2 as the restriction of τ 1 to G × F n (S). Let the fundamental group of the corresponding orbit space F n (S) τ 2 be denoted by B G (S), and let

ι 2 = (p τ 2 ) # : P n (S) = π 1 (F n (S)) -→ π 1 (F n (S) τ 2 ) = B G (S) and π 2 = θ τ 2 : π 1 (F n (S) τ 2 ) -→ π 1 (F n (S) τ 2 )/ Im ((p τ 2 ) # ) ∼ = G.
For j, k ∈ {1, 2}, we may regard i and ι j as inclusions, π k as a natural projection, and τ 2 as the restriction of τ 1 to G × F n (S). The map h :

F n (S) τ 2 -→ F n (S) τ 1 that sends a τ 2 -orbit to the corresponding τ 1 orbit is well defined, satisfies h • p τ 1 = p τ 2 ,

and induces an injective homomorphism h

# : π 1 (F n (S) τ 2 ) -→ π 1 (F n (S) τ 1 )
. Using the construction of (1), we obtain the following commutative diagram of short exact sequences:

1 / / P n (S) ι 2 / / B G (S) π 2 / / _ h # G / / _ i 1 1 / / P n (S) ι 1 / / B n (S) π 1 / / Σ n / / 1. (6)
Identifying G with i(G) and π 1 (F n (S) τ 2 ) = B G (S) with Im (h # ), we see that B G (S) ⊂ π -1 1 (G), and it follows from (6) that B G (S) = π -1 1 (G).

Remark 2.3. Given a finite group G = {g 1 , g 2 , . . . , g n } of order n, the set of bijections Σ G from G to G equipped with the operation of composition is a group that is isomorphic to Σ n , and that we will identify in what follows. Note that composition in Σ G is read from right to left, but as we mentioned above, a product of permutations will be read from left to right. For all g ∈ G, the map i g : G -→ G given by i g (g ′ ) = g. g ′ for all g ′ ∈ G is a bijection, and the map i : G -→ Σ n that assigns g to i g is an embedding of G in Σ n . With respect to the homomorphism i, and with the notation of the previous paragraph, F n (S) may be described as

F n (S) = {(x g 1 , . . . , x gn ) ∈ S n | x g j ̸ = x g k if g j ̸ = g k },
and τ 2 (g, (x g 1 , . . . , x gn )) = (x g.g 1 , . . . , x g.gn ).

As for the Artin braid groups, the elements of B n (S) may be described geometrically as follows (other equivalent geometric constructions may be found in [START_REF] Birman | Braids, Links, and Mapping Class Groups[END_REF][START_REF] Hansen | Braids and coverings: selected topics[END_REF][START_REF] Murasugi | A study of braids[END_REF]

* c π(β)(n) )
, where the symbol * denotes concatenation of paths. This induces a group structure on the set of n-braids, and the resulting group is isomorphic to B n (S). Up to this isomorphism, we may identify π 1 and π. The pure braid group P n (S) (resp. the group B G (S)) consists of those n-braids whose permutation is the identity (resp. belongs to the subgroup G of Σ n ).

The following theorem is the main result of this section. It establishes a connection between the Borsuk-Ulam property and the braids groups of a surface, thus generalising [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero[END_REF]Theorem 7].

Theorem 2.4. Let (M, m 1 ) be a pointed, pathwise-connected CW -complex, and suppose that there exists a proper free cellular action τ : G × M -→ M , where G is a finite group of order n ≥ 2 that we embed in Σ n as in Remark 2.3. Let (S, s 1 ) be a pointed surface, where S is either R 2 or a compact surface without boundary different from S 2 and RP 2 .

(a) Let α ∈ [M, m 1 ; S, s 1 ] be a pointed homotopy class. Suppose that there exist homomorphisms φ : π 1 (M, m 1 ) -→ P n (S) and ψ : π 1 (M τ , p τ (m 1 )) -→ B G (S) for which the following diagram is com-mutative:

π 1 (M, m 1 ) φ / / (pτ ) # α # ) ) P n (S) ι 2 (p 1 ) # / / π 1 (S, s 1 ) π 1 (M τ , p τ (m 1 )) ψ / / θτ & & B G (S) π 2 | | G, (7)
where p 1 : F n (S) -→ S denotes projection onto the first coordinate. Then α does not have the Borsuk-Ulam property with respect to (G, τ ). Conversely, if G is an Abelian group and α does not have the Borsuk-Ulam property with respect to (G, τ ), then there exist homomorphisms φ : π 1 (M, m 1 ) -→ P n (S) and ψ : π 1 (M τ , p τ (m 1 )) -→ B G (S) for which the diagram ( 7) is commutative. (b) Let β ∈ [M, S] be a free homotopy class, and let α ∈ [M, m 1 ; S, s 1 ] be a pointed homotopy class such that every representative map of α is also a representative map of β when we forget the basepoints. If β has the Borsuk-Ulam property with respect to (G, τ ), then so does α. Conversely, if G is an Abelian group and α has the Borsuk-Ulam property with respect to (G, τ ), then so does β.

Remarks 2.5. By applying [17, Chapter V, Theorem 4.3 and Corollary 4.4], if S is either R 2 or a compact surface without boundary different from S 2 and RP 2 and G is an Abelian group, Theorem 2.4 provides us with an algebraic criterion to decide whether a homotopy class has the Borsuk-Ulam property or not. The cases where S = S 2 or RP 2 is the subject of work in progress, and the techniques required are somewhat different. In the general case, for the time being we have not been able to remove the restriction that G is Abelian in the converse of parts (a) and (b) in the statement of Theorem 2.4.

Proof of Theorem 2.4.

In what follows, for all g ∈ G and x ∈ M , we will denote τ (g, x) simply by gx. With respect to Remark 2.3, by permuting the elements of G if necessary, we may suppose that g 1 is its trivial element.

(a) First suppose that there exist homomorphisms φ : π 1 (M, m 1 ) -→ P n (S) and ψ : π 1 (M τ , p τ (m 1 )) -→ B G (S) for which the diagram (7) is commutative. Since F n (S) is a K(P n (S), 1)-manifold, [17, Chapter V, Theorem 4.3] and Lemma 2.2 imply that the homomorphism φ is induced by a (τ, τ 2 )-equivariant map F : M -→ F n (S), so F # = φ. For i = 1, . . . , n, let f g i : M -→ S be a map such that:

F (x) = (f g 1 (x), f g 2 (x), . . . , f gn (x)) (8)
for all x ∈ M . Since F is (τ, τ 2 )-equivariant, for all i = 1, . . . , n and x ∈ M , by ( 8) and Reamrk 2.3, we have:

(f g 1 (g i x), f g 2 (g i x), . . . , f gn (g i x)) = F (g i x) = τ 2 (g i , F (x)) = (f g i .g 1 (x), f g i .g 2 (x), . . . , f g i .gn (x)) (9)
Now let f = f g 1 . Since g 1 is the trivial element of G, we see from the first coordinate of (9) that f g i (x) = f (g i x) for all i = 1, . . . , n and x ∈ M . We conclude from (8) that

F (x) = (f (x), f (g 2 x), . . . , f (g n x)).
Since F (x) ∈ F n (S), then by definition, the pointed homotopy class [f ] does not have the Borsuk-Ulam property. Again, by [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle[END_REF] and [17, Chapter V, Theorem 4.3], we have α = [f ], and thus α does not have the Borsuk-Ulam property as required.

Conversely, suppose that G is an Abelian group, and that α ∈ [M, m 1 ; S, s 1 ] does not have the Borsuk-Ulam property with respect to (G, τ ). Then there exists a map f : (M, m 1 ) -→ (S, s 1 ) such that α = [f ], and for which the elements f (g 1 x), . . . , f (g n x) are pairwise distinct for all x ∈ M . For i = 1, . . . , n, let f g i : M -→ S be the map defined by f g i (x) = f (g i x). Then the map F : M -→ F n (S) given by F (x) = (f g 1 (x), . . . , f gn (x)) for all x ∈ M is well defined, and it satisfies p 1 • F = f . We claim that F is (τ, τ 2 )-equivariant. Indeed, using the fact that G is Abelian, for all g ∈ G and x ∈ M , we have:

F (τ (g, x)) = F (gx) = (f g 1 (gx), . . . , f gn (gx)) = (f (g 1 (gx)), . . . , f (g n (gx)))
= (f ((g. g 1 )x), . . . , f ((g. g n )x)) = (f g.g 1 (x), . . . , f g.gn (x)) = τ 2 (g, (f g 1 (x), . . . , f gn (x))) = τ 2 (g, F (x)).

If φ : π 1 (M, m 1 ) -→ P n (S) is the homomorphism induced by F , then (p 1 ) # • φ = (p 1 ) # • F # = (p 1 • F ) # = (f g 1 ) # = f # = α # ,
π 1 (M, m 1 ) φ ′ / / (pτ ) # α ′ # ) )
P n (S)

ι 2 (p 1 ) # / / π 1 (S, s 1 ) π 1 (M τ , p τ (m 1 )) ψ ′ / / θτ & & B G (S) π 2 { { G. (10)
By [17, Chapter V, Theorem 4.3 and Corollary 4.4], there exists γ ∈ π 1 (S, s 1 ) such that the homomorph-

isms α # , α ′ # : π 1 (M, m 1 ) -→ π 1 (S, s 1 ) satisfying α # (v) = γα ′ # (v)γ -1 for all v ∈ π 1 (M, m 1 )
. Further, by [START_REF] Birman | Braids, Links, and Mapping Class Groups[END_REF]Theorem 1.4], there exists b ∈ P n (S) such that (p 1 ) # (b) = γ. We define φ :

π 1 (M, m 1 ) -→ P n (S) (resp. ψ : π 1 (M τ , p τ (m 1 )) -→ B G (S)) by φ(v) = bφ ′ (v)b -1 for all v ∈ π 1 (M, m 1 ) (resp. ψ(w) = ι 2 (b)ψ ′ (w)ι 2 (b) -1 for all w ∈ π 1 (M τ , p τ (m 1 ))).
Using the commutativity of [START_REF] Gonc ¸alves | The Borsuk-Ulam property for maps from product of two surfaces into a surface[END_REF], it follows that the diagram ( 7) is also commutative, and therefore α does not have the Borsuk-Ulam property as a pointed homotopy class.

The following corollary will be used in the proof of Propositions 4.2-4.4, which is the restriction of Theorem 2.4 to the case where S = R 2 .

Corollary 2.6. Let M be a compact surface without boundary, let G be a finite Abelian group of order n ≥ 2, and let τ : G × M -→ M be a free action. Then (M, G, τ ; R 2 ) does not have the Borsuk-Ulam property if and only if there exists a homomorphism ψ : π 1 (M τ ) -→ B G (R 2 ) such that the following diagram is commutative:

π 1 (M τ ) ψ / / θτ # # B G (R 2 ) π 2 { { G. (11)
Proof. First of all, since R 2 is contractible then there is only one homotopy class α ∈ [M, R 2 ], and α # : π 1 (M ) -→ π 1 (R 2 ) is trivial. So by Theorem 2.4, (M, G, τ ; R 2 ) does not have the Borsuk-Ulam property if and only if there exist homomorphisms φ : π 1 (M ) -→ P n (R 2 ) and ψ :

π 1 (M τ ) -→ B G (R 2 )
for which the diagram ( 7) is commutative. In particular, if (M, G, τ ; R 2 ) does not have the Borsuk-Ulam property then the lower triangle of ( 7) is the commutative diagram [START_REF] Hansen | Braids and coverings: selected topics[END_REF]. Conversely, given the commutative diagram [START_REF] Hansen | Braids and coverings: selected topics[END_REF], the exactness of ( 1) and the upper row of ( 6) allow us to obtain [START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle[END_REF].

3 The group B Z n (R 2 )
Let n ≥ 2. We denote the cyclic group of order n by Z n = {0, 1, . . . , n -1}. With respect to Remark 2.3 and the paragraph that precedes it, we fix an embedding i : Z n -→ Σ n of Z n in Σ n , given by i(1) = (1, n, . . . , 2), and we identify B Zn (S) with its image in B n (S) under h # . The aim of this section is to give a presentation of the group B Zn (R 2 ). Recall that if n ∈ N, the Artin braid group B n on n strings is generated by the Artin generators σ 1 , . . . , σ n-1 , subject to the following Artin relations:

σ i σ j = σ j σ i if |i -j| ≥ 2, where 1 ≤ i, j ≤ n -1 (12) σ i σ i+1 σ i = σ i+1 σ i σ i+1 for i = 1, . . . , n -2. (13)
Geometrically, σ i and its inverse correspond to the braids illustrated in Figure 1.

• • • • • • • • • • • • 1 i -1 i i + 1 i + 2 n 1 i -1 i i + 1 i + 2 n σ i σ -1 i Figure 1:
The braid σ i and its inverse.

Theorem 3.1. The group B Zn (R 2 ) admits the following presentation: generating set: {g} ∪ {A i,j } 1≤i<j≤n . relations:

(I) A -1 r,s A i,j A r,s =          A i,j if r < s < i < j or i < r < s < j A r,j A i,j A -1 r,j if r < i = s < j A r,j A s,j A i,j A -1 s,j A -1 r,j if i = r < s < j A r,j A s,j A -1 r,j A -1 s,j A i,j A s,j A r,j A -1 s,j A -1 r,j if r < i < s < j. (II) gA i,j g -1 = A i+1,j+1 if 1 ≤ i < j ≤ n -1 A 1,n • • • A n-1,n A 1,i+1 A -1 n-1,n • • • A -1 1,n if 1 ≤ i ≤ n -1 and j = n. (III) g n = n j=2 j-1 i=1 A i,j .
Remarks 3.2.

(a) In terms of the Artin generators of B n , we take g = σ 1 • • • σ n-1 , and:

A i,j = σ j-1 • • • σ i+1 σ 2 i σ -1 i+1 • • • σ -1 j-1 for 1 ≤ i < j ≤ n. (14)
One may check using ( 12) and ( 13) that:

A i,j = σ -1 i • • • σ -1 j-2 σ 2 j-1 σ j-2 • • • σ i for 1 ≤ i < j ≤ n. (15)
Geometrically, A i,j corresponds to the braid illustrated in Figure 2.

(b) The element g n that appears in relation (III) is the well-known full-twist braid, denoted by ∆ 2 n , that generates the centre of B n (R 2 ) if n ≥ 3. (c) With respect to the lower row of (6), we have π 2 (g) = (1, n, . . . , 2). Taking S = R 2 and G = Z n , where we identify Z n with the cyclic subgroup ⟨(1, n, . . . , 2)⟩ of Σ n in the manner described at the beginning of this section, it follows that g ∈ B Zn (R 2 ). (d) Considering the permutation homomorphism π 2 : B Zn (R 2 ) -→ Z n of the upper row of ( 6), and up to the identification of B Zn (R 2 ) (resp. Z n ) with Im (h # ) (resp. with Im (i)), we have π 2 (g) = 1 and π 2 (A i,j ) = 0 for all 1 ≤ i < j ≤ n. Further, for all a ∈ B Zn (R 2 ), there exist unique 0 ≤ m ≤ n -1 and w ∈ P n (R 2 ) for which a = wg m .

• • • • • • • • • 1 i -1 i j j + 1 n Figure 2: The element A i,j of B n .
Proof of Theorem 3.1. We apply the method of [12, Chapter 10, Proposition 1] to the short exact sequence of the upper row of ( 6), where we take S = R 2 , G = Z n , and g and A i,j , where 1 ≤ i < j ≤ n, as in Remarks 3.2(a). We identify the cyclic subgroup ⟨(1, n, . . . , 2)⟩ of Σ n with Z n as in Remarks 3.2(c). With respect to the upper row of (6), it follows that g is a coset representative of the generator 1 of Z n . Since (A i,j ) 1≤i<j≤n is a generating set of P n (R 2 ) by [11, Chapter 1, Lemma 4.2,], it follows from [12, Chapter 10, Proposition 1] that {g} ∪ {A i,j } 1≤i<j≤n is a generating set for B Zn (R 2 ). From the same proposition, the following three types of relation form a complete set of relations of B Zn (R 2 ): (i) the relations emanating from P n (R 2 ). By [11, Chapter 1, Lemma 4.2], these are the relations (I) of the statement.

(ii) the conjugates of the generators of P n (R 2 ) by g, rewritten in terms of the generators of P n (R 2 ). Suppose first that 1 ≤ i < j ≤ n -1. Using ( 12) and ( 13), we see that that gσ k g -1 = σ k+1 for all k = 1, . . . , n -2, and it follows from ( 14) that gA i,j g -1 = A i+1,j+1 , which is the first relation of (II).

Proof of Proposition 4.2. We will define a homomorphism ψ : π 1 (M τ ) -→ B Zn (R 2 ) for which the diagram [START_REF] Hansen | Braids and coverings: selected topics[END_REF] is commutative, from which it will follow by Corollary 2.6 that (M, Z n , τ ; R 2 ) does not have the Borsuk-Ulam property. With respect to the cases (I)-(III), let ψ : π 1 (M τ ) -→ B Zn (R 2 ) be the map defined on the generators of π 1 (M τ ) as follows:

(a) ψ(a i ) = g b i for i = 1, . . . , 2m, where g ∈ B Zn (R 2 ) is as defined in Remark 3.2(a), and 0 ≤ b i ≤ n-1 is such that θ τ (a i ) = (1, n, . . . , 2) b i . (b) ψ( δ) = 1 in cases (II) and (III).

(c) ψ(v) = g w , where 0 ≤ w ≤ n -1 is such that θ τ (v) = (1, n, . . . , 2) w in case (III).

Noting that θ τ ( δ) is trivial by hypothesis and π 2 (g) = (1, n, . . . , 2), one may check in each of the three cases (I)-(III) that ψ defines a homomorphism for which the diagram ( 11) is commutative.

Although the conclusion of Proposition 4.3 is the same as that of Proposition 4.2, its proof is somewhat different in nature. We first give a geometric proof of Proposition 4.3 in a specific case. This will then enable us to give an algebraic proof in the general case. We regard T 2 as the quotient space R 2 /Z 2 . Notationally, we shall not distinguish between an element of R 2 and its image under the canonical projection in T 2 . Let x and y denote the pointed homotopy classes of the loops in T 2 given by (t, 0) t∈[0,1] and (0, t) t∈[0,1] respectively. Then π 1 (T 2 ) = ⟨x, y | xyx -1 y -1 ⟩. ) for all (a, b) ∈ T 2 . Then τ defines a free action of Z 4k on T 2 such that the orbit space is the Klein bottle K 2 , and for which induced short exact sequence (1) associated to the covering p τ : T 2 -→ K 2 is of the form:

1 / / π 1 (T 2 ) = ⟨x, y | xyx -1 y -1 = 1⟩ (pτ ) # / / π 1 (K 2 ) = ⟨u, v | uvuv -1 = 1⟩ θτ / / Z 4k / / 1, (16) where:
(p τ ) # :

x -→ uv 2k y -→ v 4k and θ τ :

u -→ 2k v -→ 1. ( 17
)
Moreover, (T 2 , Z 4k , τ ; R 2 ) does not have the Borsuk-Ulam property.

Proof. Let k ∈ N, and let τ : T 2 -→ T 2 be the map defined by τ (a, b) = (-a, a + b + 1 4k ) for all (a, b) ∈ T 2 . For each 0 ≤ i ≤ 4k, we have:

τ i (a, b) = (a, b + i 4k ) if i is even (-a, a + b + i 4k ) if i is odd. (18)
One may check that if 1 ≤ i < j ≤ 4k, then τ i (a, b) ̸ = τ j (a, b) (note that if a = 1/2 and a + j-i 4k ∈ Z then i and j have the same parity), and since τ 4k = Id T 2 , it follows that τ defines a free action of Z 4k on T 2 . Let g : T 2 -→ K 2 be the double covering map whose non-trivial associated deck transformation τ 1 : T 2 -→ T 2 is given by τ 1 (a, b) = (-a, b + 1 2 ), so g • τ 1 = g, and let u and v be the homotopy classes determined by the loops (g(t, 0)) t∈[0,1] and (g(0, t/2)) t∈[0,1] respectively. Then π 1 (K 2 ) = ⟨u, v | uvuv -1 = 1⟩, and the induced homomorphism g # : π 1 (T 2 ) -→ π 1 (K 2 ) is given by g # (x) = u and g # (y) = v 2 . Let f : T 2 -→ T 2 be defined by f (a, b) = (a, ka + 2kb) for all (a, b) ∈ T 2 . The induced homomorphism f # : π 1 (T 2 ) -→ π 1 (T 2 ) is given by f # (x) = xy k and f # (y) = y 2k . Notice that f is a 2k-sheeted covering of T 2 , and that f # is injective. By construction, the map g • f : T 2 -→ K 2 is a 4k-sheeted covering of K 2 by T 2 , and consequently the deck transformation group associated to the covering g • f is of order 4k. We claim that τ is a deck transformation for the covering map g • f . If this is the case, since τ defines a free action of Z 4k on T 2 , it follows that group of deck transformations for the covering map g • f is isomorphic to Z 4k and is generated by τ , thus the associated orbit space T 2 τ is homeomorphic to K 2 , and the corresponding quotient map p τ is equal, up to homeomorphism, to g • f . To prove the claim, for all (a, b) ∈ T 2 , we have:

(g • f )(τ (a, b)) = (g • f ) -a, a + b + 1 4k = g -a, -ka + 2k a + b + 1 4k = g -a, ka + 2kb + 1 2 = (g • τ 1 ) (a, ka + 2kb) = g(a, ka + 2kb) = (g • f )(a, b).
This proves the claim. Thus (p τ ) # : π 1 (T 2 ) -→ π 1 (K 2 ) is injective, and satisfies the first equality of [START_REF] Whitehead | Elements of homotopy theory[END_REF]. Let θ τ : π 1 (K 2 ) -→ Z 4k be the map defined by [START_REF] Whitehead | Elements of homotopy theory[END_REF]. One may check that θ τ is a well-defined surjective homomorphism. Let us show that the sequence ( 16) is short exact. To do so, it remains to show that Im((p τ ) # ) = Ker (θ τ ). First let z ∈ Im((p τ ) # ). Then there exist m, n ∈ Z such that z = (p τ ) # (x m y n ) = (uv 2k ) m (v 4k ) n , and thus θ τ (z) = m(2k + 2k) + n. 4k = 0, hence Im((p τ ) # ) ⊂ Ker (θ τ ). Conversely, let y ∈ Ker (θ τ ). Now y ∈ π 1 (K 2 ), so there exist m, n ∈ Z such that y = u m v n , and 0 = θ τ (y) = m. 2k + n. 1. Thus 2mk + n ≡ 0 mod 4k, and there exists l ∈ Z such that n = 2mk + 4lk. Since vuv -1 = u -1 , we see that v 2 uv -2 = u, so u and v 2 commute, and thus:

y = u m v n = u m (v 2k ) m (v 4k ) l = (uv 2k ) m (v 4k ) l = ((p τ ) # (x)) m ((p τ ) # (y)) l = (p τ ) # (x m y l ) ∈ Im((p τ ) # ).
Therefore Ker (θ τ ) ⊂ Im((p τ ) # ). Hence Ker (θ τ ) = Im((p τ ) # ), and we conclude that the sequence ( 16) is short exact. We conclude that the short exact sequence associated to the covering p τ : T 2 -→ K 2 given by ( 1) is [START_REF] Vick | Homology Theory: An Introduction to Algebraic Topology[END_REF]. It remains to show that (T 2 , Z 4k , τ ; R 2 ) does not have the Borsuk-Ulam property. It suffices to exhibit an explicit map h :

T 2 -→ R 2 such that h(τ i (a, b)) ̸ = h(τ j (a, b)) for each (a, b) ∈ T 2 and 0 ≤ i < j ≤ 4k -1.
First note that by [START_REF] Zieschang | Surfaces and Planar Discontinuous Groups[END_REF], the orbit of a point (a, b) ∈ T 2 by the action of Z 4k generated by τ belongs to the union of the vertical circles ((a, t)) t∈[0,1] and ((1 -a, t)) t∈[0,1] , and these two circles coincide if a ∈ {0, 1 2 }. Let C ⊂ R 2 be the annulus defined with respect to polar coordinates by C = {(r, θ) | 1 ≤ r ≤ 2 and θ ∈ [0, 2π)}, and let h : T 2 -→ R 2 be defined by h(a, b) = (r(a), 2πb) for all (a, b) ∈ T 2 (we take 0 ≤ a, b ≤ 1), where:

r(a) =      (3 -4a)/2 if 0 ≤ a ≤ 1 4 (1 + 4a)/2 if 1 4 ≤ a ≤ 3 4 (7 -4a)/2 if 3 4 ≤ a ≤ 1.
One may check that h is well defined and continuous. Further, its restriction to each circle ((a, t)) t∈[0,1] of T 2 , where 0 ≤ a ≤ 1, is injective, and that if a ∈ (0, 1) \ { 1 2 } then the two circles (h(a, t)) t∈[0,1] and (h(1 -a, t)) t∈[0,1] are disjoint. We conclude that the restriction of the map h to each orbit of the action τ of Z 4k is injective, and this completes the proof of the lemma.

Lemma 4.6. Given k ∈ N, there exist α, β ∈ B Z 4k (R 2 ) such that π 2 (α) = 2k, π 2 (β) = 1 and αβαβ -1 = 1 in B Z 4k (R 2 ).
Proof. Let θ τ : π 1 (K 2 ) -→ Z 4k be the homomorphism defined in [START_REF] Whitehead | Elements of homotopy theory[END_REF]. By Corollary 2.6 and Lemma 4.5, there exists a homomorphism ψ : π 1 (K 2 ) -→ B Z 4k (R 2 ) such that the following diagram is commutative:

⟨u, v | uvuv -1 = 1⟩ ψ / / θτ ' ' B Z 4k (R 2 ) π 2 z z Z 4k .
It then suffices to take α = ψ(u) and β = ψ(v).

Proof of Proposition 4.3. We will define a homomorphism ψ : π 1 (M τ ) -→ B Z 4k (R 2 ) for which the diagram (11) is commutative, from which it will follow from Corollary 2.6 that (M, Z 4k , τ ; R 2 ) does not have the Borsuk-Ulam property. For this, we make use of the elements α and β given in Lemma 4.6 and the element g given in Remark 3.2(a). We endow π 1 (M τ ) with one of the presentations of cases (II) and (III) given at the beginning of this section. To simplify the analysis, we write π 1 (M τ ) = δ, v, a 1 , a 2 , . . . , a 2m-1 , a 2m δv δv -1 [a 1 , a 2 ] • • • [a 2m-1 , a 2m ] = 1 , where we take v = 1 in case (II). By hypothesis, θ τ ( δ) = 2k. Let 0 ≤ z ≤ 4k -1 be such that θ τ (v) = z (so z = 0 if v = 1). For i = 1, . . . , 2m, let b i ∈ Z be such that θ τ (a i ) = b i , and let ψ(v) = β z . If z is odd, we define ψ on the remaining generators of π 1 (M τ ) as follows:

ψ : δ -→ α a i -→ g b i , where 1 ≤ i ≤ 2m. ( 19
)
If z is even, then by the surjectivity of θ τ there exist 1 ≤ j ≤ 2m and l ∈ Z odd such that θ τ (a j ) = l.

If j is odd (resp. even), let ȷ = j + 1 (resp. ȷ = j -1). Then we define ψ on the remaining generators of π 1 (M τ ) as follows:

ψ :          δ -→ α (-1) j a j -→ β l a ȷ -→ α -1 β b ȷ +2k a i -→ g b i if i ̸ = j, ȷ. (20)
Let us show that ψ extends to a homomorphism from π 1 (M τ ) to B Z 4k (R 2 ). If z is odd, then we are in case (II), and by (19), we have: = α (-1) j . β z α (-1) j β -z (β l α -1 β b ȷ +2k β -l . β -b ȷ -2k α) (-1) j+1 = α 2(-1) j . α (-1) j+1 = 1, where we have used Lemma 4.6 and the fact that z is even and l is odd. This proves that ψ is a homomorphism. One may check that π 2 • ψ = θ τ , and hence the diagram ( 11) is commutative. The result then follows from Corollary 2.6.

Proof of Proposition 4.4. Suppose on the contrary that (M, Z 4k+2 , τ ; R 2 ) does not have the Borsuk-Ulam property. Then by Corollary 2.6 there exists a homomorphism ψ : π 1 (M τ ) -→ B Z 4k+2 (R 2 ) such that the diagram (11) is commutative. Suppose that M τ is non-orientable of odd (resp. even) genus, so that π 1 (M τ ) is as in case (II) (resp. case (III)) given at the beginning of this section. By hypothesis, θ τ ( δ) = 2k + 1, where δ = c (resp. δ = u). From Theorem 3.1, for i = 1, . . . , 2m, there exist w i ∈ P 4k+2 (R 2 ) and 0 ≤ b i < n, w δ ∈ P 4k+2 (R 2 ) (resp. w v ∈ P 4k+2 (R 2 ) and 0 ≤ b < n) such that for i = 1, . . . , 2m, θ τ (a i ) = b i and ψ(a i ) = w i g b i , ψ( δ) = w δ g 2k+1 , and if M τ is of even genus, θ τ (v) = b and ψ(v) = w v g b . The evaluation map ε : P 4k+2 (R 2 ) -→ Z defined on the generating set (A i,j ) 1≤i<j≤n of P 4k+2 (R 2 ) by ε(A i,j ) = 1 for all 1 ≤ i < j ≤ n extends to a well-defined homomorphism that satisfies ε(gwg -1 ) = ε(w) for all w ∈ P 4k+2 (R 2 ). For all i = 1, . . . , m, it follows that:

ε • ψ([a 2i-1 , a 2i ]) = ε w 2i-1 . g b 2i-1 w 2i g -b 2i-1 . g b 2i w -1 2i-1 g -b 2i . w -1 2i = ε([w 2i-1 , w 2i ]) = 0. (21)

Theorem 1 . 1 .

 11 Let M be a compact surface without boundary, and let τ : Z n × M -→ M be a free action. Then the quadruple (M, Z n , τ ; R 2 ) has the Borsuk-Ulam property if and only if the following conditions are satisfied: (1) n ≡ 2 mod 4.

  for example). Pick n distinct points P 1 , . . . , P n in S. A geometric n-braid is a n-tuple β = (b 1 , . . . , b n ) where b 1 , . . . , b n : [0, 1] -→ S × [0, 1] are paths (called strings of β) satisfying the following conditions: (i) b i (0) = P i × {1} for all i = 1, . . . , n. (ii) there exists a permutation π(β) ∈ Σ n , called the permutation of β, such that b i (1) = P π(β)(i) × {0} for all i = 1, . . . , n. (iii) for all t ∈ [0, 1], the intersection (S × {t}) ∩ {b 1 (t), . . . , b n (t)} consists of n distinct points. Two geometric n-braids β 0 and β 1 are said to be equivalent if there exists a continuous family of paths b s 1 , . . . , b s n : [0, 1] -→ S × [0, 1], s ∈ [0, 1], such that h 0 = β 0 , h 1 = β 1 , and h s = (b s 1 , . . . , b s n ) is an n-braid for all s ∈ [0, 1]. This defines an equivalence relation on the set of geometric n-braids, and the equivalence class of a geometric n-braid is called simply an n-braid. Notationally, we do not distinguish a geometric n-braid from its equivalence class. If β = (b 1 , . . . , b n ) and γ = (c 1 , . . . , c n ) are geometric n-braids, then we define the product of β and γ by βγ = (b 1 * c π(β)(1) , . . . , b n

  and applying Lemma 2.2 to the pointed homotopy class [F ], there exists a homomorphism ψ :π 1 (M τ , p τ (m 1 )) -→ B G (S) for which ψ • (p τ ) # = ι 2 • φ and π 2 • ψ = θ τ ,which yields the commutative diagram[START_REF] Gonc ¸alves | The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle[END_REF]. (b) Clearly, if β has the Borsuk-Ulam property, then α does too because every representative map of α is also a representative map of β. Conversely, assume that G is an Abelian group, and suppose that β does not have the Borsuk-Ulam property. So there exists a map f : M -→ S such that β = [f ] and for which the cardinality of the set {f (g 1 x), . . . , f (g n x)} is equal to n for all x ∈ M . By[START_REF] Vick | Homology Theory: An Introduction to Algebraic Topology[END_REF] Lemma 6.4], there exists a homeomorphism H : S -→ S that is homotopic to the identity such that H(f (m 1 )) = s 1 . Thus the map H •f is a representative map of β, the cardinality of the set {H •f (g 1 x), . . . , H •f (g n x)} is equal to n for all x ∈ M , and the pointed homotopy class α ′ = [H • f ] ∈ [M, m 1 ; S, s 1 ] does not have the Borsuk-Ulam property with respect to (G, τ ) (as a pointed homotopy class). By part (a), there exist homomorphisms φ ′ : π 1 (M, m 1 ) -→ P n (S) and ψ ′ : π 1 (M τ , p τ (m 1 )) -→ B G (S) such the following diagram is commutative:

Lemma 4 . 5 .

 45 Given k ∈ N, let τ : T 2 -→ T 2 be the map defined by τ (a, b) = (-a, a + b + 1 4k

ψ( δv δv - 1 [a 1 , a 2 ]

 112 • • • [a 2m-1 , a 2m ]) = ψ(uvuv -1 ) = α. β z αβ -1 = α. α -1 = 1,where we have used Lemma 4.6 and the fact that z is odd. If z is even, then by (20), we have:ψ( δv δv -1 [a 1 , a 2 ] • • • [a 2m-1 , a 2m ]) = ψ( δv δv -1 [a j , a ȷ ] (-1) j+1 )

Acknowledgements

The work on this paper started during the Ph.D thesis [13] of the third author who was supported by the CNPq project n o 140836 and the Capes/COFECUB project n o 12693/13-8, and was completed during his Postdoctoral Internship at IME-USP from March 2020 to August 2021 that was supported by the Capes/INCTMat project n o 88887.136371/2017-00-465591/2014-0. The first author is partially supported by the Projeto Temático FAPESP, grant n o 2016/24707-4: Topologia Algébrica, Geométrica e Diferencial.

Assume now that 1 ≤ i ≤ n -1 and j = n. One may check (by induction on n for example) that

,n , and it follows from this and (15) that:

which is the second relation of (II).

(iii) the lift of the relation n. 1 = 0 of Z n in terms of generators of P n (R 2 ). Since g n = ∆ 2 n by Remarks 3.2(b), relation (III) of the statement follows by noting that ∆ 2 n = n j=2 j-1 i=1 A i,j .

4 Proof of Theorem 1.1

Let M be a compact surface without boundary, and let Z n = {0, 1, 2, . . . , n -1} denote the cyclic group of order n ≥ 2. If τ : Z n × M -→ M is a free action, to simplify the notation, we will consider τ to be the continuous map τ : M -→ M defined by τ = τ (1, •), that satisfies τ n = Id and for which the cardinality of the set {x, τ (x), τ 2 (x), . . . , τ n-1 (x)} is equal to n for all x ∈ M . Then the orbit space M τ is also a compact surface without boundary. In what follows, we will distinguish the following three possibilities, and in each case, we define an element δ of π 1 (M τ ):

]⟩, and δ = u.

Remark 4.1. Let G = Z n , and consider the homomorphism θ τ : π 1 (M τ ) -→ Z n defined by [START_REF] Birman | Braids, Links, and Mapping Class Groups[END_REF]. If M is a compact surface without boundary, then θ τ ( δ) = 0 if M τ is orientable or if n is odd. In other words, if θ τ ( δ) ̸ = 0, then M τ is non-orientable, n is even, and θ τ ( δ) = n/2. Since the target of the homomorphism θ τ : π 1 (M τ ) -→ Z n is Abelian, the homomorphism θ τ factors through the Abelianisation π 1 (M τ ) Ab via the homomorphism that we denote (θ τ ) Ab : π 1 (M τ ) Ab -→ Z n , and for which θ τ ( δ) = (θ τ ) Ab (δ).

In order to establish Theorem 1.1, by Remark 4.1, it suffices to prove the following three propositions.

Proposition 4.2. Let M be a compact surface without boundary that admits a free action τ of Z n . With the notation of (I)-(III) above, suppose that θ τ ( δ) = 0. Then (M, Z n , τ ; R 2 ) does not have the Borsuk-Ulam property. Proposition 4.3. Let M be a compact surface without boundary that admits a free action τ of Z n . With the notation of (I)-(III) above, suppose that M τ is non-orientable, n = 4k for some k ∈ N, and θ τ ( δ) = 2k. Then (M, Z n , τ ; R 2 ) does not have the Borsuk-Ulam property. Proposition 4.4. Let M be a compact surface without boundary that admits a free action τ of Z n . With the notation of (I)-(III) above, suppose that M τ is non-orientable, n = 4k + 2 for some k ∈ N, and θ τ ( δ) = 2k + 1. Then (M, Z n , τ ; R 2 ) has the Borsuk-Ulam property.

To prove Propositions 4.2-4.4, we will apply Corollary 2.6 to the case where G is the finite cyclic group {0, 1, 2, . . . , n -1}. In what follows, we shall identify G with the subgroup of Σ n generated by the n-cycle (1, n, . . . , 2), and the generator 1 of G with this n-cycle.

Applying (21) and relation (III) of Theorem 3.1, if M τ is of odd genus, we obtain:

while if M τ is of even genus, we see that:

Combining the two cases of odd and even genus, we see that:

It follows from relation (III) of Theorem 3.

Taking equation ( 22) modulo 2, we obtain a contradiction, and hence (M, Z 4k+2 , τ ; R 2 ) has the Borsuk-Ulam property.

5 Two cases where G is the symmetric group

Let N be a compact surface without boundary, and let G be a non-trivial finite group for which there exists a surjective homomorphism θ : π 1 (N ) -→ G. Let ι : Ker (θ) -→ π 1 (N ) denote the inclusion. By standard covering space arguments, there exists a compact surface without boundary M and a free action τ : G × M -→ M such that the associated exact sequence given by ( 1) is of the following form:

We say that the surface M and the action τ arise from θ. This construction provides us with a method to construct a compact surface without boundary on which G acts freely. We will consider examples where G is either Σ n or Z n , where n > 2.

Let N 1 = T 2 # T 2 be the connected sum of two copies of T 2 , let N 2 = RP 2 # T 2 be the connected sum of RP 2 and T 2 , and let θ 1 :

be given by:

Note that θ 1 and θ 2 define surjective homomorphisms. For i = 1, 2, let M i and τ i : Σ n × M i -→ M i be the compact surface without boundary and the free action arising from θ i respectively.

The following result suggests that there are many possibilities to be explored in the generalisation of the Borsuk-Ulam theorem to actions of finite groups on surfaces. Proposition 5.1. Let n > 2. With the above notation, we have the following: (a) the quadruple (M 1 , Σ n , τ 1 ; R 2 ) does not have the Borsuk-Ulam property. (b) for the quadruple (M 2 , Σ n , τ 2 ; R 2 ), there does not exist a homomorphism ψ for which the lower triangle of the diagram (2) is commutative. (c) the quadruple (M 2 , Z n , τ Zn ; R 2 ), where the action of Z n is given by the restriction of the action of Σ n corresponding to the action of the cyclic subgroup ⟨(1, n . . . , 2)⟩, does not have the Borsuk-Ulam property.

Remark 5.2. If n = 2, a stronger version of Proposition 5.1 may be found in [START_REF] Gonc ¸alves | The Borsuk-Ulam theorem for surfaces[END_REF]. Note that the quadruple (M 1 , Σ 2 = Z 2 , τ 1 ; R 2 ) does not have the Borsuk-Ulam property, while the quadruple (M 2 , Σ 2 = Z 2 , τ 2 ; R 2 ) does have the Borsuk-Ulam property.

Proof of Proposition 5.1. By construction, first note that for

) be the homomorphism defined on the generators of π 1 (M τ 1 ) by ψ(a 1 ) = ψ(b 1 ) = σ 1 and ψ(a 2 ) = ψ(b 2 ) = g, where g is as defined in Remarks 3.2(a). Using ( 23) and ( 24), we may check that π • ψ = θ τ 1 , and Corollary 2.6 implies that (M 1 , Σ n , τ 1 ; R 2 ) does not have the Borsuk-Ulam property. (b) Suppose on the contrary that there exists a homomorphism ψ for which the lower triangle of the diagram (2) is commutative. From the lower short exact sequence of ( 6) and (25), there exist x, y, z ∈ P n (R 2 ) such that ψ(c) = xσ 1 , ψ(a 1 ) = yg and ψ(b 1 ) = zg, where g is as defined in Remarks 3.2(a). Taking the image of the relation of π 1 (M τ 2 ) and using [START_REF] Li | Artin braid groups and homotopy groups[END_REF], we obtain:

Note that σ 1 xσ -1 1 , gzg -1 and gy -1 g -1 belong to

) -→ Z be the evaluation homomorphism defined in the proof of Proposition 4.4. By [14, Lemma 3.1] and Theorem 3.1, we have ε(σ 1 A i,j σ -1 1 ) = ε(gA i,j g -1 ) = ε(A i,j ) = 1 for all 1 ≤ i < j ≤ n. So applying ε to (26), we obtain 0 = 2ε(x) + 1 in Z, which yields a contradiction. (c) If n ̸ ≡ 2 mod 4, the result is clear by Theorem 1.1. So let us assume that n ≡ 2 mod 4, so n = 4k+2 for some k ∈ Z. We claim that the covering space M 2 is non-orientable, therefore M τ Zn is nonorientable. To see that M 2 is non-orientable, for i = 1, . . . , 2k+1, the transposition (1, 2) is conjugate in Σ 4k+2 to the transposition (i, 2k +1+i), so there exists

Since c represents a non-orientable loop in N 2 , 2k + 1 is odd and a 1 is orientable, it follows that w represents a non-orientable loop in N 2 . One may check that w ∈ Ker (θ 2 ), from which it follows that M 2 is non-orientable, and thus (M 2 ) τ Zn is also non-orientable. We claim that the homomorphism (θ τ Zn ) Ab : H 1 ((M 2 ) τ Zn , Z) -→ Z 4k+2 induced by θ 2 satisfies (θ τ Zn ) Ab (δ) = 0, where δ is the unique non-trivial torsion element of H 1 ((M 2 ) τ Zn ). This being the case, it follows from Theorem 1.1 that the Borsuk-Ulam property does not hold for the quadruple (M 2 , Z n , τ Zn ; R 2 ). To prove the claim, if p : (M 2 ) τ Zn -→ N 2 is the covering map such that p•p τ Zn = p τ 2 , it suffices to show that δ ∈ Ker ((p # ) Ab ), where (p # ) Ab : π 1 ((M 2 ) τ Zn ) Ab -→ π 1 (N 2 ) Ab (or equivalently the homomorphism p * : H 1 ((M 2 ) τ Zn ) -→ H 1 (N 2 ) on the level of first homology) is the homomorphism induced by p. To see this, observe that the induced homomorphism p * : H 2 (N 2 ) -→ H 2 ((M 2 ) τ Zn ) in cohomology with integral coefficients is trivial since H 2 (N 2 ) and H 2 ((M 2 ) τ Zn ) are isomorphic to Z 2 , and the homomorphism p * is multiplication by (4k + 1)!, which is the cardinality of Σ n divided by n. Since n > 2, the homomorphism p * is trivial. Using the universal coefficient formula, it follows that the torsion of H 1 ((M 2 ) τ Zn ) is sent to zero, and the result follows from Theorem 1.1.

An immediate consequence of the Proposition 5.1(a) is the following. Corollary 5.3. Given any finite group G, there exists a quadruple (M, G, τ ; R 2 ) for which the Borsuk-Ulam property does not hold.

Given a finite group G, it is not true in general that there exists a quadruple (M, G, τ ; R 2 ) satisfying the Borsuk-Ulam property. Theorem 1.1 provides many examples of this. For example, if G is cyclic of odd order, the quadruple (M, G, τ ; R 2 ) does not satisfy the Borsuk-Ulam property. It is thus a natural problem to classify the finite groups G for which there exists a quadruple (M, G, τ ; R 2 ) satisfying the Borsuk-Ulam property.


Ionel Sorin Ciuperca

José Ignacio Tello

Lack

Ignacio Tello

Lack of contact in a lubricated system

Keywords: lubricated systems, Reynolds variational inequality, global solutions, sationnary solutions. A.M.S. subject classification. 35J20, 47H11, 49J10

come

Introduction

Lubricated contacts are widely used in mechanical systems to connect solid bodies that are in relative motion. A lubricant fluid is introduced in the narrow space between the bodies with the purpose of avoiding direct solid-to-solid contact. This contact is said to be in the hydrodynamic regime, and the forces transmitted between the bodies result from the shear and pressure forces developed in the lubricant film.

We consider one of the simplest lubricated systems which consists of two rigid surfaces in hydrodynamic contact. The bottom surface, assumed planar and horizontal moves with a constant horizontal translation velocity and a vertical given force F > 0 is applied vertically on the upper body.

The wedge between the two surfaces is filled with an incompressible fluid. We suppose that the wedge satisfy the thin-film hypothesis, so that a Reynolds-type model can be used to describe the problem.

We denote by Ω the two-dimensional domain in which the hydrodynamic contact occurs. We assume that Ω is open, bounded and with regular boundary ∂Ω. Without lost of generality we consider 0 ∈ Ω. We assume that the upper body, the slider, is allowed to move only by vertical translation. The normalized distance between the surfaces is given by h(x, t) = h 0 (x) + η(t)

where η(t) > 0 represents the vertical translation of the slider and h 0 : Ω -→ [0, ∞[ describes the shape of the slider and is a given function satisfying

(1.1)

h 0 ∈ C 1 ( Ω), min x∈Ω h 0 (x) = h 0 (0) = 0.
The mathematical model we study considers the possible cavitation in the thin film, so the (normalized) pressure "p" of the fluid satisfies the Reynolds variational inequality (see [START_REF] Frêne | Lubrification hydrodynamique[END_REF]):

(1.2)

Ω h 3 ∇p • ∇(ϕ -p) ≥ Ω h ∂ ∂x 1 (ϕ -p) -η ′ (t)
Ω (ϕ -p), ∀ϕ ∈ K where K = ϕ ∈ H 1 0 (Ω) : ϕ ≥ 0 , and "∇" denotes the gradient with respect to the variables x ∈ Ω. Without lost of generality we assume the velocity of the bottom surface is oriented in the direction of the x 1 -axis and its normalized value is equal to 1. The equation of motion of the slider is

(1.3) η ′′ = Ω pdx -F (second Newton Law)
completed with the initial conditions:

(1.4)

η(0) = η 0 (1.5) η ′ (0) = η 1 ,
where η 0 > 0, η 1 ∈ IR are given data. The unknowns of the problem are the pressure p(x, t) and the vertical displacement of the slider η(t). It is known that for any given C 1 function η(t) the problem (1.2) is well posed (see for instance [START_REF] Kinderlehrer | An Introduction to Variational Inequalities and Their Applications[END_REF]).

The system (1.2)-(1.5) is equivalent to the following Cauchy problem for a second order ordinary differential equation in η:

(1.6)    η ′′ = G(η, η ′ ) η ′ (0) = η 1 , η(0) = η 0 , where G :]0, ∞[×IR -→ IR is given by G(β, γ) := Ω q(x)dx -F,
and q ∈ K (depending on β and γ) is the unique solution to (1.7)

     Ω (h 0 + β) 3 ∇q • ∇(ϕ -q) ≥ Ω h 0 ∂ ∂x 1 (ϕ -q) -γ Ω (ϕ -q) ∀ϕ ∈ K.
The main goal of the paper is to give sufficient conditions on the shape h 0 of the slider to obtain global existence on time to (1.6), i.e. there is no contact solid-to-solid for t < ∞. We also study the existence of steady states of the problem. Another interesting physical question which we adress here is to see if there exists a "barrier" value

η b > 0 such that η(t) ≥ η b , ∀ t > 0.
We prove the existence of η b for two of the three cases studied. Third case (the so called "flat case"), we prove that η tends to 0 as t → ∞.

The main ideas of these results are the following: when the distance between the surfaces decreases (i.e. η ′ ≤ 0) there exists a lower bound of the force exerted by the pressure of the fluid on the upper body. This lower bound admits an expression of the form F S + F D , where F S is a "spring-like" force and F D is a "damping force" (see Corollary (3.1) and Remark 3.1). F S depends only on the position η(t) and represents the force exerted by the pressure of the fluid for the stationary position in an auxiliary sub-domain U of Ω.

F D is of the form F D = -η ′ d where d is a "dumping" coefficient and depends only on η. The global existence of the solution η is a consequence of the velocity of blow up of d when η tends to 0. The existence of a "barrier" η b is based on the fact that F S blows up when η tends to 0. In the "flat case" the force F S is equal to zero, which explains the non existence of a barrier.

The present work is related to different articles on the fluid-rigid interaction problems (see for example [START_REF] Conca | Exsitence of solutions for the equations modelling the motion of a rigid body in a viscous fluid[END_REF], [START_REF] Desjardins | Existence of weak solutions for the motion of rigid bodies in a viscous fluids[END_REF], [START_REF] Gérard-Varet | Regularity issues in the problem of fluid structure interaction[END_REF], [START_REF] Hesla | Collision of smooth bodies in viscous fluids: a mathematical investigation[END_REF] and [START_REF] Hillairet | Lack of collision between solid bodies in a 2D incompressible viscous flow[END_REF], for a non-exhaustive bibliography on this subject). These papers concern the study of the motion of one or many rigid bodies inside a domain Q ∈ IR n , n = 2, 3, filled with an incompressible fluid with constant viscosity. The mathematical model is a coupled system between Navier-Stokes equations modeling the fluid and second Newton Law to describe the rigid bodies positions. A relevant problem in this context is the so called "non-collision" problem, where the question is to know if this body will touch the boundary ∂Q of the fluid in finite time. In [START_REF] Hillairet | Lack of collision between solid bodies in a 2D incompressible viscous flow[END_REF] Hillairet consider the particular case where Q is the half-plane IR×IR + and the rigid body is a disk which moves only along the vertical axis. He proves that in absence of external forces the solution is defined globally in time. He also shows that the disk remains all the time "far" from the boundary. In [START_REF] Gérard-Varet | Regularity issues in the problem of fluid structure interaction[END_REF] Gérard-Varet and Hillairet consider a more general shape of the rigid body in a general domain Q in presence of gravity. They prove the existence of a global in time solution of the problem, but now the rigid body can go the boundary of the domain as t goes to infinity. Similar results are given by Hesla in [START_REF] Hesla | Collision of smooth bodies in viscous fluids: a mathematical investigation[END_REF].

The main difference between the above mentioned works and the present one is the obtention in this study of a "barrier" value η b > 0 for any exterior force F . We can explain this difference by the high shear and pressure that develop in a lubricant fluid film, due especially to the relative motion of the closed surfaces. An interesting open question is to see if similar "barrier" results can be obtained for situations when the thin film hypothesis is not satisfied in the fluid (so the full Navier-Stokes equations must be used in the place of Reynolds models), but relative horizontal motion exists between the two surfaces.

Fluid-rigid interaction problems in lubrication where also considered in [START_REF] Díaz | A note on some inverse problems arising lubrication theory[END_REF] where Reynolds equation is used in the place of Reynolds variational inequality in the particular "flat" case. We also mention the papers [START_REF] Buscaglia | Existence of equilibria in articulated bearings[END_REF], [START_REF] Buscaglia | Existence of equilibria in articulated bearings in presence of cavity[END_REF] and [START_REF] Ciuperca | On the existence of solutions of equilibria in lubricated journal bearings[END_REF], where the existence of steady states is studied for lubricated systems with two degrees of freedom.

The contents of the paper are the following: In Section 2 we precise the hypothesis on h 0 and present the main results of the paper. In Section 3 we give some preliminary results and Section 4 is devoted to the proof of the theorems of Section 2.

Main results

We begin by the local in time existence and uniqueness result, for which the minimal hypothesis (1.1) is sufficient.

Theorem 2.1. The function G is locally Lipschitzian, so we have the existence and uniqueness of solution to (1.6)

locally in time.

Let [0, T [ be the maximal interval of existence of solution to (1.6)

, so η ∈ C 2 ([0, T [).
The main goal of the paper is to prove that T = +∞. It is equivalent to prove that for any fixed T > 0 there exists m > 0 and M > 0 (depending eventually on T ) such that

(2.8) m ≤ η(t) ≤ M, for all t ∈ [0, T [ |η ′ (t)| ≤ M.
Moreover, we are interested to know if there exists such constants m and M independent on T .

In order to study the existence of steady states and global existence of solutions to (1.6) we consider three different cases depending on the shape of the slide h 0 .

Case I. Line contact

We assume that h is equal to 0 only in the line {x 1 = 0} i.e.

h 0 (0, x 2 ) = 0 for all x 2 ∈ IR such that (0, x 2 ) ∈ Ω and h 0 (x 1 , x 2 ) > 0 for all (x 1 , x 2 ) ∈ Ω, x 1 = 0.
We also assume that there exists α ≥ 1 such that (2.9)

h 0 (x 1 , x 2 ) ∼ |x 1 | α when x 1 → 0.
More precisely there exists a neighborhood W of 0 and a function h 1 regular enough on the closure W of W with h 1 > 0 on W such that

h 0 (x 1 , x 2 ) = |x 1 | α h 1 (x 1 , x 2 ) in W.

Case II. Point contact

We assume that h 0 is equal to 0 only in the point {x = 0}, i.e.

h 0 (0) = 0 and h 0 (x) > 0 for all x ∈ Ω -{0}.

We also assume that there exists α ≥ 1 such that (2.10) h 0 (x) ∼ |x| α when x → 0 that is, there exist W and h 1 as in Case I such that

h 0 (x) = |x| α h 1 (x) in W (where | • | is the euclidian norm in IR 2 ).

Case III. Flat slides

We assume that h is flat, i.e.

(2.11)

h 0 = 0 on Ω which implies h(x, t) = η(t).
The results concerning the existence of steady states for cases I and II are enclosed in the following theorem: Theorem 2.2. Let h 0 satisfy assumption 2.9 in case I or 2.10 in case II for α satisfying

(2.12) α > 1 in Case I (line contact) α > 3 2 in Case II (point contact) .
Then there exists at least one stationary solution η > 0 of the Cauchy problem (1.6), i.e. G(η, 0) = 0.

Remark 2.1. The problem of uniqueness of the stationary solution is a difficult one. In [START_REF] Buscaglia | Existence of equilibria in articulated bearings in presence of cavity[END_REF] the authors proved the uniqueness of solutions to the 1-dimension problem for a particular function h 0 .

Results of global existence and barrier functions are presented in the following theorem: Theorem 2.3. We assume that h 0 satisfy assumption 2.9 in case I or 2.10 in case II for α satisfying

(2.13) α ≥ 3 2 in Case I (line contact) α ≥ 2
in Case II (point contact) , then T = +∞. Moreover, there exist constants m 0 , M 0 and M 1 such that 0 < m 0 ≤ M 0 and M 1 ≥ 0 satisfying ∀t ≥ 0:

m 0 ≤ η(t) ≤ M 0 |η ′ (t)| ≤ M 1 , for t ≥ 0.
Remark 2.2. For the one-dimensional problem, i.e. Ω is an interval of IR, the results are the same than in the case II (line contact) for the two-dimensional problem.

Some relevant questions concerning the dynamical system (1.6) remain open: -Uniqueness of solution for the steady states.

-Stability of the steady states.

-Existence of periodic solutions.

-The Attractor of the dynamical system. Theorem 2.4. We assume that h 0 ≡ 0 (Case III), then T = +∞, moreover there exist M 0 , M 1 > 0 such that

0 < η(t) ≤ M 0 |η ′ (t)| ≤ M 1 , for t ≥ 0 and η(t) → 0 as t → +∞. Moreover there exist t 0 ≥ 0, a > 0 and b ∈ IR with t 0 + b > 0, such that η(t) ≥ a √ t + b ∀ t ≥ t 0 .
In addition, no stationary solution exist for the system (1.6).

Remark 2.3. The same result can be obtained for the corresponding one-dimensional problem.

3. Some preliminary results on the function G

3.1.

Results for h 0 satisfying (1.1) (all cases are included). In this subsection we proof some preliminary results on G under the minimal hypothesis (1.1). Let V 1 be defined as follows

(3.14) V 1 = sup x∈Ω - ∂h 0 ∂x 1 (x) It is clear that V 1 ≥ 0. Lemma 3.1. i) There exists c 1 > 0 such that G(β, γ) ≤ c 1 β 3 -F ∀β > 0, γ ≥ 0. ii) G(β, γ) = -F ∀β > 0, γ ≥ V 1 .
Proof. i) We take ϕ = 0 and ϕ = 2q in (1.7) to have

Ω (h 0 + β) 3 |∇q| 2 = Ω h 0 ∂q ∂x 1 -γ Ω q.
We use the inequalities h 0 + β ≥ β, γ ≥ 0, q ≥ 0 to obtain

β 3 Ω |∇q| 2 ≤ Ω h 0 ∂q ∂x 1 .
We use the Poincaré inequality and the proof of case i) ends.

ii) The inequality (1.7) can be written:

Ω (h 0 + β) 3 ∇q • ∇(ϕ -q) ≥ - Ω ∂h 0 ∂x 1 + γ (ϕ -q) dx, ∀ϕ ∈ K. Since γ + ∂h 0 ∂x 1 ≥ 0, ∀ x ∈ Ω
we have that q = 0 which gives the result.

The rest of the results enclosed in this section concern the function G(β, γ) when γ ≤ 0. We begin by a general result on variational inequalities.

Lemma 3.2. Let a ∈ L ∞ (Ω) such that inf Ω a > 0. Let f ∈ H -1 (Ω)
and q ∈ K be the solution of the problem

(3.15) Ω a∇q • ∇(ϕ -q) ≥ < f, ϕ -q >,
for all ϕ ∈ K.

Let U ⊂ Ω arbitrary and open, and let r ∈ H 1 0 (U) the solution to

(3.16) U a∇r • ∇ψ = < f, ψ >, for all ψ ∈ H 1 0 (U).
Then q ≥ r on U.

Proof. We consider ψ ∈ H 1 0 (U), ψ ≥ 0 arbitrary, and we extend it to Ω by 0 and denote the extended function by ψ which belongs to K. For simplicity we omit the tilde. We take ϕ = q + ψ in (3.15) to obtain (3.17)

U a∇q • ∇ψ ≥ < f, ψ >,
Let us denote ξ = qr. From (3.16) and (3.17) we have

(3.18) U a∇ξ • ∇ψ ≥ 0,
for any ψ ∈ H 1 0 (U), ψ ≥ 0. On the other hand we have ξ ≥ 0 on ∂U. From the maximum principle we obtain ξ ≥ 0 on U which proves the lemma.

The following result is a consequence of the above lemma and nonnegativity of the solution q to (1.7) in Ω.

Corollary 3.1. Let us denote for any open set U ⊂ Ω and any β > 0 by q 1β and q 2β the solutions to the following problems

(3.19)    -∇ • [(h 0 + β) 3 ∇q 1β ] = -∂h 0 ∂x 1 on U q 1β = 0, on ∂U and (3.20)    -∇ • [(h 0 + β) 3 ∇q 2β ] = 1 on U q 2β = 0, on ∂U
respectively. We then have

(3.21) G(β, γ) ≥ U q 1β dx -γ U q 2β dx -F.
for all β > 0, γ ∈ IR and U ⊂ Ω open.

Remark 3.1. The expressions U q 1β dx and U q 2β dx represent the force "F S " and the damping coefficient "d" respectively, as we described in the Introduction.

3.2.

The case of non-horizontal slider. In this subsection we assume h 0 = 0 and also that h 0 satisfies the hypothesis of Cases I or II (line contact and point contact case respectively). We prove the existence of a sub-domain U ⊂ Ω such that the averages of the corresponding functions q 1β and q 2β are "large" in some sense when β is small. We denote by ρ ≥ 0 and θ ∈ [0, 2π] the polar coordinates of (x 1 , x 2 ).

∂h 0 ∂x 1 ≤ -c 2 β 1-1/α on B l,β .

b) Case II. (Contact point)

There exists θ 0 ∈ ]0, π 2 [, c 2 , β 0 > 0 and the sector B p,β defined by

B p,β = {(x 1 , x 2 ) ∈ IR 2 ; β 1/α ≤ ρ ≤ 2β 1/α ; π -θ 0 ≤ θ ≤ π + θ 0 } such that for any 0 < β ≤ β 0 : ∂h 0 ∂x 1 ≤ -c 2 β 1-1/α on B p,β .
Proof. a) We have for x 1 ≤ 0

∂h 0 ∂x 1 = -α(-x 1 ) α-1 h 1 +(-x 1 ) α ∂h 1 ∂x 1 = (-x 1 ) α-1 h 1 (x) -α -x 1 ∂h 1 ∂x 1 h 1 (

x) .

Since h 1 > 0 on W we obtain

x 1 ∂h 1 ∂x 1 h 1 (x) -→ 0 when x -→ 0
and the result is obvious. b) For any x in W -{0} we have

∂h 0 ∂x 1 = α|x| α-1 x 1 |x| h 1 + |x| α ∂h 1 ∂x 1 = |x| α-1 h 1 α x 1 |x| + |x| ∂h 1 ∂x 1 h 1 . Now we can chose θ 0 ∈]0, π 2 [ such that x 1 |x| < -1 2 if π -θ 0 ≤ θ ≤ π + θ 0 (choose for example θ 0 = π 6
). On the other hand we have

|x| ∂h 1 ∂x 1 h 1 -→ 0 when x -→ 0
which proves the lemma.

Lemma 3.4. Let us consider q 1β , q 2β the solutions to (3.19)- (3.20) where U is given by U := B l,β in case I, U := B p,β in case II, with B l,β and B p,β defined in Lemma 3.3. Then there exists β 0 , c 3 , c 4 > 0 such that for any β ∈]0, β 0 ] we obtain

(3.22) B l,β q 1β (x)dx ≥ c 3 β 2(1/α-1) in case I B p,β q 1β (x)dx ≥ c 3 β 3/α-2 in case II moreover (3.23) B l,β q 2β (x)dx ≥ c 4 β 3(1/α-1) in case I B p,β q 2β (x)dx ≥ c 4 β 4/α-3 in case II.
Proof. From (3.20) we deduce

(3.24) U q 2β dx = U (h 0 + β) 3 |∇q 2β | 2 dx.
From the equality

U (h 0 + β) 3 ∇q 2β • ∇ϕ = U ϕ, for all ϕ ∈ H 1 0 (U)
and Cauchy-Schwarz inequality, we get

(3.25) U (h 0 + β) 3 |∇q 2β | 2 dx ≥ sup ϕ∈H 1 0 (U ), ϕ =0 U ϕdx 2 U (h 0 + β) 3 |∇ϕ| 2 .
It suffices to find appropriate test functions ϕ ∈ H 1 0 (U), ϕ = 0 such that the term

( U ϕdx) 2 U (h 0 + β) 3 |∇ϕ| 2 is large enough.
Proof of (3.23) Case I: Line contact. We choose

ϕ(x 1 , x 2 ) = ψ 1 x 1 β 1/α ψ 2 (x 2 )
with

ψ 1 ∈ D(] -2, -1[), ψ 1 ≥ 0, ψ 1 ≡ 0 and ψ 2 ∈ D(] -δ 2 , δ 2 [), ψ 2 ≥ 0, ψ 2 ≡ 0. Then B l,β ϕ dx = -β 1/α -2β 1/α δ 2 -δ 2 ψ 1 x 1 β 1/α ψ 2 (x 2 )dx 1 dx 2 = β 1/α -1 -2 ψ 1 (y 1 )dy 1 δ 2 -δ 2 ψ 2 (x 2 )dx 2 and B l,β (h 0 + β) 3 |∇ϕ| 2 dx = -β 1/α -2β 1/α δ 2 -δ 2 (h 0 +β) 3 1 β 2/α |ψ ′ 1 x 1 β 1/α ψ 2 (x 2 )| 2 + |ψ 1 x 1 β 1/α ψ ′ 2 (x 2 )| 2 dx .
It is easy to show that

B l,β (h 0 + β) 3 |∇ϕ| 2 dx ≤ c ′ 2 β 3-1/α
where c ′ 2 > 0 is a constant independent on β. (3.25) implies

B l,β (h 0 + β) 3 |∇q 2β | 2 dx ≥ c 3 β 3(1/α-1)
and thanks to (3.24) we obtain (3.23) 1 .

Case II: Contact point. We choose ϕ(

x 1 , x 2 ) = ψ 3 ( ρ β 1/α )ψ 4 (θ) with ψ 3 ∈ D(]1, 2[), ψ 3 ≥ 0, ψ 3 ≡ 0, and ψ 4 ∈ D(]π -θ 0 , π + θ 0 [), ψ 4 ≥ 0, ψ 4 ≡ 0.
In polar coordinates, we have

B p,β ϕ dx = 2β 1/α β 1/α π+θ 0 π-θ 0 ψ 3 ρ β 1/α ψ 4 (θ)ρdρdθ = β 2/α 2 1 ψ 3 (ρ 1 )ρ 1 dρ 1 π+θ 0 π-θ 0 ψ 4 (θ) dθ and B p,β (h 0 + β) 3 |∇ϕ| 2 dx = B p,β (h 0 +β) 3 1 β 2/α ψ ′ 3 ρ β 1/α 2 |ψ 4 (θ)| 2 + 1 ρ 2 ψ 3 ρ β 1/α 2 |ψ ′ 4 (θ)| 2 .
We easily obtain

B p,β (h 0 + β) 3 |∇ϕ| 2 dx ≤ c ′ 2 β 3 ,
where c ′ 2 > 0 is a constant independent of β. From (3.25) we obtain

B p,β (h 0 + β) 3 |∇q 2β | 2 dx ≥ c 3 β 4/α-3 ,
and by (3.24) we get (3.23) 2 .

Proof of (3.22) From lemma 3.3 we have

- ∂h 0 ∂x 1 ≥ c 2 β 1-1/α on B l,β (in case I)
and

- ∂h 0 ∂x 1 ≥ c 2 β 1-1/α on B p,β (in case II).
By maximum principle we deduce the inequality

q 1β ≥ c 2 β 1-1/α q 2β , on B l,β (respectively B p,β ).
By (3.23) the proof ends.

The following corollary is a consequence of Corollary 3.1 and Lemma 3.4. Corollary 3.2. For any β ∈ ]0, β 0 ] and γ ≤ 0 we have

G(β, γ) ≥ c 3 β 2(1/α-1) -γc 4 β 3(1/α-1) -F, in case I or G(β, γ) ≥ c 3 β 3/α-2 -γc 4 β 4/α-3 -F, in case II.
with β 0 , c 3 , c 4 as in Lemma 3.4.

Proof of the main results

We consider η(t) the solution of the Cauchy problem (1.6) defined on the maximal interval [0, T [. 4.1. Bounds on η for the non-horizontal slider case. In this subsection we assume that h 0 satisfies the hypothesis of Cases I or II (line contact and point contact case respectively). We prove that η and η ′ are bounded and η reminds "far" from 0. We first prove in Proposition 4.1 that η ′ admits an upper bound and the same for η in Proposition 4.2. These results are needed to prove the existence of lower bounds for η ′ (Propostion 4.3) and η (Proposition 4.4) Let V 2 be defined by (4.26)

V 2 = max{η 1 + 1, V 1 }
for V 1 as in (3.14), then we have:

Proposition 4.1. η ′ (t) < V 2 ∀t ∈ [0, T [.
Proof. We argue by the contrary and assume that t 1 > 0 is the first point such that η ′ (t 1 ) = V 2 , which implies η ′′ (t 1 ) ≥ 0 which contradicts Lemma 3.1 ii) where η ′′ (t 1 ) = -F.

We introduce two energies E 1 , E 2 :]0, +∞[×IR → IR defined by

E 1 (β, γ) = 1 2 γ 2 + F β and E 2 (β, γ) = 1 2 γ 2 + F β + c 1 2β 2 for c 1 as in Lemma 3.1.
The energies E 1 and E 2 are used in the following lemma when η(t) is non-increasing or non-decreasing respectively.

Lemma 4.1. For any t ∈ [0, T [ we have i) d dt E 1 (η(t), η ′ (t)) ≤ 0 if η ′ (t) ≤ 0 ii) d dt E 2 (η(t), η ′ (t)) ≥ 0 if η ′ (t) ≥ 0.
Proof. i) We multiplying the equation

η ′′ + F = G(η, η ′ ) + F
by η ′ and use the inequality G(η, η ′ ) + F ≥ 0 to obtain the result. ii) From Lemma 3.1 i) we have

η ′′ - c 1 η 3 + F ≤ 0.
We multiply by η ′ to end the proof.

Let D 1 and D 2 be defined by

D 1 := c 1 F 1/3
and

D 2 := 2 max η 0 , D 1 , 1 F 1 2 η 2 1 + F η 0 + c 1 2η 2 0 , 1 F 1 2 V 2 2 + F D 1 + c 1 2D 2 1 . Proposition 4.2. η(t) < D 2 ∀t ∈ [0, T [.
Proof. By the contrary we assume that t 3 > 0 is the first time such that (4.27)

η(t 3 ) = D 2 .
Then, it results that η ′ (t 3 ) > 0 or η ′ (t 3 ) = 0 . In the last case, since

η(t 3 ) > D 1 , Lemma 3.1 i) implies η ′′ (t 3 ) = G(η(t 3 ), 0) < 0.
So, in both cases, since η ∈ C 2 , there exists

t 1 ∈ [0, t 3 [ such that η ′ (t) ≥ 0, ∀t ∈ [t 1 , t 3 ],
where t 1 is the smallest number with this property. Two options concerning t 1 are possible: Option 1: t 1 = 0. In this case, we have

η ′ (t) ≥ 0, ∀t ∈ [0, t 3 ].
From Lemma 4.1 ii) we obtain

E 2 (η(t 3 ), η ′ (t 3 )) ≤ E 2 (η 0 , η 1 )
which implies

η(t 3 ) ≤ 1 F 1 2 η 2 1 + F η 0 + c 1 2η 2 0
and contradicts (4.27). Option 2:

t 1 ∈ ]0, t 3 [. We have in this case η ′ (t 1 ) = 0, η ′ (t) ≥ 0 ∀t ∈ [t 1 , t 3 ] which implies η ′′ (t 1 ) ≥ 0. From Lemma 3.1 i) we obtain c 1 η 3 (t 1 ) ≥ F that is η(t 1 ) ≤ D 1 . Let t 2 ∈ [t 1 , t 3 ] be a time such that η(t 2 ) = D 1 .
From Lemma 4.1 ii) and Proposition 4.1 we have

E 2 (η(t 3 ), η ′ (t 3 )) ≤ E 2 (η(t 2 ), η ′ (t 2 )) ≤ 1 2 V 2 2 + F D 1 + c 1 2D 2 1
which implies

η(t 3 ) ≤ 1 F 1 2 V 2 2 + F D 1 + c 1 2D 2 1
and contradicts (4.27) and the proof ends.

We define V 3 as follows (4.28)

V 3 := max 1 -η 1 , 2 2F D 2 , 2 η 2 1 + 2F η 0 . Proposition 4.3. η ′ (t) > -V 3 , ∀t ∈ [0, T [.
Proof. We argue by the contrary and assume that t 2 ∈]0, T [ is the first time such that

(4.29) η ′ (t 2 ) = -V 3 .
We have two options:

Option I. η ′ (t) ≤ 0, ∀t ∈ [0, t 2 ]. From Lemma 4.1 i) we have E 1 (η(t 2 ), η ′ (t 2 )) ≤ E 1 (η 0 , η 1 ) which implies 1 2 |η ′ (t 2 )| 2 ≤ 1 2 η 2 1 + F η 0 and contradicts (4.

29).

Option II: There exists

t 1 ∈ ]0, t 2 [ such that η ′ (t 1 ) = 0 and η ′ (t) ≤ 0 ∀t ∈ [t 1 , t 2 ]. Then E 1 (η(t 2 ), η ′ (t 2 )) ≤ E 1 (η(t 1 ), 0) which combined with Proposition 4.2 implies 1 2 |η ′ (t 2 )| 2 ≤ F D 2
and contradicts (4.29).

The most difficult part is to obtain a lower bound of η (Proposition 4.4). Before we remark that from Corollary 3.2 we have By definition of D 3 we have:

D 3 := 2 3 min η 0 , c 3 F 1/s 1 , (4.34)        D 4 = 1 2 min{η 0 , β 0 , c 3 F 1/s 1 , D -s 2 3 + s 2 c 4 V 3 -1/s 2 } if s 2 > 0 and D 4 = 1 2 min{η 0 , β 0 , c 3 F 1/s 1 , D 3 e -V 3 /c 4 } if s 2 = 0.
(4.38) η ′ (t 1 ) ≤ 0, η ′ (t 2 ) ≤ 0 c 3 η(t) -s 1 > F, ∀t ∈ [t 1 , t 2 ].
We first see

(4.39) η ′ (t) ≤ 0, ∀t ∈ [t 1 , t 2 ].
Suppose that (4.39) is false, then there exists τ ∈]t

η ′′ ≥ -c 4 η ′ η -1-s 2 on [t 1 , t 2 ].
Case i): s 2 > 0.

We integrate (4.41) to deduce

η ′ (t) ≥ η ′ (t 1 ) + c 4 s 2 η(t) -s 2 - c 4 s 2 η(t 1 ) -s 2 , ∀t ∈ [t 1 , t 2 ]
and thanks to (4.39) and Proposition 4.3 applied for t = t 1 we obtain

c 4 s 2 η(t 2 ) -s 2 ≤ c 4 s 2 η(t 1 ) -s 2 + V 3 . Since η(t 1 ) = D 3 it results η(t 2 ) ≥ D -s 2 3 + s 2 c 4 V 3 -1/s 2
which contradicts (4.36).

Case ii): s 2 = 0. We integrate (4.41) to obtain

η ′ (t) ≥ η ′ (t 1 ) + c 4 log 1 η(t) -c 4 log 1 η(t 1 ) , ∀t ∈ [t 1 , t 2 ]
which implies

c 4 log 1 η(t 2 ) ≤ c 4 log 1 η(t 1 ) + V 3 . Then η(t 2 ) ≥ D 3 e -V 3 /c 4
which contradicts (4.36).

4.2. Bounds on η for the flat case. We consider the case h 0 ≡ 0. Let us introduce the auxiliary function w defined as the unique solution to the problem By maximum principle we have w > 0 on Ω which implies C(Ω) > 0.

In the following, for any real number z we denote z + = max{z, 0} (positive part) and z -= -min{z, 0} (negative part). We have the identity z = z +z -.

Lemma 4.2. η satisfies the following differential equation

η ′′ = C(Ω) (η ′ ) - η 3 -F.
Proof. For h 0 ≡ 0 the inequality (1.7) becomes (4.43)

β 3 Ω ∇q • ∇(ϕ -q) ≥ -γ Ω (ϕ -q), ∀ ϕ ∈ K.
The required result is a direct consequence of the following facts -if γ ≥ 0 the solution of (4.43) is q = 0 -if γ < 0 the solution of (4.43) is q = -γw β 3 . The bounds on η and η ′ can be summarized in the following proposition Proposition 4.5. The following inequalities are valid: I) For η 1 ≤ 0 and t ∈ ]0, T [ we have

Ia) -η 2 1 + 2F η 0 ≤ η ′ (t) < 0 Ib) η 0 C(Ω) C(Ω)+2η 2 0 F t-2η 2 0 η 1 1/2 ≤ η(t) ≤ η 0 .
II) For any η 1 > 0 we define t 0 = η 1 F and η0 = η 0 + η 2 1 2F , then t 0 < T and we have IIa) η(t) = -1 2 F t 2 + η 1 t + η 0 for t ∈ [0, t 0 ] IIb) -√ 2F η0 ≤ η ′ (t) < 0 for t ∈ ]t 0 , T [

IIc) η0 C(Ω) C(Ω)+2η 2 0 F (t-t 0 ) 1/2
≤ η(t) ≤ η0 for t ∈ ]t 0 , T [.

Proof. I)

We assume that η 1 ≤ 0. Then, Lemma 4.2 implies η ′′ (0) = -F and therefore there exists a point t 1 ∈ ]0, T ] such that η ′ (t) < 0, ∀ t ∈ ]0, t 1 [, where t 1 denotes the largest element with this property. We now prove (4.44) η ′ (t) < 0, ∀ t ∈ ]0, T [, which is equivalent to assert t 1 = T . In order to prove (4.44) we argue by contradiction and assume t 1 < T and η ′ (t 1 ) = 0 which implies η ′′ (t 1 ) ≥ 0 and contradicts η ′′ (t 1 ) = -F which is obtained from Lemma 4.2 and proves (4.44). (4.44) implies (4.45)

η ′′ = -C(Ω) η ′ η 3 -F on [0, T [.
We multiply by η ′ to obtain that 1 2 (η ′ ) 2 +F η is a non-increasing function on [0, T ]. This completes the proof of the double inequality in Ia). Since η is a non-increasing function, we deduce the inequality of the right-hand side of Ib). Now we integrate (4.2) over [0, t[ to obtain

η ′ = η 1 + C(Ω) 2η 2 - C(Ω) 2η 2 0 -F t.
Thanks to η ′ < 0 of Ia) we obtain C(Ω) 2η 2 < C(Ω) 2η 2 0 + F tη 1 on ]0, T [ which completes the proof of Ib). II) we assume that η 1 > 0. Then, Lemma 4.2 implies η ′′ = -F for t ∈ [0, t 0 ] which proves IIa). Since η ′ (t 0 ) = 0 and η(t 0 ) = η0 the proofs of IIb) and IIc) are similar to the proofs of Ia) and Ib) respectively.

4.3.

Proofs of the theorems. Proof of Theorem 2.1. Let us fix β > 0 and γ ∈ IR and take β > 0 and γ ∈ IR such that ( β, γ) are close enough to (β, γ). We denote by q ∈ K the solution to the Reynolds inequality

Ω h 0 + β 3 ∇q • ∇(ϕ -q) ≥ Ω h 0 ∂ ∂x 1 (ϕ -q) -γ Ω (ϕ -q), ∀ϕ ∈ K
which can be written in the form

Ω (h 0 + β) 3 ∇q • ∇(ϕ -q) ≥ Ω h 0 ∂ ∂x 1 (ϕ -q) -γ Ω (ϕ -q)+ + (β -β) Ω A β, β(x)∇ q • ∇(ϕ -q) (4.46)

Lemma 3. 3 .

 3 a) Case I. (Line contact) There exist δ, β 0 , c 2 > 0 and B l,β defined by B l,β :=] -2β 1/α , -β 1/α [ × ]δ, δ[ such that for any 0 < β ≤ β 0 we have

( 4 . 4 α

 44 30) G(β, γ) ≥ c 3 β -s 1c 4 γβ -1-s 2 -F, ∀β ∈]0, β 0 ], ∀γ ≤ 0where β 0 , c 3 , c 4 were defined in Lemma 3in Case II, Notice that s 1 > 0 and s 2 ≥ 0. Let D 3 and D 4 > 0 be defined by(4.33)

2 in

 2 Case I (line contact) or α ≥ 2 in Case II (point contact) we have η(t) > D 4 , ∀t ∈ [0, T [. Proof. By the contrary we assume t 2 ∈ ]0, T [ is the first time such that (4.36) η(t 2 ) = D 4 .Notice that η 0 > D 3 > D 4 . Let t 1 ∈]0, t 2 [ be the last point where (4.37) η(t 1 ) = D 3 .

  1 , t 2 [ such that η ′ (τ ) > 0.Let τ 1 be the supremum of τ ∈]t 1 , t 2 [ satisfying η ′ (τ ) > 0. It is clear that τ 1 < t 2 and it is a local maximum of η which implies

	(4.40)	η ′ (τ 1 ) = 0 η ′′ (τ 1 ) ≤ 0.
	Then from (4.30) and (4.38) we have
		η ′′ (τ 1 ) = G(η(τ 1 ), 0) ≥	c 3 η(τ 1 ) s 1 -F > 0
	which contradicts (4.40). Then (4.39) is proved.
	Combining (4.30) and (4.38) we deduce
	(4.41)

where A β, β is uniformly bounded in β. We take ϕ = q in (1.7), ϕ = q in (4.46) and we add both inequalities to get

By Poincaré inequality the proof ends. Proof of Theorem 2.2. Let us introduce the function g :]0, +∞[→ IR defined by

From Theorem 2.1, it is clear that g is continuous. Lemma