Éphane St

Sabourau
email: sabourau@lmpt.math.univ-tours.fr

SYSTOLIC VOLUME OF HYPERBOLIC MANIFOLDS AND CONNECTED SUMS OF MANIFOLDS

Keywords: systole, systolic volume, connected sums, hyperbolic manifolds

published or not. The documents may come

Systolic volume of hyperbolic manifolds and connected sums of manifolds

Introduction

Consider a nonsimply connected closed n-manifold M endowed with a Riemannian metric g. The systole of (M, g), denoted sys(M, g), is defined as the length of the shortest noncontractible loop in M . Define the systolic volume of M as σ(M ) = inf g vol(M, g) sys(M, g) n (1.1)

where g runs over the space of all metrics on M . The systolic volume is a homotopy invariant, cf. [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF]. Furthermore, in a large number of cases, the systolic volume of a manifold depends only on its homology class in the corresponding Eilenberg-MacLane space (cf. [START_REF] Babenko | Géométrie systolique des variétés de groupe fondamental Z2, Sémin[END_REF] and [START_REF] Babenko | Nature topologique des systoles unidimensionnelles[END_REF] for a more precise statement).

A closed oriented n-manifold M is said to be essential if there is a map Φ : M → K(π, 1) to some K(π, 1)-space such that Φ * [M ] is nonzero in H n (π; Z). Note that closed oriented aspherical manifolds are essential. Furthermore, the connected sum of an essential manifold with any closed oriented manifold is essential.

M. Gromov [START_REF] Gromov | Filling Riemannian manifolds[END_REF] proved that the systolic volume of essential manifolds is bounded away from zero (cf. [Gu] for a recent extension). More precisely, there exists a positive constant C n such that every essential n-manifold M satisfies σ(M ) ≥ C n .

(1.2) Conversely, I. Babenko [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF] showed that a closed oriented manifold with positive systolic volume is essential.

The systolic inequality (1.2) can be improved by taking into account the topology of the manifold. For instance, M. Gromov [START_REF] Gromov | Filling Riemannian manifolds[END_REF]6.4.C' & 6.4.D'], [START_REF] Gromov | Systoles and intersystolic inequalities[END_REF]3.C.3] showed that the systolic volume of a closed manifold M is bounded from below in terms of its simplicial volume ||M || and its simplicial height h(M ) (cf. Section 2 for a definition). Specifically, there exist two positive constants C n and C n depending only on n such that every closed n-manifold M satisfies

σ(M ) ≥ C n ||M || log n (1 + ||M ||) , (1.3) σ(M ) ≥ C n h(M ) exp(C n log h(M ))
.

(1.4)

When M is aspherical, n = 3, the author [START_REF] Sabourau | Systolic volume and minimal entropy of aspherical manifolds[END_REF] showed that the simplicial volume can be replaced by the minimal entropy in (1.3).

In the two-dimensional case, the inequality (1.3), improved in [START_REF] Katz | Entropy of systolically extremal surfaces and asymptotic bounds[END_REF], yields a lower bound on the systolic volume of surfaces (an asymptotic lower bound for general 2-complexes can be found in [RS]). The examples of hyperbolic surfaces with large systoles obtained in [START_REF] Buser | On the period matrix of a Riemann surface of large genus. With an appendix[END_REF] (cf. [START_REF] Katz | Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups[END_REF] for other examples and generalizations in higher dimension) provide a similar upper bound on the systolic volume. Specifically, there exists an explicit positive constant c such that every closed orientable surface Σ k of genus k satisfies

π k (log k) 2 σ(Σ k ) c k (log k) 2 .
(1.5)

The surface Σ k can be described as a connected sum of k tori or as a surface admitting a hyperbolic metric (of area 4π(k -1)). In higher dimension, no manifold satisfies these two features, but we can separately study the systolic volume of the manifolds satisfying one or the other.

The first result we obtain deals with connected sums.

Theorem A. Let M be a closed oriented essential n-manifold. Then, the systolic volume of the connected sums

# k M = M # . . . #M of k copies of M is unbounded. More precisely, σ(# k M ) ≥ C n k exp(C n √ log k) (1.6)
where C n and C n are two positive constants depending only on n.

We can also replace # k M by the connected sum M 1 # • • • #M k of k closed oriented essential n-manifolds, cf. Theorem 3.1 for a slightly more general version. If M is non-essential then # k M is also non-essential and its systolic volume vanishes, cf. [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF].

Theorem A provides a partial answer to a question raised in [Gr96, p. 330] and [BB05, Q 1 ] asking for the asymptotic behaviour of σ(# k M ). In [START_REF] Babenko | Géométrie systolique des sommes connexes et des revêtements cycliques[END_REF], I. Babenko and F. Balacheff show that, for every closed n-manifold M , there exists a constant C(M ) depending only on the topology of M such that

σ(# k M ) ≤ C(M ) k √ log log k √ log k .
This estimate provides a sublinear upper bound on σ(# k M ). They also conjecture that for every closed essential n-manifold M

σ(# k M ) ∼ k (log k) n .
Example 1.1. The systolic volume of k copies of n-tori goes to infinity with k, i.e., σ(T n # • • • #T n ) → ∞. This result does not follow from the inequality (1.3) since the simplicial volume ||T n || of the torus is zero for n ≥ 3, cf. [START_REF] Gromov | Volume and bounded cohomology[END_REF]. When n = 2, this result follows from the estimate (1.5).

The second result we obtain deals with hyperbolic manifolds. Theorem B. Let {M i } be a sequence of infinitely many, non-homeomorphic, closed hyperbolic n-manifolds. Then, the systolic volume of the M i 's is unbounded, that is

lim i→∞ σ(M i ) = ∞. (1.7)
This result is already known for n = 2 and n ≥ 4. In the two-dimensional case, it follows from the asymptotic bound (1.5). When n ≥ 4, H. C. Wang's theorem [START_REF] Wang | Topics on totally discontinuous groups, Symmetric spaces[END_REF] (see [START_REF] Burger | Counting hyperbolic manifolds[END_REF] for a quantitative version) asserts that there are only finitely many closed hyperbolic n-manifolds with bounded simplicial volume (recall that the simplicial volume is proportional to the hyperbolic volume, cf. [START_REF] Thurston | Three-dimensional geometry and topology[END_REF], [START_REF] Gromov | Volume and bounded cohomology[END_REF]). Combined with the inequality (1.3), this implies Theorem B for n ≥ 4. Since the conclusion of H. C. Wang's theorem does not hold for n = 3, the three-dimensional case requires other arguments. This paper is organized as follows. In Section 2, we introduce the notions of relative systoles and simplicial heights. We also recall a very general systolic inequality due to M. Gromov, which relates the systolic volume of a manifold to its simplicial height, generalizing the inequality (1.4). This systolic inequality plays a central role in the proofs of Theorems A and B. In Section 3, we introduce some constructions and notations about the connected sums of manifolds. In Section 4, we bound from below the simplicial height of the connected sum # k M by k, which leads to Theorem A. The proof of Theorem B proceeds as follows. In [START_REF] Soma | Non-zero degree maps to hyperbolic 3-manifolds[END_REF], T. Soma proved a finiteness result for closed orientable hyperbolic manifolds dominated by the same manifold. Recently, G. Besson, G. Courtois and S. Gallot [START_REF] Besson | Inégalités de Milnor-Wood géométriques[END_REF] found another proof of this result based on the deformation of representations in the isometry group of the hyperbolic space. A straightforward generalization of their arguments shows that T. Soma's theorem applies to cycles, cf. Sections 5 and 6. We deduce then that the simplicial height of a sequence of closed orientable hyperbolic manifolds is unbounded, which leads to Theorem B.

Acknowledgement. Part of this work was completed during the program "Spaces of negative curvature" at the Centre Interfacultaire Bernoulli, where the author tried to construct a counterexample to Theorem B, see Example 6.3. The author would like to thank the Centre for its hospitality and the participants, including F. Balacheff, C. Croke, H. Parlier and A. Reid, for several helpful discussions.

Systolic volume and simplicial height

Let π be a discrete group. A geometric cycle representing a homology class α in H n (π; Z) is a map Ψ : X -→ K(π, 1) from a closed oriented ndimensional Riemannian pseudomanifold (X, g) (cf. [START_REF] Spanier | Algebraic topology[END_REF] for a definition) to a K(π, 1)-space such that Ψ * [X] = α where [X] is the fundamental class of X. Denote by ψ : π 1 (X) → π the homomorphism induced by Ψ and by ψ the map induced by ψ between the conjugacy classes. The ψ-systole of (X, g), denoted sys ψ (X, g), is defined as the length of the shortest loop of X whose image by ψ is nontrivial. The ψ-systolic volume of (X, g) is defined as

σ ψ (X, g) = vol(X, g) sys ψ (X, g) n .
(2.1)

Here, the volume of an n-dimensional Riemannian polyhedron (X, g) is defined as the sum of the n-volumes of all the n-simplices of X. Thus, the n-volume agrees with the n-dimensional Hausdorff measure. We clearly have

σ ψ (X) ≤ σ(X). (2.2)
The systolic volume of a homology class α ∈ H n (π; Z) is defined as

σ(α) = inf X σ ψ (X, g), (2.3)
where X = (X, Ψ, g) runs over all the geometric cycles representing α. M. Gromov showed in [Gr83, §6] that there exists a positive constant c n such that every nontrivial homology class α ∈ H n (π; Z) satisfies

σ(α) ≥ c n (2.4) By definition, if M is a closed oriented essential manifold, there exists a map Φ : M → K(π, 1) such that Φ * [M ] is nonzero in H n (π, Z). Thus, σ(Φ * [M ]) ≤ σ(M ).
(2.5)

In particular, every closed essential n-manifold M satisfies σ(M ) ≥ c n .

Example 2.1. Given a closed Riemannian manifold M , the homology modulo torsion, denoted

H * (M ; Z) R , is the image of H * (M ; Z) in H * (M ; R).
Suppose there exist n cohomology classes in H 1 (M ; R) whose cup product is nonzero in H n (M ; R). Then, the natural map

Φ : M → K(H 1 (M ; Z) R , 1) H 1 (M ; R)/H 1 (M ; Z) R
to the Jacobi variety takes the fundamental class of M to a nonzero class in homology. The inequality (2.4) yields a systolic inequality for the homology systole (modulo torsion) defined as sys H (M ) = sys φ (M ) where φ :

π 1 (M ) → H 1 (M ; Z) R is the natural homomorphism. Namely, sys H (M ) n ≤ c n vol(M ).
Since the stable systole, cf.

[CK03] for a definition, satisfies stsys(M ) ≤ sys H (M ), this leads to the inequality,

stsys(M ) n ≤ c n vol(M )
where the constant c n depends only on n (in particular, it is independent of the (first) Betti number of M ). Compare with [START_REF] Bangert | Stable systolic inequalities and cohomology products[END_REF], [START_REF] Ivanov | Generalized degree and optimal Loewner-type inequalities[END_REF] where more general systolic inequalities have been established for the stable systole but with a constant depending on the Betti numbers of M . See also [START_REF] Bangert | Stable systolic inequalities and cohomology products[END_REF], [START_REF] Bangert | An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm[END_REF], [START_REF] Ivanov | Generalized degree and optimal Loewner-type inequalities[END_REF], [START_REF] Bangert | Boundary case of equality in optimal Loewner-type inequalities[END_REF], [KR] for examples of sharp systolic inequalities for the stable and conformal systoles along with a description of the extremal metrics and [START_REF] Croke | Universal volume bounds in Riemannian manifolds[END_REF] for a general account on the subject.

By using regularization techniques on geometric cycles, M. Gromov also showed that the lower bound (2.4) can be improved by taking into account the topology of the representatives of the homology class α as follows.

Definition 2.2 ([Gr96, 3.C.3]). The simplicial height of a homology class α ∈ H n (π; Z), denoted h(α), is the minimal possible number of simplices of an oriented pseudomanifold X which admits a map Ψ : X → K(π, 1) to some K(π, 1)-space such that Ψ * [X] = α where Ψ * : H n (X; Z) → H n (π; Z) is the homomorphism induced by Ψ. The simplicial height of a closed orientable manifold M , denoted h(M ), is defined as the simplicial height of the image of its fundamental classs under the homomorphism Φ * : H n (M ; Z) → H n (π 1 (M ); Z) induced by its classifying map.

The following estimate, to compare with (1.4), improves the bound (2.4).

Theorem 2.3 ([Gr83, 6.4.C'] & [Gr96, 3.C.

3]).

There exist two positive constants C n and C n depending only on n such that every nontrivial homology class α ∈ H n (π; Z) satisfies

σ(α) ≥ C n h(α) exp(C n log h(α)) .
(2.6) 3. Relative systolic volume of connected sums of manifolds

Fix n ≥ 3. Let Φ i : M i -→ K i = K(π i , 1
), 1 ≤ i ≤ k, be maps between closed oriented n-dimensional manifolds M i to aspherical spaces K i with fundamental group π i . The maps Φ i induce homomorphisms φ i : π 1 (M i ) -→ π i between the fundamental groups and homomorphisms Φ i, * : H n (M i ; Z) -→ H n (π i ; Z) between the n-th homology groups. Note that the K i are K(π i , 1)spaces.

Let M be the connected sum of M 1 , M 2 , . . . , M k , that is

M = M 1 #M 2 # . . . #M k ,
and K be the bouquet of the aspherical spaces K 1 , K 2 , . . . , K k , that is

K = K 1 ∨ K 2 ∨ • • • ∨ K k .
The space K is also aspherical. Since n ≥ 3, the fundamental groups of M and K are respectively isomorphic to the following free products

π 1 (M ) = π 1 (M 1 # . . . #M k ) π 1 (M 1 ) * • • • * π 1 (M k ) π 1 (K) = π 1 (K 1 ∨ • • • ∨ K k ) π 1 * • • • * π k .
We will denote by π the fundamental group of K. Thus, K a K(π, 1)space with

π π 1 * π 2 * • • • * π k . Define a map Φ : M -→ K as follows Φ : M 1 # . . . #M k -→ M 1 ∨ • • • ∨ M k -→ K 1 ∨ • • • ∨ K k . (3.1)
The map Φ first contracts the attaching spheres of

M 1 # . . . #M k into a point, then takes each component M i of the bouquet M 1 ∨ • • • ∨ M k to the corresponding component K i of K 1 ∨ • • • ∨ K k through Φ i .
The homomorphism φ : π 1 (M ) -→ π induced by Φ between the fundamental groups agrees with the homomorphism

φ 1 * • • • * φ k from π 1 (M 1 ) * • • • * π 1 (M k ) to π 1 * • • • * π k which takes every element x of π 1 (M i ) to φ i (x) in π 1 * • • • * π k .
Using the previous notations, we can state the following result, whose proof occupies the next section.

Theorem 3.1. Let Φ i : M i -→ K i , 1 ≤ i ≤ k, be maps as above. Suppose that, for every i, the class

Φ i * [M i ] is nontrivial in H n (π i ; Z). Then, σ φ (M 1 # . . . #M k ) ≥ C n k exp(C n √ log k) , (3.2)
where C n and C n are two positive constants depending only on n.

Remark 3.2. The assumptions of Theorem 3.1 mean exactly that the manifolds M i are essential. Using the inequality (2.2), Theorem 3.1 yields Theorem A.

Remark 3.3. It should be noted, cf. [Ba95, [START_REF] Babenko | Géométrie systolique des sommes connexes et des revêtements cycliques[END_REF][START_REF] Babenko | Nature topologique des systoles unidimensionnelles[END_REF], that adding a non-essential manifold to an essential manifold does not change the systolic volume. More precisely, let M and N be two closed oriented n-manifolds with M essential and N non-essential then σ(M #N ) = σ(M ).

Proof of Theorem A

Let us prove Theorem 3.1, which implies Theorem A.

Using the notations of the previous section, the n-th homology group H n (π; Z) of the bouquet K of the K(π i ; Z)-spaces K i is given by

H n (π; Z) k i=1 H n (π i ; Z).
By construction, the map Φ, cf. (3.1), takes the fundamental class of M to the sum of the

Φ i * [M i ], that is Φ * [M ] = Φ 1 * [M 1 ] ⊕ • • • ⊕ Φ k * [M k ] H n (π; Z).
Let us show that the simplicial height of Φ * [M ] is bounded from below by k. Consider a map Ψ :

X -→ K from an oriented n-dimensional pseu- domanifold X to K such that Ψ * [X] = Φ * [M ], (4.1)
where Ψ * : H n (X; Z) → H n (π; Z) is the homomorphism beween the n-th homology groups induced by Ψ. Denote by ψ : π 1 (X) -→ π the homomorphism between the fundamental groups induced by Ψ.

By Grushko's theorem [Ma77, §7.6], the image G = ψ(π 1 (X)) of ψ decomposes into a free product of subgroups

G = F p * H 1 * • • • * H r ⊂ π
such that F p is free of rank p, while every H i is nontrivial, non isomorphic to Z and freely indecomposable. Furthermore, given another decomposition of this sort, say G = F q * H 1 * • • • * H s , one necessarily has p = q, r = s and, after reordering, H i is conjugate to H i . STEP 1. Our first step amounts to drop the free factor F p in the Grushko decomposition of G.

The homomorphism ψ : π 1 (X) → π factors through the following commutative diagram G π 1 (X) ψ / / < < y y y y y y y y y π where π 1 (X) → G is the epimorphism induced by ψ and G → π is the monomorphism given by the inclusion. Therefore, the map Ψ factors, up to homotopy, through the classifying maps induced by the above epimorphism and monomorphism, according to the following commutative (up to homotopy) diagram

K X Ψ / / > > | | | | | | | | K where K is a K(G, 1)-space.
The composite Ψ : X → K → K of these classifying maps and the map Ψ take [X] to the same class, that is

Ψ * [X] = Ψ * [X]. (4.2) Since G = F p * H with H = H 1 * • • • * H r , the Eilenberg-MacLane space K = K(G, 1) can be realized (up to homotopy equivalence) as K = K ∨ p i=1 S 1 , where K is a K(H, 1)-space. The projection of K to K defines a map K → K . Consider the natural composite Θ : X → K → K → K
where the map K → K is the classifying map induced by the inclusion homomorphism H → G. By construction, we have Θ * [X] = Ψ * [X]. From (4.1) and (4.2), this relation can be written as

Θ * [X] = Φ * [M ].
Furthermore, the image of the homomorphism θ :

π 1 (X) → G induced by Θ agrees with H, that is θ(π 1 (X)) = H.
This shows that, for our purpose, we can drop the free factor in Grushko's decomposition. STEP 2. Consider now another natural composite

Θ j : X Θ -→ K → K j ,
where the second map K → K j is the projection of K to K j . Note that the homomorphism p j : π → π j induced by K → K j is trivial on each free factor π i of π with i = j.

Since Θ * [X] = Φ * [M ], the homomorphism between the n-th homology groups induced by Θ j takes [X] to the nontrivial class Φ j * [M j ]. Therefore, the homomorphism

θ j : π 1 (X) θ -→ H p j -→ π j induced by Θ j is nontrivial. STEP 3. By Kurosh's theorem [Ma77, §7.5], the subgroup H i of π = π 1 * • • • * π k is conjugate in π to
a subgroup of π j with j = τ (i). This property fails if H i is a nontrivial free subgroup of π. Indeed, one can define a homomorphism from F p to π by taking each element of a natural basis of F p to any element of π. This justifies our first step.

Thus, if j is different from τ (i), then p j (H i ) is trivial. Therefore, the map τ from {1, • • • , r} to {1, • • • , k} is surjective, otherwise there would be a trivial homomorphism θ j : π 1 (X) → H → π j with j / ∈ Im τ , contradicting Step 2. Thus, we have r ≥ k. STEP 4. We can now conclude the proof of Theorem 3.1, which implies Theorem A. By Grushko's theorem (cf. [Ma77, §7.6]), we have rank G ≥ r. Thus,

rank π 1 (X) ≥ rank G ≥ r ≥ k.
Since the number of 1-simplices in X bounds from above the rank of π 1 (X), we obtain

h(Φ * [M ]) = h(Ψ * [X]) ≥ k,
where the simplicial height h is defined in Section 2. Using (2.6), we deduce

σ φ (M 1 # . . . #M k ) ≥ C n k exp(C n √ log k) .
Hence Theorem 3.1.

Deformations of representations in hyperbolic spaces

Let P be a finite simplicial n-complex and P be its universal cover (on which π 1 (P ) acts by deck transformations). Let ρ : π 1 (P ) → Isom(H n ) be a representation from π 1 (P ) to the isometry group of the hyperbolic nspace H n . Choose a piecewise smooth ρ-equivariant map f : P → H n . The pullback f * (ω) of the volume form ω on H n descends to a simplicial form on P , still denoted by f * (ω). Define the ρ-volume of an n-cycle c of P as follows:

vol(ρ, c) = c f * (ω).
The ρ-volume of an n-cycle is independent of the choice of f . Indeed, two piecewise smooth ρ-equivariant maps f 0 and f 1 from P to H n are ρequivariantly homotopic. Thus, the pullbacks f * t (ω) of the volume form ω on H n by this ρ-equivariant homotopy f t passe to the quotient on P . Their quotients on P define a homotopy between f * 0 (ω) and f * 1 (ω) (viewed as simplicial forms on P ). Thus, these two forms lie in the same cohomology class through the De Rham isomorphism for simplicial complexes, cf. [START_REF] Swan | Thom's theory of differential forms on simplicial sets[END_REF]. Therefore, the ρ-volume vol(ρ, c) of an n-cycle c of P is well-defined (see [START_REF] Dunfield | Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds[END_REF] for further details).

The following proposition generalizes to finite simplicial complexes a result established in [START_REF] Reznikov | Rationality of secondary classes[END_REF] for closed oriented 3-manifolds and in [BCG05, §6] for closed oriented n-manifolds. Note that if c is not a cycle but only a chain, the ρ-volume of c depends on the ρ-equivariant map f and the conclusion of the proposition fails.

Proposition 5.1. Let c be an n-cycle of a finite simplicial n-complex P . There exists a finite set Λ ⊂ R such that the ρ-volume of c lies in Λ for every representation ρ : π 1 (P ) → Isom(H n ).

Proof. We adapt the arguments of [BCG05, §6], developing only the points that differ from that reference.

Since the space of representations R from π 1 (P ) to Isom(H n ) is an algebraic variety, it has finitely many path-connected components. For our purpose, it is therefore enough to show that vol(., c) is constant on each of these path-connected components.

Let ρ t : π 1 (P ) → Isom(H n ), t ∈ R, be a C 1 -family of representations. By [BCG05, Lemme 6.2], there exists a C 1 -family of nondegenerate ρ tequivariant affine maps f t : P → H n defined for |t| small enough. Here, a map from a simplicial complex to H n is said to be affine if it is affine along the geodesic arcs of every (Euclidean) simplex. Such a map is said to be nondegenerate if the image of every simplex is a nondegenerate (geodesic) simplex of H n . Strictly speaking, this result has been stated in [START_REF] Besson | Inégalités de Milnor-Wood géométriques[END_REF] for closed manifolds, but the proof extends to finite polyhedra.

Fix an orientation on every n-simplex of P . Let c = i∈I k i s i be an n-cycle of P , where the s i are n-simplices of P . Let F be an oriented 2codimensional simplex of P . An n-simplex s of P with F as an (n -2)-face has exactly two (n -1)-faces meeting along F . The barycenters of these two faces can be joined by a segment of s. The link around F is defined as the union of all these segments. Thus, the link around F is a (possibly multiconnected) graph Γ whose edges are in one-to-one correspondence with the n-simplices of P with F as a face. The orientation of F along with the orientations of the n-simplices of P induce an orientation on the edges of Γ.

Let s j , j ∈ J F ⊂ I, be the n-simplices s i of c with F as a face. Denote by e j the (oriented) edges of Γ corresponding to the n-simplices s j of P . Since c is a cycle, the 1-chain e = j∈J F k j e j of Γ is a cycle too.

The geodesic simplex F t := f t (F ) of H n is endowed with the orientation of F . Since H n is an oriented manifold, the link around F t is a circle C t endowed with the induced orientation. The map f t induces a map ϕ t : Γ → C t between the links. The homomorphism ϕ t * : H 1 (Γ, Z) → H 1 (C t , Z) Z induced by ϕ t takes the homology class of the 1-cycle e to an integral multiple k of the fundamental class of C t . Since f t is nondegenerate for |t| small enough, the integer k, also noted deg F f t , does not depend on t for |t| small enough.

Given an n-simplex s of P with F as a face, let θ(t; F, s) be the nonnegative hyperbolic dihedral angle of f t (s) at F t . Define i as 1 if the restriction of f t to the n-simplex s i (which is an affine map) preserves the orientation and as -1 otherwise. As previously, since f t is nondegenerate for |t| small enough, the value i does not depend on t for |t| small enough.

By adapting the proofs of the lemmas 6.3 and 6.4 in [START_REF] Besson | Inégalités de Milnor-Wood géométriques[END_REF], one can show the following Lemma 5.2 (Compare with [BCG05, Lemmes 6.3 & 6.4]). We have

2π deg F f t = ± j∈J F j k j θ(t; F, s j ).
In particular,

d dt   j∈J F j k j θ(t; F, s j )   = 0
for |t| small enough.

We can now conclude as in [START_REF] Besson | Inégalités de Milnor-Wood géométriques[END_REF]§6]. By definition, vol(ρ t , c)

= c= P k i s i f * t (ω) = i∈I i k i vol(f t (s i )).
Its derivative can be expressed using Schläfli's differential formula, cf. [START_REF] Milnor | The Schläfli differential inequality[END_REF],

d dt vol(ρ t , c) = i∈I i k i   - 1 n -1 F ⊂s i d dt (θ(t; F, s i )) vol n-2 (F t )   = - 1 n -1 F ⊂c d dt   j∈J F j k j θ(t; F, s j )   vol n-2 (F t )
for |t| small enough. By Lemma 5.2, this expression is equal to zero. Therefore, vol(., c) is constant on the path-connected components of the space of representations R. Hence the result. 6. Proof of Theorem B Proposition 6.1. Let {M i } be a sequence of infinitely many, non-homeomorphic, closed oriented hyperbolic n-manifolds. Then, the sequence {h(M i )} of their simplicial heights is unbounded.

Proof. We follow the arguments developed in [BCG05, Théorème 6.6], which provide an alternate proof to a result of T. Soma [START_REF] Soma | Non-zero degree maps to hyperbolic 3-manifolds[END_REF].

We argue by contradiction and assume that the simplicial height of the manifolds M i is bounded by an integer N , cf. Definition 2.2. Let Ψ i : P i → M i be a map from a finite simplicial complex P i with at most N simplices which induces an epimorphism Ψ i * : H n (P i , Z) → H n (M i , Z) between the nth homology groups. Such maps exist by assumption since M i is aspherical, that is a K(π 1 (M i ), 1)-space. For every integer i, choose a family of n-cycles c 1 i , . . . , c m i i of P i inducing in homology a basis of the Z-module H n (P i , Z). For every pair i, j such that P i = P j (notice that there are finitely many simplicial complexes P i ), we can further impose that c k i = c k j for every k = 1, . . . , m i (= m j ). Since the homomorphism Ψ i * :

H n (P i , Z) → H n (M i , Z) is nonzero, there is an n-cycle c i = c k i i of P i such that Ψ i * [c i ] = d i [M i ] where d i is a nonzero integer. The properties of the simplicial volume, cf. [Gr81], imply that ||[c i ]|| ≥ ||Ψ i * [c i ]|| = |d i | • ||M i ||.
Since M i is a closed hyperbolic manifold, its simplicial volume, which is proportional to its volume cf. [START_REF] Gromov | Volume and bounded cohomology[END_REF], is bounded away from zero, cf. [START_REF] Thurston | Three-dimensional geometry and topology[END_REF]. Thus, the sequence {d i } is bounded (recall that ||[c i ]|| takes finitely many values). Let ρ i : π 1 (P i ) → Isom(H n ) be the representation induced by Ψ i . By construction, the relation vol(ρ i , c i ) = d i vol M i holds for every i. By Proposition 5.1, the volume of the closed hyperbolic manifolds M i takes finitely many values (recall that d i is nonzero and bounded). Therefore, there are finitely many closed hyperbolic manifolds M i , cf. [START_REF] Thurston | Three-dimensional geometry and topology[END_REF].

Remark 6.2. To cover the nonorientable case, one can use Z 2 homology coefficients in the definition of the simplicial height, cf. Definition 2.2. However, the example of a nonorientable closed hyperbolic 3-manifold admitting maps of Z 2 -degree 1 onto infinitely many closed hyperbolic manifolds, cf. [BW96], [START_REF] Rong | Degree one maps of Seifert manifolds and a note on Seifert volume[END_REF], shows that the conclusion of Proposition 6.1 fails in this case.

We can now conclude.

Proof of Theorem B. Combined with the inequality (1.4), Proposition 6.1 implies Theorem B in the orientable case. In the nonorientable case, we consider the orientable double Riemannian cover M → M . Since the quotient map from M to M is a local isometry which induces an injective homomorphism between the fundamental groups, we have sys(M ) ≥ sys(M ). On the other hand, vol(M ) = 2 vol(M ). Hence σ(M ) ≤ 2σ(M ). Now, since only finitely many hyperbolic n-manifolds have the same volume when n ≥ 3, no subsequence of {M i } is constant. By Theorem B (orientable case), the systolic volume of the M i 's, and so of the M i 's, is unbounded. Hence the result.

Example 6.3 (Systolic volume of tubes). Let T 2 be a flat 2-torus with systole . For every prime class c ∈ π 1 (T 2 ) Z 2 , we can fill T 2 into a solid 3-torus T 3 c such that the kernel of the π 1 -homomorphism induced by the inclusion ∂T 3 c ⊂ T 3 c is generated by c. For every prime c ∈ π 1 (T 2 ), we can construct a metric g on T 3 c such that (C1) g agrees with the initial flat metric on ∂T 3 c = T 2 ; (C2) sys(∂T 3 c , g) = . Is it possible to do so while controlling the volume of the filling? More precisely, is there a constant V 0 such that, for every prime c ∈ π 1 (T 2 ), there exists a metric g on ∂T 3 c = T 2 satisfying the conditions (C1) and (C2) with vol(∂T 3 c , g) ≤ V 0 ? This question, which naturally arises when one tries to construct a counterexample to Theorem B (see below), has a negative answer: there is no such uniform upper bound on the volume of ∂T 3 c (for a metric satisfying (C1) and (C2)). Specifically, let c n ∈ π 1 (T 2 ) be a sequence of infinitely many distinct prime classes and g n be a sequence of metrics on T 3 cn satisfying (C1) and (C2), then vol(g n ) → ∞. To prove this result, we argue by contradiction and consider a complete hyperbolic 3-manifold M of finite volume with one cusp. Cut this cusp along a horosphere far enough and remove the noncompact convex component. One obtains a compact hyperbolic 3-manifold with boundary a flat torus T 2 along which we glue a solid 3-torus T 3 c . These Dehn surgeries yield infinitely many, non-homeomorphic, closed 3-manifolds M c admitting a hyperbolic metric, cf. [START_REF] Thurston | Three-dimensional geometry and topology[END_REF]. By construction, the systole of M c agrees with and the volume of M c is less than vol(M ) + V . Thus, the systolic volume of M c is bounded. Hence a contradiction with Theorem B.


Éphane St

Sabourau
email: sabourau@gargan.math.univ-tours.fr.

GLOBAL AND LOCAL VOLUME BOUNDS AND THE SHORTEST GEODESIC LOOPS

HAL is

Introduction

The relations between the volume of complete Riemannian manifolds and the length of their shortest nontrivial geodesic loops under no curvature assumption are studied in this paper. We present and compare two lower bounds on the global volume, one of which admits a local version.

Previous curvature-free estimates have been first obtained with the injectivity radius. Namely, M. Berger proved in [START_REF] Berger | Une borne inférieure pour le volume d'une variété riemannienne en fonction du rayon d'injectivité[END_REF] the isoembolic theorem Vol(M ) ≥ C n inj(M ) n for all complete Riemannian manifolds with a sharp positive constant C n . A local version was then established by C. Croke in [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF] VolB(x 0 , R) ≥ C n R n for all R ≤ 1 2 inj(M ) for all complete Riemannian manifolds but with a nonsharp constant.

Replacing the notion of injectivity radius with one of local geometric contractibility, M. Gromov extended Berger's global volume estimate in [START_REF] Gromov | Filling Riemannian manifolds[END_REF]. In the same way, an extension of Croke's local version was then established by R. Greene and P. Petersen in [START_REF] Greene | Little topology, big volume[END_REF].

Other results have been obtained by M. Gromov and C. Croke, who compared the volume with the length of the shortest nontrivial closed geodesic, noted scg(M ). In [START_REF] Gromov | Filling Riemannian manifolds[END_REF], M. Gromov proved that every 1-essential closed Riemannian manifold M satisfies Vol(M ) ≥ C n scg(M ) n for some positive constant C n depending only on the dimension n of M . Recall that an n-dimensional manifold M is by definition k-essential if there exists a continuous map ψ : M -→ K into a K(π, k) space such that ψ * [M ] = 0 where [M ] ∈ H n (M ) is the fundamental class of M . The homology coefficients are in Z, if M is orientable, and in Z 2 , otherwise. For the two-dimensional sphere, the previous inequality still holds as it was proved by C. Croke in [START_REF] Croke | Area and the length of the shortest closed geodesic[END_REF]. These two statements are the only results known in this direction and no counter-example exist so far.

Throughout this paper, we will consider geodesic loops rather than closed geodesics. By definition, a geodesic loop (based at a point * ) is a geodesic arc with endpoints * . In particular, a closed geodesic is a geodesic loop whose tangent vectors at its endpoints agree. The length of the shortest nontrivial geodesic loop of a Riemannian manifold is denoted by sgl(M ).

The first theorem we will prove provides lower bounds on the diameter, on the volume of the whole manifold and on the volume of sufficiently small balls.

Theorem A. Let M be a complete Riemannian manifold. Then there exists positive constant C n , C n and C n depending only on the dimension n of M such that

Vol(M ) ≥ C n sgl(M ) n Diam(M ) ≥ C n sgl(M ) VolB(x 0 , R) ≥ C n R n for every ball B(x 0 , R) of radius R ≤ 1 2 sgl(M ) in M .
Explicit formulas for C n , C n and C n can be calculated. Note that scg(M ) may be greater than sgl(M ) and yield no universal lower bound on the volume of "small" balls of M . A counter-example is given by any Riemannian manifold on which a long and thin spike is glued.

The following result, introducing the lengths of geodesic double loops, refines the global volume estimate for 2-essential manifolds but does not admit any local version. The two geodesic loops which compose Φ 0 may agree. In this case, since Φ 0 is minimal, this latter reduces to a closed geodesic run twice and we get lower bounds on the volume and the diameter in terms of scg(M ).

Unfortunately, even after slight perturbations of the metric, the double loop Φ 0 might still be composed of two distinct geodesic loops. This will be discussed in more details at the end of this paper.

The example of a long and thin cone glued to a Riemannian manifold shows that 0 (M ) provides no universal lower bounds on the volume of "small" balls. This shows the difference in nature between the shortest geodesic loop and the double loop Φ 0 . Note that considering multi-loops formed of k nontrivial geodesic loops of the same length based at the same point (with k fixed) instead of double loops does not really change the problem since their shortest representants have comparable lengths.

In [START_REF] Anderson | Ricci curvature bounds and Einstein metrics on compact manifolds[END_REF], M. Anderson proved a compactness theorem for metrics whose Ricci curvature, diameter, volume and length of the shortest geodesic loop are bounded. From Theorem A, it turns out that the required bound on the volume can be dropped. In [START_REF] Rotman | Upper bounds on the length of the shortest closed geodesic on simply connected manifolds[END_REF], an upper bound similar to the second inequality of Theorem A has also been established by R. Rotman for simply connected Riemannian manifolds with nontrivial second homology group and with nonnegative curvature. These topological and geometrical assumptions on the manifold can actually also be dropped.

Similar curvature-free estimates were independently obtained in [START_REF] Nabutovsky | Volume, diameter and the minimal mass of a stationary 1-cycle[END_REF] by A. Nabutovsky and R. Rotman who found upper bounds on the minimal mass of nontrivial stationnary one-cycles (see also [START_REF] Nabutovsky | Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem[END_REF] for estimates depending on the curvature).

In the first section, we present a map extension criterion based on a simplex filling process and derive Theorem A using Greene and Petersen's theorem (see [START_REF] Greene | Little topology, big volume[END_REF]). In the second section, we introduce a variational calculus on the double loop space and describe the structure of critical points. Then, we prove Theorem B using filling radius estimates developped by M. Gromov in [START_REF] Gromov | Filling Riemannian manifolds[END_REF].

The author would like to thank I. Babenko and C. Croke for several helpful conversations and comments.

Extension process and short geodesic loops

In this part, we construct a map extension process and prove Theorem A.

1.1. Preliminaries. The notion of retraction of loop spaces and the definition of ordered complex are presented in this section.

Let (X, d) be a metric space, more simply noted X, and * be a point in X. The loop space of X based at * , noted Ω * X, is by definition the space of continuous loops γ : [0, 1] -→ X with endpoints * . It is endowed with the C 0 -topology. Let κ ≥ 0, we define Ω κ * X as the space of loops based at * of length less than κ. The length of a loop γ of a metric space is given by

L(γ) = sup m-1 i=0 d(γ(s i ), γ(s i+1 )) | (s i ) 0≤i≤m subdivision of [0, 1]
The Birkhoff curve-shortening process with fixed endpoints is defined on the loop space of Riemannian manifolds (cf. [START_REF] Klingenberg | Lectures on closed geodesics[END_REF] for a precise definition). It gives rise to a continuous length non-increasing homotopy γ t ∈ Ω * M from a fixed loop γ ∈ Ω * M to a geodesic loop based at * . Furthermore, geodesic loops are the only fixed points of this process. Therefore, if κ < sgl(M ), there exists no nontrivial critical point for the length functional in Ω κ * X. In this case, the Birkhoff process gives rise to a retraction from any loop space Ω κ * X into the singleton formed by the point curve * .

This property of retraction of loop spaces can also be set out for general metric spaces. Furthermore, it may be weakened, requiring that only finite dimensional subspaces of the loop space be retractible. In the rest of this part κ 0 ≥ 0 and m ∈ N are fixed and the metric spaces X considered satisfy the following property : For every point * ∈ X, every continuous map f : P -→ Ω κ 0 * X defined on any simplicial complex of dimension ≤ m, there exists a continuous map F : P × [0, 1] -→ Ω κ 0 * X such that -F agrees with f on P × {0} -F reduces to the constant loop * on P × {1} -L(γ p,t ) ≤ L(γ p,s ) whenever 0 ≤ s ≤ t ≤ 1 where γ p,t = F (p, t).

An ordered complex is, by definition, a simplicial complex endowed with a total order relation on the set of its vertices. For instance, a complex whose vertices are numbered is ordered by the natural order of N. A subcomplex of an ordered complex will be endowed with the order induced by restriction. In particular, a k-dimensional simplex ∆ of P will be considered as a

(k + 1)-uplet (v 0 , v 1 , . . . , v k ) with v 0 < v 1 < • • • < v k .
Note that the order induced by restriction is compatible with the inclusion. More precisely, let P ⊂ P ⊂ P be subcomplexes, the order on P induced by the inclusion P ⊂ P is the same as the one induced by P ⊂ P .

Extension criterion.

In this section, we state an extension criterion for maps defined on the vertices of simplicial complexes. We also treat the case where the extension is prescribed on a subcomplex.

Let P be a locally finite n-dimensional simplicial complex and f : P 0 -→ X be a map from the 0-skeleton P 0 of P to X.

Fix α n = 1 4.3 n-1 -1 .
Proposition 1.1. If the distance between the images of any pair of adjacent vertices is less than ρ with ρ < α n κ 0 , then f extends to a continuous map F : P -→ X. Furthermore, the diameter of the image by F of any simplex of P is less than ρ n := 2.3 n ρ.

Proof. Let us number the vertices of P by natural integers and consider the induced order. Every (k + 1)-dimensional simplex ∆ k+1 of P is endowed with the order induced by restriction and identifies in a natural way with the (k + 1)-dimensional standard ordered simplex ∆ k+1 0 = (x 0 , . . . , x k+1 ). The points x i represent the vertices of the simplex.

The construction of the extension F : P -→ X carries out successively on the skeleton of P . We will denote by ϕ : ∂∆ k+1 -→ X the restriction of F to the boundary ∂∆ k+1 of a (k + 1)-dimensional simplex ∆ k+1 of P . The simplex ∆ p I = (x i 0 , . . . , x ip ) ∆ p 0 , where

I = {i 0 , . . . , i p } is a subset of {0, . . . , k + 1} with i 0 < • • • < i p and p ≤ k, represents a p-face of ∆ k+1
0 . It identifies in a natural way with the p-dimensional standard ordered simplex ∆ p 0 .

We define F on the 1-skeleton of P by mapping each edge of P to a minimizing arc between the images of the endpoints by f . In particular, the lengths of the images of the edges of P are bounded from above by ρ.

Suppose now that we have defined by induction an extension F : P k -→ X of f on the k-skeleton P k of P . We also suppose that the restrictions of ϕ to the k-faces

∆ k I of ∆ k+1 0 , noted ϕ I,k : ∆ k I ∆ k 0 -→ X or more generally ψ : ∆ k 0 ∆ k I -→ X, satisfy the extension condition (E k ) which follows: Let x ∈ ∆ k-1 0 = (x 0 , . . . , x k-1 ) ⊂ ∆ k 0 ,
Given z ∈ ∆ k-2 0 = (x 0 , . . . , x k-2 ) ⊂ ∆ k 0 , let x be a point lying in the ray x k-1 z contained in ∆ k-1 0
arising from x k-1 and joining z (see figure). This ray decomposes into three segments zz , z z and z x k-1 of equal lengths.

z z z x k-1 x k
For any given x running through zz , there exists a point w in zx k-1 which divides zx k-1 into the same ratio as x divides zz . The map ψ takes then the ray x k x to the path ψ(x k z) ∪ ψ(zw) ∪ ψ(wx) where ψ(x k z), ψ(zw) and ψ(wx) are given by induction as images of segments lying in some (k -1)faces of ∆ k-1 0 . Similarly, for any given x running through z z , there exists a point w in x k-1 x k which divides x k-1 x k into the same ratio as x divides z z . The map ψ takes the ray x k x to the path ψ(

x k z) ∪ ψ(zx k-1 ) ∪ ψ(x k-1 w) ∪ ψ(wx k-1 ) ∪ ψ(x k-1 x). For x = z , we get the path γ ∪ ψ(x k x k-1 ) ∪ ψ(x k-1 z ) where γ is a loop based at ψ(x k ), image by ψ of the triangle x k zx k-1 of sides x k z, zx k-1 and x k-1 x k . For x dividing z x k-1 into a ratio t, the ray x k x is sent by ψ to a path γ t ∪ ψ(x k x k-1 ) ∪ ψ(x k-1 x
) where γ t is a homotopy of loops based at ψ(x k ) with γ 0 = γ and γ 1 is the constant loop ψ(x k ). Note that the loops γ and γ t lying in the image of ∆ k 0 are continuously parametrized by z and will therefore be noted γ ∆ k I ,z and γ ∆ k I ,z,t , or more simply γ z and γ z,t .

Thus, the images of the rays x k x, where x runs along x k-1 z, give rise to a continuous family of loops γ z which contract into ψ(x k ) through a continuous family of homotopies of loops γ z,t based at ψ(x k ). Note that, reciprocally, such a family of homotopies γ z,t permits us to define the images of the rays x k x where x runs along x k-1 z using the restriction of ϕ to the (k -1)-faces of ∂∆ k 0 .

The maps ψ are thus defined by induction from the restriction of ϕ to the 1-skeleton of ∂∆ k+1 0 and some continuous families of homotopies γ z,t . Furthermore, we assume that the images by ψ of the rays x k x where x lies in ∆ k-1 0 are of length bounded from above by k := (2.3 k-1 -1)ρ. This concludes the statement of the extension condition (E k ). Now, we want to define an extension Φ of ϕ to ∆ k+1 0 as a cone over ∆ k 0 = (x 0 , . . . , x k ) based at x k+1 , which satisfies the extension condition (E k+1 ).

Let x ∈ ∆ k 0 . The ray of ∆ k+1 0 emanating from x k+1 and joining x is noted x k+1 x. By definition, the images by the extension Φ of the rays x k+1 x where x lies in ∆ k 0 are parametrized proportionally to arclength.

Let z be the base point in ∆ k-1 0 of the ray of ∆ k 0 arising from x k and passing through x where x ∈ ∆ k 0 , x = x k . The image by ϕ of the triangle x k+1 zx k of ∆ k+1 0 of sides x k+1 z, zx k and x k x k+1 is noted γ z .

We want to define the images by Φ of the rays arising from x k+1 in the same manner as the restrictions ψ of ϕ to the faces of ∂∆ k+1 0 are defined (see above). For that, we need to construct a continuous family of homotopies γ z,t as it arises in the extension condition (E k ).

This family must be defined for (z, t)

∈ ∆ k-1 0 ×[0, 1]
. It must also contract the loops γ z,0 = γ z based at ψ(x k+1 ) into the constant loops γ z,1 ≡ ψ(x k+1 ) through a homotopy of loops γ z,t based at ψ(x k+1 ).

Since Φ extends ϕ, the images by Φ of the rays x k+1 x where x ∈ ∂∆ k 0 are given by ϕ. Therefore, the family γ z,t with (z, t) ∈ ∂∆ k-1 0 × [0, 1] must agree with the homotopies γ ∆,z,t where ∆ is a k-face of ∂∆ k+1 0 containing x k and x k+1 .

The family of homotopies γ z,t is thus defined for (z, t)

∈ ∆ k-1 0 × ∂[0, 1] ∪ ∂∆ k-1 0 × [0, 1] =: Σ k-1 S k-1
. By construction, the lengths of the loops γ z,t for (z, t) ∈ Σ k-1 are bounded from above by 2 k + ρ < κ 0 . Therefore, the family γ z,t given by a map Σ

k-1 -→ Ω 2 k +ρ x k+1 X extends to Σ k-1 × [0, 1]/Σ k-1 × {1} ∆ k-1 0 × [0, 1]/∆ k-1 0 × {1} B k with L(γ z,t ) ≤ 2 k + ρ.
Thus, the family γ z,t extends to ∆ k-1 0 × [0, 1] and satisfies all the required conditions.

That completes the construction of the family of homotopies γ z,t and yields an extension Φ : ∆ k+1 0 -→ X of ϕ. By definition, L(Φ(x k+1 x)) ≤ 3 k + 2ρ = k+1 where x lies in ∆ k 0 . Therefore, the map F : P k -→ X extends to the (k + 1)-skeleton of P into a map, still noted F , which satisfies the extension condition (E k+1 ). We conclude then by induction.

The last statement of the proposition comes from the construction of F .

Remark. In order to extend ϕ : ∂∆ k+1 -→ X to ∆ k+1 , we only used the fact that the restrictions of ϕ to the k-faces containing the vertex x k+1 satisfy the extension condition (E k ) and that the length of the ϕ-images of the rays x k x lying in ∆ k 0 = (x 0 , . . . , x k ) are bounded from above by k . We will use this remark afterwards in the following manner.

Given a subcomplex Q of P , we number the vertices of P starting by those of Q and consider the induced order. The identification of the simplices of Q with the standard simplex yields a natural metric length structure on Q.

Let f : P 0 ∪ Q -→ X be a map and ε > 0. Assume that the restrictions of f to the simplices of Q are ε-Lipschitz and that the restriction of f to P 0 fulfills the hypothesis of Proposition 1.1. If ε is small enough then the construction of the proof of Proposition 1.1 applies with the previously defined order on P , from the remark above. The map f extends then to a continuous map F : P -→ X as in the proposition. 1.3. Volume and the shortest geodesic loop. Theorem A is established and the example of Riemannian homogeneous manifolds is presented in this section.

Proof of Theorem A. In order to apply Greene and Petersen's theorem (see [6, Theorem 1]) to derive a local volume estimate, we need to show that M satisfies a condition of local geometric contractibility (see [6, p. 274]). Fix x 0 ∈ M and 0 < r < α n sgl(M ). We want to show that the ball B(x 0 , r) is contractible in B(x 0 , 4.3 n r). Without loss of generality, we can assume that the sphere S(x 0 , r) is represented by a finite (n-1)-dimensional simplicial complex σ : Q -→ M . The identification of the simplices of Q with the standard simplex yields a natural metric length structure on Q. Since the map σ : Q -→ M can be approximated by continuous maps homotopic to σ whose restrictions to the simplices of Q are Lipschitz, we can assume that the same holds for σ. Every simplicial complex endowed with the coherent topology admits a subdivision finer than a given open covering U (see [15, p. 126]). Fix ε > 0, we apply this result to the open covering U ε of P where

U ε = {U ⊂ Q | U open, f |U is k-Lipschitz and Diam(U ∩ Q) ≤ ε k for some k > 0}.
Then we dilate the metric on Q so that every simplex of the new triangulation of Q is isometric to the standard simplex. The map σ is ε-Lipschitz with respect to this new metric on Q.

Let P be the cone over Q. We extend σ to P 0 ∪Q by sending the vertex of P \ Q to x 0 . From the remark following Proposition 1.1, the map σ extends to σ : P -→ B(x 0 , 4.3 n r). Therefore, the ball B(x 0 , r) is contractible in B(x 0 , 4.3 n r). Thus, the continuous function ρ : [0, sgl(M )[-→ R + is a local geometric contractibility function for M .

Using Theorem 1 of [START_REF] Greene | Little topology, big volume[END_REF], we can conclude that there exists

C n > 0 such that VolB(R) ≥ C n R n for every ball B(R) of radius R in M .
The following well-known statement refines Theorem A for Riemannian homogeneous manifolds.

Lemma 1.2. The geodesic loops of Riemannian homogeneous manifolds agree with the simple closed geodesics.

Proof. Given a geodesic loop γ based at x 0 ∈ M , the first variation formula shows that d dt L(γ t ) = g( γ(1), X(x 0 )) -g( γ(0), X(x 0 )) = 0 where γ t is the variation of γ induced by the Killing vector field X. Since the Killing vector fields generate the tangent space of homogeneous Riemannian manifolds, we have γ(1) = γ(0). Therefore, γ is closed.

As a corollary, there exists

C n > 0 such that Vol(M ) ≥ C n scg(M ) n
for every Riemannian homogeneous manifold. This inequality contrasts with the case of the multi-dimensional systoles. Indeed, M. Katz constructs, in appendix D of [START_REF] Gromov | Metric structures for Riemannian and non-Riemannian spaces[END_REF], a sequence of homogeneous metrics on S 3 × S 3 such that Vol(gn) sys 3 (gn) 2 → 0, where sys

3 (g) = inf{Vol g (S) | S is a 3-cycle such that [S] = 0 ∈ H 3 (S 3 × S 3 )}.

Volume and short geodesic loops

In this part, we first present some filling estimates. Then, we define a functional on the double loop space and study the structure of some of its critical points. Finally, the proof of Theorem B is presented.

2.1. Generalities. In this section, we recall the definition of the filling radius introduced by Gromov in [START_REF] Gromov | Filling Riemannian manifolds[END_REF] and state general filling estimates.

Let M be a complete Riemannian manifold and L ∞ (M ) be the space of all Borel functions on M with the sup-norm. Because of the possible noncompactness of M , we consider unbounded functions on M ; therefore, the distance between two of them may be infinite. However the natural embedding i : M → L ∞ (M ) defined by i(x)(.) = dist M (x, .) is an isometry between metric spaces. Considering M isometrically embedded in L ∞ (M ), we define U r (M ) as the r-tubular neighborhood of M in L ∞ (M ).

Definition. The filling radius of M , denoted FillRad(M ), is the infinimum of positive reals r such that there exists a locally finite (n + 1)dimensional chain c in U r (M ) whose boundary ∂c, contained in M , represents the fundamental class of M in H n (M ). That means the homomorphism H n (M ) -→ H n (U r (M )) induced by the inclusion map vanishes. Here, the homology is with noncompact supports and the coefficients are in Z if M is orientable and in Z 2 otherwise. where c n is a positive constant depending only on the dimension of M .

And the second one was proved by M. Katz in [START_REF] Katz | The filling radius of two-point homogeneous spaces[END_REF].

Theorem 2.2. Let M be a closed Riemannian manifold of diameter Diam(M ), then

FillRad(M ) ≤ 1 3 Diam(M ) 2.2.
Morse theory on the double loop space. In this section, we describe the double loop space and give a precise definition of L 0 (M ).

Let M be a complete Riemannian manifold. All the functional spaces will be endowed with the C 0 -topology. Let C be the space of piecewise smooth curves γ : [0, 1] -→ M . We recall that ΛM and ΩM designate respectively the free loop space and the pointed loop space. The spaces Γ

= {Φ = (γ 1 , γ 2 ) ∈ C × C | γ 1 (0) = γ 1 (1) and γ 2 (0) = γ 2 (1)} and Γ = {Φ = (γ 1 , γ 2 ) ∈ C × C | γ 1 (0) = γ 2 (1)
and γ 1 (1) = γ 2 (0)} identify naturally with ΛM × ΛM and ΛM . Let Γ be the union of Γ and Γ and Γ 0 ⊂ Γ be the set of point curves. The intersection of Γ and Γ , noted X, represents the double loop space.

Next proposition will not be used afterwards. We state it because describing the relative homotopy of Γ it sets up the basis of a variational calculus on the double loop space. Proposition 2.3. Let M be a simply connected closed manifold. Then, π 1 (Γ, Γ 0 ) is naturally isomorphic to π 2 (M ).

Proof. Let us consider the natural projection

p : Γ -→ M × M Φ -→ (γ 1 (0), γ 2 (0))
This projection gives rise to two Serre fibrations with fiber ΩM × ΩM :

p : Γ = ΛM × ΛM -→ M × M and p : Γ = ΛM -→ M × M.
Let ∆ M be the diagonal of M × M . We have X = p -1 (∆) = ΛM × ΛM ∩ ΛM and p : X = Γ ∩ Γ -→ ∆ is the restricted fibration. The space Γ may be written as ΛM × ΛM X ΛM . From Van Kampen's theorem, π 1 (Γ) is naturally isomorphic to π 1 (ΛM × ΛM ) * π 1 (X) π 1 (ΛM ). Since M is simply connected, the exactness of the homotopy sequences of the fibrations implies that the groups π 1 (ΛM × ΛM ) and π 1 (X) are isomorphic to π 1 (ΛM ) ⊕ π 1 (ΛM ). Furthermore, the inclusion X → ΛM × ΛM induces an isomorphism π 1 (X) -→ π 1 (ΛM × ΛM ). Therefore, π 1 (Γ) is naturally isomorphic to π 1 (ΛM ). Since M is simply connected, π 1 (Γ) and π 1 (ΛM ) are respectively isomorphic to π 1 (Γ, Γ 0 ) and π 2 (M ). Hence the result.

We introduce the functional L on Γ defined by L(Φ) = max{L(γ 1 ), L(γ 2 )} where Φ = (γ 1 , γ 2 ) ∈ Γ. The restriction of L to X induces a functional L X on the double loop space. Let L 0 (M ) be the infinimum of the positive local minima of L X .

The following lemma describes the structure of the local minima of L X . Lemma 2.4. Let Φ ∈ X be a positive local minimum of L X . Then, Φ is formed either of a nontrivial closed geodesic to which a loop is attached or of two distinct nontrivial geodesic loops γ 1 and γ 2 of the same length based at the same point. In the second case, the differences v 1 and v 2 of the tangent vectors at the endpoints of γ 1 and γ 2 point to different directions. That is g(v 1 , v 2 ) ≤ 0 where v i = γi (1) -γi (0) is different from 0.

Proof. Let γ 1 and γ 2 be the two loops forming Φ. Suppose that none of them is a closed geodesic of null index. In this case, the loops γ i are homotopic to a shorter closed curve. If L(γ 1 ) > L(γ 2 ) then there exists a L X -decreasing deformation of Φ (which possibly slightly increase the length of γ 2 ). This is impossible because Φ is a local minimum of L X . Switching the roles of γ 1 and γ 2 , we get L(γ 1 ) = L(γ 2 ). The loops γ 1 and γ 2 are geodesic, otherwise the Birkhoff curve-shortening flow with fixed endpoints would decrease the length of one of them. This would contradict the inequality L(γ 1 ) = L(γ 2 ) for the local minima of L X . If g(v 1 , v 2 ) > 0, the first variation formula shows that it would be possible to decrease simultanuously the lengths of γ 1 and γ 2 through a local flow. Hence another contradiction.

Remark. When M is compact, L 0 (M ) is positive. Furthermore, L 0 (M ) is reached either by the length of a nontrivial closed geodesic of null index or by half the length of a geodesic double loop formed of two distinct nontrivial geodesic loops of the same length based at the same point. Therefore, we have 0 (M ) ≤ 2 min{scg(M ), L 0 (M )}. . Suppose that δ < 1 24 0 (M ). We are going to construct a continuous map F : P -→ K which agrees with ψ • σ on ∂P . This will lead to a contradiction since ψ * [M ] = 0. Therefore, M does not bound in U δ (M ) with δ < 1 24 0 (M ), hence the inequality FillRad(M ) ≥ 1 24 0 (M ).

Subdividing P if necessary, we can assume that the diameter of the image by σ of the simplices of P is less than ε > 0 with ε < 1 12 0 (M ) -2δ. We first define a map f : P 0 ∪ Q -→ M with f |Q = σ by sending each vertex p i of P to a nearest point of σ(p i ) in M , as we wish. Since i : M → U δ (M ) is isometric, every pair p i , p j of adjacent vertices of P satisfies

d M (f (p i ), f (p j )) ≤ d L ∞ (f (p i ), σ(p i )) + d L ∞ (σ(p i ), σ(p j )) + d L ∞ (σ(p j ), f (p j )) ≤ 2δ + ε =: ρ < 1 24 0 (M )
We extend the map f to P 1 by taking the edges of P to minimizing segments joining the image of their endpoints. Let ∆ 2 be a 2-simplex of P . By construction, the length of the image of ∂∆ 2 is less than 3ρ < scg(M ). Therefore, the Birkhoff curve-shortening process defines a map from the disk D ∆ 2 to M , which sends ∂D = ∂∆ 2 onto its image. This construction yields an extension f :

P 2 ∪ Q -→ M of σ.
We want now to extend it to P 3 . This is possible if the image of the boundary ∂∆ 3 of every 3-simplex ∆ 3 of P bounds a ball in M . Let ∆ i be the faces of ∆ 3 . The boundaries ∂∆ i converge to point curves through homotopies (c i t ) 0≤t≤1 , given by the Birkhoff curve-shortening process, which define the images of ∆ i . The curves c i 0 agree with the images of ∂∆ i by f and the curves c i 1 reduce to point curves. Note also that L(c i t ) < scg(M ). Using homotopies to reparametrize the loops if necessary, we can assume that c 1 0 (s) and c 2 0 (s) agree and run through their common edge for 2 3 ≤ s ≤ 1. The family Φ t = (c 1 1-t , c 2 1-t ) ∈ Γ , 0 ≤ t ≤ 1, extends to Γ through the homotopy Φ t (s) = (c 1 0 ( s t ), c 2 0 ( s t )) ∈ Γ where 1 ≤ t ≤ 3 2 and 0 ≤ s ≤ 1. The same goes for c 3 t and c 4 t . This yields two homotopies of Γ. We use now homotopies to reparametrize the final curves so that they agree. Putting these two homotopies together gives rise to a one-parameter family of curves Φ t ∈ Γ, 0 ≤ t ≤ 3, starting and ending at point curves.

Let us show now that the class it induces in π 1 (Γ, Γ 0 ) is trivial. The family decomposes into three parts :

-(Φ t ) 0≤t≤1 and (Φ t ) 2≤t≤3 , which join in Γ point curves to double loops.

-(Φ t ) 1≤t≤2 , which join in Γ the two double loops Φ 1 and Φ 2 . Since L(Φ 1 ) and L(Φ 2 ) are < 3ρ < L 0 (M ), the two double loops Φ 1 and Φ 2 converge to point curves through L-non-increasing homotopies Φ 1 t and Φ 2 t of X. In particular, each of the loops which compose them converge to point curves through homotopies given by Φ 1 t and Φ 2 t . The length of the loops of these homotopies is less than 3ρ < scg(M ). Therefore, they are homotopic to the c i t 's. Indeed, two homotopies starting from the same loop and ending at point curves form a path in ΛM with endpoints in Λ 0 M . Such a path composed of loops of length < scg(M ) is homotopically trivial in π 1 (ΛM, Λ 0 M ) from the minimax principle of Fet and Lyusternik (see [START_REF] Klingenberg | Lectures on closed geodesics[END_REF]). Since we want to show that the family Φ t bounds in M , we can assume that (Φ t ) 0≤t≤1 and (Φ t ) 2≤t≤3 agree with Φ 1 t and Φ 2 t . Under this assumption, Φ t lies in Γ ΛM . It gives rise to a path in ΛM which starts and ends at point curves with L(Φ t ) < 6ρ < scg(M ). As previously, this implies that this path is homotopically trivial. Therefore, the image of the boundary of every 3-simplex of P bounds a ball in M . The map f extends to P 3 .

From obstruction theory, the map F : P 3 ∪Q f -→ M ψ -→ K extends to P . Hence the result.

Remark. Generically, i.e., for a metric lying in a dense open set of the space of metrics endowed with the C 2 topology, closed geodesics are nondegenerate and have different lengths. This property does not hold for geodesic loops. It is not possible to get rid of the double loops formed of two distinct nontrivial geodesic loops of the same length by slightly perturbing the metric.

We only sketch the arguments. Let us consider a metric which has two distinct geodesic loops of the same length based at the same point x 0 . We suppose that these geodesic loops are non-degenerate and that their index is null. This construction may be carried out by modifying the metric on a tubular neighborhood of a figure-eight curve. These two geodesic loops vary continuously with their base-point in the neighborhood of x 0 . Locally, the set of base-points for which the two geodesic loops have equal length forms a hypersurface. This hypersurface is stable and does not disappear by slightly perturbing the metric. Therefore, on open sets of metrics, points do exist through which two geodesic loops of the same length pass. Given a metric with three distinct (non-degenerate) geodesic loops of the same length based at the same point, the previous arguments suggest that this triple loop configuration might disappear after some perturbations of the metric.

  Theorem B. Let M be a 2-essential complete Riemannian manifold. Then there exists positive constants c n and c n depending only on the dimension n of M such that Vol(M ) ≥ c n 0 (M ) n Diam(M ) ≥ c n 0 (M ) where 0 (M ) is the length of the shortest double loop Φ 0 among the double loops formed of two nontrivial geodesic loops of the same length based at the same point.

1 0

 1 we designate by x k x the ray of ∆ k 0 emanating from x k and joining ∆ k-at x. The images by ψ of the rays x k x for x lying in ∆ k-1 0 are parametrized proportionally to arclength. They are given by induction on k from the restriction of ϕ to the 1-skeleton of ∂∆ k+1 0 in the following way:

  The two following results, which provide upper bounds on the filling radius of Riemannian manifolds, will permit us to derive Theorem B The first one was established by M. Gromov in[7, p. 41]. Theorem 2.1. Let M be a complete Riemannian manifold of dimension n, then FillRad(M ) ≤ c n Vol(M ) 1 n

2. 3 .FillRad

 3 Volume and double geodesic loops. Now, we derive Theorem B from Theorems 2.1 and 2.2, the previous remark and the following inequality. Proposition 2.5. Let M be a 2-essential complete Riemannian manifold, then By definition, the fundamental class [M ] of M vanishes in U δ (M ) ⊂ L ∞ (M ) where δ > FillRad(M ). Therefore, there exists a continuous map σ : P -→ U δ (M ) from an (n + 1)-dimensional simplicial complex P such that the restriction σ : ∂P -→ M represents [M ] in H n (M ). Since M is 2-essential, there exists a continuous map ψ : M -→ K into a K(π, 2) space, which induces a nontrivial morphism ψ * : H n (M ) -→ H n (K). The image of Q := ∂P by ψ • σ represents ψ * [M ] in H n (K)