Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps

A free homotopy decomposition of any continuous map from a compact Riemmanian manifold $\mathcal{M}$ to a compact Riemannian manifold $\mathcal{N}$ into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in $W^{s,p} (\mathcal{M}, \mathcal{N})$, with $sp = m = \dim \mathcal{M}$. In particular, when the fundamental group $\pi_1 (\mathcal{N})$ acts trivially on the homotopy group $\pi_m (\mathcal{N})$, the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form $$\iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}}\frac{1}{d_\mathcal{M} (x, y)^{2 m}} \, \mathrm{d} x \, \mathrm{d} y.$$ When $m \ge 2$, the estimates scale optimally as $\varepsilon \to 0$. Linear estimates on the Hurewicz homorphism and the induced cohomology homomorphism are also obtained.

where ω S m is the volume form on the sphere S m normalized so that´S m ω S m = 1 and f ♯ ω S m is the pullback of the form ω S m by the map f . In view of the classical inequality between the geometric and quadratic means, we have |f ♯ ω S m | = |det Df | ω S n ≤ |Df | m m m/2 ω S m everywhere on the sphere S m ; this implies then the integral estimate on the degree (see [4,Remark 0 where we have defined the Sobolev energy E 1,p for p ∈ [1, +∞) by This estimate (1.2) remains valid under the weaker assumption that the map f lies in the Sobolev space W 1,m (S m , S m ) of weakly differentiable maps whose weak derivative satisfies the integrability condition that´S m |Df | n < +∞; the degree of f is then understood in the sense of maps of vanishing mean oscillation (VMO) [11]. By the classical Hölder inequality, the estimate (1.2) implies that the degree of f can also be controlled by the L p norms of its derivative Df for p ∈ (m, +∞].
Although the integral formula (1.1) does not have a clear sense when the map f does not have some kind of derivatives, the natural counterpart of the integral estimate (1.2) still holds for fractional Sobolev maps: for every p ∈ (m, +∞), there exists a constant C m,p such that for every map f ∈ W m/p,p (S m , S m ), one has [4, theorem 0.6] (see also [6, [42]): they have proved that for every ε ∈ (0, 2(1 + 1 m+1 )), there exists a constant C ε,m such that for every map f ∈ C (S m , S m ), one has (1.5) |deg(f )| ≤ C ε,m( x,y)∈S m ×S m |f (y)−f (x)|>ε 1 |y − x| 2m dx dy .
In view of the definition (1.4), the gap potential estimate (1.5) implies the fractional Sobolev estimate (1.3). If m ≥ 2, the constant can be taken to satisfy C ε,m ≤ C m ε m [43].
The gap potentials on the right-hand side of (1.5) also appeared in estimates on lifting of maps into the circle [39, theorem 2] and were showed to characterize as ε → 0 Sobolev spaces [35,36,38,44] and provide a property stronger than VMO [10]. .
Compared to the corresponding estimate of the topological degree (1.2), a power 1 + 1 3 applied to the integral appears, related to the Whitehead formula for the Hopf invariant [57]. No fractional counterpart to (1.6) seems to be know (see open problem 5 below). Rivière's bound (1.6) extends straightforwardly to a its higher-dimensional counterpart which is a homotopy invariant for maps from the sphere S 2n−1 into S n [3, proposition 17.22], resulting in an estimate (1.7) |deg H f | ≤ C ˆS 2n−1 |Df | 2n−1 The Hopf invariant takes nontrivial values when n is even [3, proposition 17.22] but is not necessarily injective (when n ∈ {1, . . . , 20}, it is injective if and only if n ∈ {2, 6} [55]). In all cases, only finitely many homotopy classes share the same value of the Hopf invariant and thus any set of maps which is bounded in the Sobolev space W 1,2m−1 (S m , S 2m−1 ) is contained in finitely homotopy classes of maps. By a theorem of Jean-Pierre Serre [54], all the other classes of continuous maps between spheres of different dimension consist only of finitely many homotopy classes; thus in general a bounded set of maps in W 1,3 (S m , S n ) is contained in finitely many homotopy classes.
1.3. Estimates on free homotopy decompositions. The results outlined above for maps between spheres raise the question whether sets which are bounded Sobolev norms are contained in finitely many homotopy classes of maps.
When s ∈ (0, 1), p ∈ (1, +∞) and sp > m, the classical Morrey-Sobolev embedding (see for example [7, theorem 9.12; 58, lemma 6.4.3]) ensures that sets which are bounded in energy in W s,p (S m , N ) are also bounded in the space of Hölder-continuous functions C 0,s−m/p (S m , N ), and thus by the Ascoli compactness criterion and the local invariance of homotopy classes, they are contained in finitely homotopy classes.
A slighly more subtle case is W 1,1 (S 1 , N ): although there is no compact embedding in the set of continuous maps, each map is homotopic to a map whose Lipschitz constant is controlled; hence bounded sets are contained in finitely many homotopy classe.
In the general case of W s,p (S m , N ) with sp = m and p > 1 with an arbitrary target manifold N , such a control turns out to be impossible. In order to construct infinitely many non-homotopic maps whose Sobolev energies remain bounded, we rely on the following definition: Definition 1.1 (Free homotopy decomposition). A map f ∈ C (S m , N ) has a free homotopy decomposition into the maps f 1 , . . . , f k ∈ C (S m , S m ) whenever there exists a map g ∈ C (S m , N ) homotopic to f on S m and nontrivial geodesic balls B ρ 1 (a 1 ), . . . , B ρ ℓ (a k ) ⊂ S m such that g is constant on S m \ k i=1 B ρ i (a i ) and for every i ∈ {1, . . . , k}, its restriction g|B ρ i (a i ) is homotopic to some f i ∈ F on S m ≃B ρ i (a i )/∂B ρ i (a i ).
The map g is well defined on the quotientB ρ i (a i )/∂B ρ i (a i ) ≃ S m because it is constant on ∂B ρ i (a i ).
The free homotopy decomposition appears in the construction of harmonic and polyharmonic maps that are known in many instances to generate through free homotopy decomposition all the homotopy classes [18, theorem 14; 50, theorem 5.5].
The free homotopy decomposition is an invariant under homotopies of the maps, but is not in general a faithful invariant: for example if N = (S 1 × S 2m ∪ S m × S m+1 )/S 2m , then there are two maps into which infinitely many homotopy classes decompose freely (see proposition 2.4 below).
The next result shows that maps that have the same free homotopy decomposition satisfy up to homotopy the same fractional Sobolev bound. Theorem 1.2 (Bound on the Sobolev energy by free homotopy decomposition). Let m ∈ N * and N be a connected Riemannian manifold. If f ∈ C (S m , N ) has a free homotopy decomposition into f 1 , . . . , f k ∈ C (S m , N ), then for every s ∈ (0, 1] and p ∈ [m, +∞) In particular, theorem 1.2 implies that all the homotopy classes that decompose freely into the maps f 1 , . . . , f k satisfy the same energy bound; if there are infinitely many such homotopy classes then there are infinitely many nonhomotopic map satisfying the same energy bound.
The proof of theorem 1.2 is performed by gluing together the maps f 1 , . . . , f k with an arbitrarily small energetic cost of gluing, performed through conformal transformations by Mercator projections. Theorem 1.2 does not cover the case s = p = m = 1. This is consistent with our observation that a Sobolev energy bound gives a control on the homotopy classes.
By taking the phenomenon described in theorem 1.2 into account, it has been proved that for every λ > 0, there exists a finite set F and k ∈ N such that every map f ∈ (W s,p ∩ C )(S m , N ) satisfying E s,m/s (g) ≤ λ has a free homotopy decomposition into k maps of the set F for m = 1, s = The critical case sp = m for estimates can be seen as a limiting case between the classical continuous picture of homotopy classes in the supercritical sp > m and the combination of collapses and appearance of homotopy classes in the subcritical case sp < m [8,9,[23][24][25][26]56].
Our main result shows that these estimates are in fact consequences of a stronger gap potential estimate similar to (1.5). Theorem 1.3 (Free homotopy decompositions controlled by a gap potential). Let m ∈ N * and N be a compact Riemannian manifold. If ε > 0 is small enough, then there is a constant C > 0 such that for every λ > 0, there exists a finite set In fact it can be observed that under the assumptions of theorem 1.3 any measurable map that satisfies the integrability condition with ε small enough has a small mean oscillation on small scales [10, proposition 1; 41] and therefore can be associated naturally and uniquely to a homotopy class of continuous maps from S m to N (see [11, (8), remark 7 and lemma A.5]).
The appearance of free homotopy decompositions in which the way of gluing the k maps together is arbitrary and uncontrolled can be thought of as a topological bubbling phenomenon, which is a topological version of the geometric bubbling phenomenon in conformally invariant geometric problems [13,45,50]. In many cases however, theorem 1.3 implies that maps satisfying a bound on the gap potential can only belong to finitely many homotopy classes. Theorem 1.4 (Finitely many homotopy classes under a gap potential bound). Let m ∈ N * and N be a compact Riemannian manifold. If m = 1 and every conjugacy class of π 1 (N ) is finite or m ≥ 2 and every orbit of the action of π 1 (N ) on π m (N ) is finite, and if ε > 0 is small enough, then for every λ > 0, there exists a finite set The assumptions of theorem 1.4 are satisfied in particular when π 1 (N ) is finite, if m = 1 and π 1 (N ) is abelian or if m ≥ 2 and the action of π 1 (N ) on π m (N ) is trivial.
In particular, under the assumptions of theorem 1.4, the homotopy group π m (N ) endowed with the norm naturally induced by a Sobolev energy satisfier a sufficient condition for compactness of the currents with coefficients on an abelian group [17, assumption (H), lemma 7.4 and corollary 7.5] (when m = 1, this only makes sense when the group π 1 (N ) is abelian).
When m ≥ 2, in analogy with the optimal scaling ε m when ε → 0 of estimates [43], we obtain a similar optimal scaling in ε (see theorem 5.8 below), with a different strategy of proof than [43].
The proof of theorem 1.3 is performed in a geometric setting where the sphere S m is considered as the boundary at infinity of the hyperbolic space H m+1 and the manifold N is embedded isometrically into a Euclidean space R ν . The extension of the map f by averaging at each point x ∈ H m+1 over the sphere at infinity -which is also in fact the hyperharmonic extension -provides a Lipschitz-continuous extension F : H m+1 → R ν . The set on which the values of the map F cannot be retracted to N is contained in a number of balls whose diameter and number is controlled allowing to construct the families of maps by a classical Ascoli compactness argument for continuous maps.
In view of theorem 1.2, theorem 1.3 describes sharply the homotopy classes that can be encountered under a boundedness assumption on the double integral. However, our proof exhibits a set of maps F λ by a compactness argument and gives thus double exponential bound of the form exp(C sinh(C ′ λ)) on the cardinal of F λ . This brings the question whether a better explicit control like the linear estimate (1.5).
When the homotopy classes can be controlled by the homology, that is, when the Hurewicz homomorphism from π m (N ) to the rational homology group H m (N ) has a finite kernel, we recover a linear control on the number of homotopy classes that satisfy a given bound (see theorem 6.1 below).
When the domain S m is replaced by a general m-dimensional manifold M , theorem 1.3 has a natural generalization, in which the corresponding homotopy classes are generated by a finite set of homotopy classes of C (M, N ) glued together with a finite number of maps taken in finitely many homotopy classes of C (S m , N ) (see section 7 below). As before, there can be in general infinitely many homotopy classes generated in this way by finitely many homotopy classes. The strategy of the proof is similar.
As perspectives of the present work, several open problems are presented in the last section of the present work (see section 8).
2. Free homotopy decomposition 2.1. Free homotopy decomposition and homotopy groups. The notion of free homotopy decomposition of definition 1.1 plays an important role in the present work. We describe here free homotopy decomposition in terms of homotopy groups.
We define f ∈ C (S m , N ) and γ ∈ π m (N ) to be homotopic whenever any representative of the relative homotopy class γ is homotopic to the map f . Since we have not fixed a base point in the homotopy between the representative in γ ∈ π m (N ) and the map f , a given map f ∈ C (S m , N ) can be homotopic to several distinct elements of π m (N ).
When m = 1, the elements of the fundamental group π 1 (N ) homotopic to a free homotopy class of maps from the circle S 1 to N form a conjugacy class of the fundamental group π 1 (N ) (see for example [27, exercise 1.1.6 and proposition 4A.2]).

Proposition 2.3.
Assume that m = 1 and every conjugacy class of π 1 (N ) is abelian, or that m ≥ 2 and every orbit of the action of π 1 (N ) on π m (N ) is finite. If k ∈ N and f 1 , . . . , f k ∈ C (S m , N ), then there exists a finite set G ⊂ C (S m , N ) such that if f ∈ C (S m , N ) has a free homotopy decomposition into f 1 , . . . , f k , then f is homotopic to some g ∈ G .
When m ≥ 2, the proof is similar and follows from the application of proposition 2.2.
2.2. Infinitely many homotopy classes sharing the same free homotopy decomposition. We now show that for some manifolds infinitely many homotopy classes can be decomposed freely into a finite set of maps. This implies in particular that the left-hand side in theorem 1.2 goes through infinitely many homotopy classes. In the one-dimensional case m = 1, examples can be provided by tori with at least two holes. The next lemma shows that a g-hole torus -or equivalently, an orientable surface of genus g -has a fundamental group which is not less complex than a free group on g generators. Lemma 2.5 (Free group in the fundamental group of g-hole tori). If N is a g-hole torus, then there exists a surjective homomorphism τ : π 1 (N ) → α 1 , . . . , α g .
The next lemma will allow us to prove in algebraic terms that maps in C (S 1 , N ) lie in different homotopy groups. Lemma 2.6 (Nonconjugacy along a conjugation orbit in a free group). If k ∈ {2, 3, . . . } and if ℓ, j ∈ N, then there exists h ∈ α 1 , . . . , α g such that Proof. If k = ℓ the statement holds with h = 1.
Conversely, it can be observed that α 1 α −ℓ 2 α k−1 1 α ℓ 2 and α 1 α −j 2 α k−1 1 α j 2 are cyclically reduced words which can be conjugate in a free group if and only the words are cyclic permutation of each other [32, theorem 1.3]. The statement can also be proved directly. We assume by contradiction that ℓ > k ≥ 0 and that there exists h ∈ α 1 , . . . , α g such that the identity holds. Then both corresponding reduced words should have the same length. Since ℓ > j ≥ 0, this means that there should be (ℓ − j) + length(h) cancellations between inverses on the left-hand side, and thus at least one cancellation at the beginning and one cancellation at the end of the word on the left-hand side. Since ℓ = 0, the cancellation on the left implies that the first letter of h is α 1 and the cancellation on the right that the first letter of h is α 2 ; this is a contradiction.
For m ≥ 2, we rely on the following construction of manifolds: Lemma 2.7 (Manifold with nontrivial action by the fundamental group). For every m ≥ 2, there exists a (2m + 1)-dimensional compact Riemannian manifold N isometrically embedded into R 2m+2 such that π 1 (N ) ≃ Z, π m (N ) ≃ Z Z and π 1 (N ) acts on π m (N ) as the translation operator.
Proof. If X S 1 ∨ S m is the CW complex obtained by the bouquet construction applied between the circle S 1 and the sphere S m , then π 1 (X ) ≃ Z, π m (X ) ≃ Z Z and π 1 (X ) acts on π m (X ) as the translation operator (see for example [27, example 4.27]).
We embed the CW complex X in R 2m+2 and we consider a neighbourhood U of X in R 2m+2 that has a smooth boundary and such that X is a retraction of U and ∂U is a retraction of U \ X . We define N ∂U .
We then observe that any Lipschitz-continuous homotopy h : with isomorphisms between the actions of π 1 (N ) on π m (N ) and of π 1 (X ) on π m (X ).
The manifold N constructed in the proof of lemma 2.7 can be described as the result of gluing S 1 × S 2m to S m × S m+1 along a trivial sphere S 2m .
Remark 2.8. When m = 2, the construction of the proof of lemma 2.7 yields a 3dimensional compact Riemannian manifold N embedded into R 4 such that π 1 (N ) is a free group on two generators.
Proof of proposition 2.4 when m ≥ 2. Let N be the manifold given by lemma 2.7. We fix a map f ∈ C (S m , N ) that is not homotopic to a constant and we choose a 0 ∈ π m (N ) homotopic to f . For each k ∈ Z, let a k be the result of the action of k ∈ Z ≃ π 1 (N ) on a 0 ∈ π m (N ). By proposition 2.3, the homotopy classes that have a free homotopy decomposition into k copies of the map f correspond to sets of the form there are infinitely many such sets.
3. Upper bound on Sobolev energies by free homotopy decomposition Theorem 1.2 will be obtained by induction from the corresponding result with k = 2: Proposition 3.1 (Estimate of Sobolev energy by free homotopy decomposition into two maps). Let m ∈ N * , N be a connected Riemannian manifold s ∈ (0, 1] and p ∈ [m, +∞).
Computations will be facilitated by parametrizing the sphere S m through its Mercator projection on the cylinder S m−1 × R. When m = 2, this corresponds to the projection used by Mercator on the cylinder to cartography the earth. The Mercator projection is a conformal transformation, and preserves thus the critical Sobolev energy.

Lemma 3.2 (Conformal derivative integrals under Mercator cylindrical projection). For every m ∈ N * and for every
(3.1) since |z| = 1. It thus follows that the mapping Υ is conformal and the identity holds.
The fractional counterpart of lemma 3.2 is an identity between the fractional integral on the sphere and a fractional integral with exponenially decaying potential in the longitudinal direction of the cylinder.

Lemma 3.3 (Conformal fractional integrals under Mercator cylindrical projection).
For every m ∈ N * , for every p ∈ (0, +∞) and for every f : S m → N , Proof. We define the Mercator projection Υ : S m−1 × R → S m as in the statement of lemma 3.2 and we observe that (3.1) holds and thus for every The identity follows then by a change of variable x = Υ(z, s) and y = Υ(w, t).
The proof of proposition 3.1 also relies on a construction of maps that are constant on some set.
η(λ ln |u|)u, and we observe that for every We define now for each y ∈ N , where exp b is the Riemmanian exponential map on N at b and inj N (b) is the injectivity radius of the Riemannian manifold N at the point b. We obtain the conclusion by taking λ > 0 small enough.
Proof of proposition 3.1. We choose a coordinate system so that a = (0, . . . , 0, 1) ∈ S m ⊂ R m+1 . By lemma 3.4, for every ε > 0, there exists maps Θ ± : N → N that are constant in a neighborhood of the point f ± (∓a). It follows then that g Up to a homotopy, we can consider that the map f is constant in a neighborhood of the equator we observe that Ψ is a homeomorphism and that the maps Ψ and Φ are homotopic.

It follows then that
We now consider the mapsg ± : We observe that there exists We define now for every λ ∈ (0, +∞) the map g λ : S m → N for each (y, t) ∈ S m ⊂ R m ×R, By construction, the map g λ is homotopic to h • Ψ −1 on S m , which in turn is homotopic to the map f on S m . It remains to estimate its Sobolev energy E s,p (g λ ).
If s = 1, we have by lemma 3.2, The conclusion then follows by letting λ → 0 and ε → 0.
If 0 < s < 1, we have by lemma 3.3, 2 ) 2 + |w − z| 2 m dt dw ds dz . and for every λ > 0, The same estimate still holds when m = 1. We have thuŝ We have then, under the changes of variables σ = t − s and τ = t + λ, and we reach thus the conclusion, by taking λ > 0 and ε > 0 arbitrarily small.

4.
Estimates of free homotopy decomposition on the sphere 4.1. Extension. In order to prove theorem 1.3, we first extend the map f on the sphere S m to a map F on the ball B m+1 taking its value into the ambient space, by relying on the next proposition which provides a suitably controlled extension. When we endow the ball B m+1 with the Poincaré metric of the hyperbolic space H m+1 , that is, if we consider the metric defined as quadratic form for z ∈ B m+1 and v ∈ R m+1 by we obtain uniform estimates on the measure of the set on which the function F is far frow the set of values on the boundary f (S m ).

Proposition 4.1 (Extension to the hyperbolic space).
Let m ∈ N * . There exists a constant C > 0 such that for every ν ∈ N * and every function In this statement, the oscillation of the function f is defined as In Euclidean terms, the estimates of proposition 4.1 read in view of the definition of the Poincaré metric (4.1) as follows: for every z ∈ B m+1 , and for every δ > ε, When the function f is bounded, the latter inequality (4.2) is a direct consequence of the work of Jean Bourgain, Haïm Brezis and Nguyên Hoài-Minh [5, lemma 2.1]. When f ∈ W s,p (S m , R ν ) with s ∈ (0, 1) and sp = m, the assertion (iii) in proposition 4.1 with ε = δ 2 implies that This inequality (4.3) can be obtained when s + 1  . Since in the sequel we will work with the Poincaré ball model of the hyperbolic space, the proof uses the hyperharmonic extension as in [14,48]; this construction corresponds to the harmonic extension in the two-dimensional case m+1 = 2 [30, §2] and to the biharmonic extension when m + 1 = 4 [47].
Proof of proposition 4.1. We define the function F : B m+1 → R ν to be the hyperharmonic extension of the function f , defined for each z ∈ B m+1 by [1, §V] The hyperharmonic extension is equivariant under the action of the conformal transformations of the ball and of the sphere, both corresponding to the group of (m + 1)dimensional Möbius transformation preserving the unit ball: if T : B m+1 → B m+1 is a conformal transformation, then F • T is the hyperharmonic extension of f • T .
The assertion (i) holds since by conformal invariance for every z ∈ B m+1 , In order to prove the assertion (ii), we first note that the Möbius transformations preserving the ball are exactly the isometries of the hyperbolic space in the Poincaré disk model [1,§II], and thus, in view of the equivariance of the hyperharmonic extension, it is sufficient to consider the case z = 0. We have then for every x ∈ S m since´S m y dy = 0, and thus For the assertion (iii), we observe that for every x ∈ S m and r ∈ [0, 1), we have by We deduce therefrom that for every ε > 0 We next observe that, by the triangle inequality, for every x, y ∈ S m and r ∈ [0, 1), we have |y − x| ≤ |y − rx| + |rx − x| = |y − rx| + |y| − |rx| ≤ 2|y − rx| .
Therefore, we have We define the set For each x ∈ S m , we set (with the convention that ρ δ (x) 0 if rx ∈ A δ for every r ∈ [0, 1)) and, since m ≥ 1, we compute that Since ρ δ (x)x ∈ A δ , we deduce from (4.5) that and we conclude that Remark 4.2. The proof of proposition 4.1 controls in fact the hyperbolic volume of the star-shaped hull A ⋆,0 δ of the set A δ with respect to 0 defined as the smallest subset which is starshaped with respect to 0 and contains A δ . By invariance under the Möbius group that models the isometries of the hyperbolic space in the Poincaré ball model, the volume of the starshaped hull A ⋆,x δ of the set A δ with respect to any x ∈ H m+1 is also controlled.
We assume now that ℓ > 1 and that the conclusion holds for ℓ − 1.

4.3.
Proof of the theorem. Theorem 1.3 will follow from the following slightly stronger statement, involving a truncated fractional integral.
Theorem 4.6 (Estimate on free homotopy decomposition by a truncated fractional energy). If ε > 0 is small enough, there is a constant C > 0 such that for every λ > 0, there exists a finite set fas a free homotopy decomposition into f 1 , . . . , f k ∈ F λ with k ≤ Cλ.
Proof of theorem 4.6. We apply proposition 4.1 to f . We define for each δ > 0 the sets N δ {y ∈ R ν | dist(y, N ) < δ} and Since N is a smooth submanifold of R ν , there exists δ * > 0 and a Lipschitz-continuous retraction Π : N δ * → N , that is, one has for every y ∈ N δ * , Π(y) ∈ N and for every y ∈ N , Π(y) = y. By the estimate (ii) in proposition 4.1, we observe that if a ∈ A δ * , then for every If we take ρ δ * 2m diam(N ) , we have We consider now a maximal set of points A ⊆ A δ * such that if a, b ∈ A and a = b then d H m+1 (a, b) ≥ 2ρ. By construction, we have On the other hand the balls (B H m+1 ρ (a)) a∈A are disjoint and thus by (4.7), we have By the invariance of the volume of balls in the hyperbolic space, we deduce that We conclude by defining the set (a i ) ≃ S m to some map g i : S m → N whose Lipschitz constant is controlled by C 2 sinh(C 3 λ). By the Ascoli compactness theorem, there exists a finite set of maps F λ ⊂ C (S m , S m ) such that any map from S m to N whose Lipschitz constant does not exceed C 2 sinh(C 3 λ) is homotopic to some map in F λ . In particular, for every i ∈ {1, . . . , k}, there exists a map f i ∈ F λ which is homotopic to g i on S m and thus tõ We consider now a ball B H m+1 2ρ (a * ) ⊂Ũ and a mapF ∈ C (U, N ) such thatF =F in ) ⊂ H m+1 , and we observe that ∂Ȗ ∩ B m+1 is homeorphic to S m+1 and thatF | ∂Ȗ ∩B m+1 has a free homotopy decomposition into f 1 , . . . , f k , and hence by homotopy invariance, f also has a free homotopy decomposition into f 1 , . . . , f k .
We deduce now theorem 1.3 from theorem 4.6.
Proof of theorem 1.3. We note that, since the map f : S m → N is bounded, we have |y − x| 2m dy dx , and the conclusion then follows from theorem 4.6.
We will observe in the sequel that when m ≥ 2, an estimate of the form (4.8) holds without any boundedness assumption on the map f and with a constant of the order of ε (see proposition 5.5 below).
Proof of theorem 1.4. This follows from theorem 1.3 and proposition 2.3.

Scaling and comparison of truncated fractional energies
In this section we improve the estimate of theorem 1.3 into an estimate that scales optimally with respect to ε as ε → 0. Our results are the counterpart of Nguyên Hoài-Minh's estimates on the topological degree [43], but are obtained with a different strategy.

Scaling of truncated fractional energies.
In order to improve the estimate of theorem 1.3, we first study how truncated fractional integral scale with varying values of the truncation in the next proposition.
When either 1 ≤ p < m, or p ≤ 1 and m > 2, then m − 1 > (p − 1) + and the estimate of proposition 5.1 improves the straightforward monotonicity estimate: if δ ≤ ε, then If the set Ω ⊂ R m is bounded and if the map f : Ω → R m is the identity, one has as ε → 0, by the change of variables r = ε(t + 1). This computation means that the scaling estimate of proposition 5.1 is optimal when 1 ≤ p < m. We do not know whether the estimate can be improved when 0 ≤ p < 1 (see open problem 3 below). The estimate will already appear to be strong enough to obtain some comparison between truncated fractional integrals of different exponents in proposition 5.5 below.
Proof of proposition 5.1. By the triangle inequality, we havë and thus by symmetry under exchange of x and ÿ We apply now the change of variable y = 2z − x and we obtain where for every x ∈ Ω, we have defined the set By combining the inequalities (5.2) and (5.3), we deduce that for every ε > 0, By iterating the estimate (5.4), we deduce that for every nonnegative integer ℓ ∈ N, If δ ∈ (0, ε), we let ℓ ∈ N be defined by the condition 2 −(ℓ+1) ε ≤ δ < 2 −ℓ ε and we conclude thaẗ In order to improve the statement of theorem 1.3, we will derive the counterpart of proposition 5.1 for spheres.
The proof of proposition 5.2 will rely on its counterpart on a convex set of the Euclidean space proposition 5.1 and on a suitable covering of the sphere by spherical caps.
Proof of proposition 5.2. Let a 0 , . . . , a m+1 ∈ S m ⊂ R m+1 be the vertices of an equilateral simplex and define for each i ∈ {0, 1, . . . , m + 1}, the spherical cap Since for every i ∈ {0, . . . , m + 1}, the spherical cap A i is diffeomorphic to a ball of R m , we have in view of proposition 5.1, We conclude by combining (5.7) and (5.8) with C = (m + 2) C 1 .

Theorem 5.4 (Free homotopy decompositions controlled by a scaled truncated Sobolev energy).
Let m ∈ N * and N be a compact Riemannian manifold. There are constants ε 0 > 0 and C > 0 such that for every λ > 0, there exists a finite set has a free homotopy decomposition into f 1 , . . . , f k ∈ F λ with k ≤ Cλ.
Proof. This follows from theorem 4.6 and proposition 5.2.

Comparison between fractional truncated energies.
In passing form theorem 4.6 to theorem 1.3 we relied on (4.8), which is not optimal when ε is small and f is the identity mapping (see (5.1)). In this section, we derive estimates that compare different gap integrals with optimal scaling on a convex subset Ω of the Euclidean space R m .

Proposition 5.5 (Comparison between truncated fractional energies).
For every m ≥ 2, p ∈ [0, m), q ∈ [0, +∞) and η ∈ (0, 1), there exists a constant C > 0 such that for every convex set Ω ⊂ R m , for every map f : Ω → N and for every ε > 0, one has In view of the asymptotics (5.1) on the integrals when f is the identity, the scaling of the estimate in proposition 5.5 is optimal and the estimate of proposition 5.5 fails when p ≥ m and p > q.
When p < q, the estimate follows from the elementary inequality: for t ≥ ε, the interest of the estimate lies essentially thus in the case q < p < m. The proof of theorem 5.8, will be relying only on the case q = 0 and p = 1.
The proof of proposition 5.5 relies on proposition 5.1 and the next lemma 5.6.
Proof of proposition 5.5. We first observe that by lemma 5.6 applied at each x, y ∈ Ω with t = d N (f (y), f (x)) and s = ε, we have Since the set Ω ⊆ R m is convex, by proposition 5.1, we have for every r ∈ (0, +∞), By combining (5.11) and (5.12), we deduce thaẗ and the estimate is satisfied.
In order to cover the case m − (1 − q) + ≤ p < m, we observe that since m ≥ 2 and q ≥ 0, the estimate holds for every p ∈ [0, 1). By iterating a second time the estimate, we obtain the estimate for each p ∈ [0, m).
The proof shows that when m = 1, the estimate of proposition 5.5 holds if p < min (1, q), in which case the estimate is in fact elementary.
The estimate of proposition 5.5 also holds when the domain is a sphere S m .
Proposition 5.7 (Comparison of truncated fractional energies on the sphere). For every m ≥ 2, p ∈ [0, m), q ∈ [0, +∞) and η ∈ (0, 1), there exists a constant C > 0 such that for every map f : S m → N and for every ε > 0, one has Proof. The proof follows the lines of the proof of proposition 5.2, relying on the covering given by lemma 5.3 and the estimate on a convex set of proposition 5.5.
We conclude this section with a scaled version of theorem 1.3.

Theorem 5.8 (Free homotopy decompositions controlled by a scaled gap potential).
Let m ∈ N \ {0, 1} and N be a compact Riemannian manifold. There are constants ε 0 > 0 and C > 0, such that for every λ > 0, there exists a finite set has a free homotopy decomposition into f 1 , . . . , f k ∈ F λ with k ≤ Cλ.
We do not know whether theorem 5.8 holds when m = 2 (see open problem 1 below).
Proof of theorem 5.8. Since m ≥ 2, by proposition 5.7, we have for every f : S m → N and every ε > 0, the conclusion then follows from theorem 5.4.

Estimates of the Hurewicz homomorphism on the sphere
The Hurewicz homomorphism is a homotopy invariant of maps that describes how a mapping from S m to N acts on the cohomology of N . For a smooth map f ∈ C 1 (S m , N ) and for a closed differential form ω ∈ C 1 ( m N ), that is, a form such that dω = 0, we define where f ♯ ω ∈ C( m S m ) denotes the pull-back by f of ω. We note that Hur acts trivially on exact forms: if ω = dη with η ∈ C 2 ( m−1 N ), then by the Stokes-Cartan theorem In particular, Hur induces a map from the homotopy group π m (N ) to the homology group H m (N , R). The Hurewicz homomorphism generalizes the degree of maps into the sphere: if N = S n , we have Hur(f ), ω S n = deg(f ); it extends more generally the degree of maps when dim N = m [19; 22, §8].
Since the Hurewicz homomorphism is invariant under homotopies, it is well-defined for maps of vanishing mean oscillation. Moreover, by standard approximation, the formula (6.1) is still valid whenever f ∈ W 1,m (S m , N ) (see [11, (19)]).
The estimate (1.2) can be generalized immediately to the Hurewicz homomorphism: Indeed, this follows from the definition of Hur in (6.1) and the fact that |f ♯ (ω)| ≤ |ω| |Df | m almost everywhere on S m . When N = S m , then (6.2) is equivalent to the degree estimate (1.2).

Theorem 6.1 (Estimate of Hurewicz homomorphism by a truncated fractional energy).
Let m ∈ N * and N be a compact Riemannian manifold. If ε > 0 is small enough, then there exists a constant C > 0 such that if f ∈ C (S m , N ), then When m ≥ 2 (6.2) can be deduced from theorem 6.1 since the right-hand side in theorem 6.1 can always be controlled by the right-hand side in (6.2) [35, lemma 2]. When m = 1, there is no such estimate [35, proposition 3], but it might be possible to refine existing Γ-convergence results [40, theorem 2] in order to deduce (6.2) from theorem 6.1.
Proof of theorem 6.1. Since N is a compact manifold embedded into R ν , there exists an open set U ⊂ R ν such that N ⊂ U and a smooth retraction Π ∈ C ∞ (U, N ). We also consider a smooth map η ∈ C ∞ c (U, R) such that η(y) = 1 if y ∈ N . Given f ∈ C (S m , N ), we let F ∈ C ∞ (B m+1 , R ν ) be given by proposition 4.1 and we compute by the Stokes-Cartan formula since d(Π ♯ ω) = Π # (dω) = 0. Hence we have, by the estimates given by proposition 4.1 We also have an estimate of the Hurewicz homomorphism with optimal scaling when m ≥ 2. Theorem 6.2 (Estimate of the Hurewicz homomorphism by a scaled gap potential). If m ≥ 2 and N is a compact Riemmanian manifold, then there exists constants C > 0 and ε 0 > 0 such that if ε ∈ (0, ε 0 ) and f ∈ C (S m , N ), then Proof. This follows from theorem 6.1 in view of proposition 5.2 and proposition 5.7.
When N = S m we recover the estimate on the degree of Nguyên Hoài-Minh [43]; the latter estimate was obtained through the John-Nirenberg estimate and seems different from our direct approach. When m = 1, the question whether theorem 6.2 holds is an open problem (open problem 1). 7. Homotopy estimates on a compact manifold 7.1. Free homotopy decompositions upon a mapping. We consider the problem of controlling the homotopy classes of maps from a general compact manifold M to another compact manifold N . The notion of free homotopy decomposition (definition 1.1) generalizes into the free homotopy decomposition upon a mapping. Since the circle S 1 is, up to a conformal transformation, the only connected compact one-dimensional Riemannian manifold, we assume throughout this section that dim(M) = m ≥ 2.  N ) upon the map f 0 ∈ C (M, N ) whenever there exist maps g, g 0 ∈ C (M, N ) and nondegenerate topologically trivial balls B ρ 0 (a 0 ), . . . , B ρ ℓ (a k ), such that g is homotopic to f , g 0 is homotopic to f 0 , g = g 0 on M \ B ρ 0 (a 0 ), g 0 is constant onB ρ 0 (a 0 ), and for every i ∈ {1, . . . , k}, Since the definition of free homotopy decomposition upon a mapping (definition 7.1) is invariant under homotopies, the condition that the map f 0 is constant on some nondegenerate topologically trivial ball can always be satisfied. Free decompositions upon a given mapping on a manifold are thus not more complex than a collection of homotopy classes of maps on a sphere relative to some point. The free homotopy decompositions into given maps upon a given map can be precisely identified and enumerated by obstruction cohomology classes with local groups [2, §4.2; 28, Chapter VI].
Proof of proposition 7.3. If the map f is homotopic to h, it follows directly from the definition of free decompositions definition 1.1 and definition 7.1 that f has a free homotopy decomposition into f 1 , . . . , f k upon f 0 .
Conversely, let us assume that f ∈ C (M, N ) has a free homotopy decomposition into f 1 , . . . , f k upon f 0 , and let g, g 0 ∈ C (M, N ) and the balls B ρ 0 (a 0 ), . . . , B ρ k (a k ) be given by definition 7.1. Without loss of generality, we can assume that B ρ 0 (a 0 ) = B ρ/2 (a).  N ) such that every map f ∈ C (S m , N ) that has a free homotopy decomposition into f 1 , . . . , f k upon f 0 is homotopic to some g ∈ G .
Since by assumption for every i ∈ {1, . . . , k} the set {β i · γ i | β i ∈ π 1 (N , b)} is finite, the set Γ is finite and we can constructG ⊂ C (B ρ (a), N ) as a finite set of mappings taking the constant value b on ∂B ρ (a) and such that under the identification S m ≃ B ρ (a)/∂B ρ (a), every element of Γ is homotopic to some map inG . We define now  N ) and has a free homotopy decomposition into f 1 , . . . , f k ∈ F λ upon f 0 ∈ F λ 0 , with k ≤ Cλ. In view of remark 7.2, theorem 1.3 corresponds to the particular case M = S m in theorem 7.5.
In order to follow in the proof of theorem 7.5 the same strategy as in the proof of theorem 1.3, we construct a Riemmanian manifold that is the counterpart of the hyperbolic space H m+1 for S m . We define the manifold M ⋆ M × (0, +∞) and we endow it with a metric g M ⋆ defined as a quadratic form for each point (x, t) ∈ M ⋆ and each tangent vector where g is the metric of the original manifold M. When M = R m , the manifold M ⋆ is the Poincaré half-space model of the hyperbolic space. The formula (7.1) shows that the manifold M ⋆ is conformally equivalent to the Riemannian cartesian product M × (0, +∞).
Remark 7.6. The manifold M ⋆ is in fact a warped product: if M ⋄ M × R is endowed with the metric g M ⋄ defined as a quadratic form for each ( We verify immediately that (i) holds by construction. The second assertion (ii) follows from the observations that the functionΦ is bounded, thatΦ(x, t, y) = 0 if d M (x, y) ≥ t and that for some constant C 1 > 0, for every x, t ∈ M ⋆ = M × (0, +∞),ˆMΦ (x, t, y) dy ≥ C 1 t m .
For the last assertion (iii) we observe that the map and that if (x j ) j∈N is any sequence in M and if (t j ) jN is a sequence in (0, +∞) converging to 0, then hence (iii) follows from the classical extreme value theorem for continuous functions.
For (ii), we note that by lemma 7.8 (i) we have for every (z, s) ∈ M ⋆ = M × (0, +∞) and every x ∈ M, and thus by differentiating with respect to (z, s) at the point (x, t), we obtain and thus we deduce from lemma 7.8 (iii) that For the last part (iii), we first observe that for each (x, t) ∈ M ⋆ ≃ M × (0, +∞), we have Hence we infer from lemma 7.8 (ii), dy .
We define now the set and, for each x ∈ M, the quantity and we compute and thus by (7.3), we conclude that dy dx .
In order to prove lemma 7.9 we need to have good estimates on the distances between points. It turns out that this distance can be computed exactly in terms of the distance on M.
(This estimate can also be proved directly by a crude lower bound on the metric on M ⋆ .) Another consequence of lemma 7.10 is the completeness of the manifold M ⋆ .
Lemma 7.11. If the manifold M is complete, then the manifold M ⋆ is complete.
It follows then from (7.5) and (7.6) that The free homotopy decomposition will be made through Lipschitz-continuous maps on spheres in M ⋆ . The next lemma ensures that the shape of these small spheres is controlled and will serve as a substitute to lemma 4.5. Proof. We observe by lemma 7.10, that for every (x, t), (y, s) ∈ M ⋆ = M × (0, +∞), It follows then that The bounds follow then from the compactness of M and the homogeneity of the metric on M ⋆ . Theorem 7.5 will follow from the following statement.
By the estimate given on F by proposition 7.7 (ii), there exists ρ > 0, independent of F , such that if x ∈ A δ * , then B ρ (x) ⊂ A δ * /2 . We consider a maximal set A ⊂ A δ * such that if for every a, b ∈ A such that a = b, one has d M ⋆ (a, b) ≥ 2ρ. This implies immediately that Since the balls B M ⋆ ρ (a) a∈A are disjoint, we have by lemma 7.12 and proposition 7.7 (iii) We have then Thanks to lemma 7.13, there exists T ∈ (0, e −2ρ ) such that if σ ≤ C 2 2ρλ and (x, t) ∈ M × (0, T ], then the exponential map, its inverse and their derivatives are controlled on B σ (x, t). By lemma 7.9, there exists balls (B M ⋆ ρ i (a i )) 1≤i≤ℓ and T ′ ∈ (0, T ) such that in view of (7.7) with the estimate This implies in particular, since T ′ ≤ T and ρ i ≥ ρ that ln T T ′ ≤ 2ρ C 2 λ ℓ ≤ 2 C 2 λ and ρ i ≤ 2ρ C 2 λ . (7.8) Since Π • F is Lipschitz continuous on M ⋆ \A δ , it follows then that the map f has a free homotopy decomposition into Π • F | ∂ Bρ 1 (a 1 ) , . . . , Π • F | ∂ Bρ ℓ (a ℓ ) upon Π • F | M×{T ′ } . Since by lemma 7.13 the exponenial map is controlled in M × (0, T ] by a bound depending on (7.8), the Lipschitz constants of the maps Π • F | ∂ Bρ 1 (a 1 ) , . . . , Π • F | ∂ Bρ ℓ (a ℓ ) upon Π • F | M×{T ′ } are bounded independently of f and the geometry of their domains are controlled by quantities depending only on λ, and thus by Ascoli's compactness theorem the maps Π • F | ∂ Bρ 1 (a 1 ) , . . . , Π • F | ∂ Bρ ℓ (a ℓ ) and Π • F | M×{T ′ } are respectively homotopic to maps taken in some finite sets F λ ⊂ C (S m , N ) and F λ 0 ⊂ C (M, N ).

Estimates of free homotopy decompositions by a scaled gap potential.
We obtain a version of theorem 7.5 that scales optimally with respect to ε, which generalizes theorem 5.4 to a general domain M. F λ ⊂ C (S m , N ) and F λ ℓ ∈ {0, . . . , m}, for every ω ∈ C ∞ ( ℓ N ) and every θ ∈ C ∞ ( m−ℓ M) such that dω = 0 and dθ = 0 the quantityˆM f ♯ ω ∧ θ .
By proposition 5.1, proposition 5.5 and lemma 7.16, the conclusion then follows.

Further problems
A first question that remains open at the end of the present work is whether estimates with optimal scaling can be proved when m = 1.
A variant of this problem would be to obtain estimates with optimal scaling on the Hurewicz homomorphism when m = 1.
The problem is already open for maps for the degree of maps from the circle S 1 to the circle S 1 , that is when N = S 1 (see [43]). It is striking that the present work and Nguyên Hoài-Minh followed quite different strategies of proof but encountered the same restriction that m > 1.
The solution of theorem 5.8 and open problem 2 could be connected to the following more technical question of scaling of truncated integrals.
Open problem 3. If p ∈ [0, 1) and m ∈ N, does there exists a constant C > 0 such that for every convex set Ω ⊂ R m and for every map f : Ω → R ν , if δ < ε, then As we have mentioned in the introduction, for every λ > 0, there exists a finite collection of maps F λ such that every f ∈ (C ∩ W 1,1 )(S 1 , N ) is homotopic to some map in F λ . The proof is done by showing that f is homotopic to a constant speed reparametrization and reduces thus the problem to Lipschitz-continuous maps to which the Ascoli theorem applies. This raises naturally the question about W m,1 (S m , N ).

Open problem 4.
Is it true that if for m ≥ 2, if λ > 0, then the maps f ∈ (C ∩ W m,1 )(S m , N ), such that´S m |D m f | ≤ λ belong to finitely many homotopy classes?
Finally, for maps from S 2n−1 to S n , the Hopf invariant can be computed through formulas that yield Rivière's estimate (1.6). The next logical step would be to obtain corresponding estimates in fractional Sobolev spaces.  Proof. The proof follow Tristan Rivière's proof [49,lemma III.1]. We construct for every k ∈ N, the map f k = ϕ k • ϕ where ψ : S 2n−1 → S n has a nontrivial Hopf degree and ϕ k : S n → S n has the property that |Dϕ k | ≤ k 1/n on S n and its Brouwer degree satisfies deg(ϕ k ) = k. It follows that |Df k | ≤ k 1/n and deg H (f k ) = k 2 . Moreover we haveẍ ,y∈S 2n−1 |f k (y)−f k (x)|≥ε dy dx A strategy that follows the proof of theorem 4.6 constructs for a given f ∈ C (S 2n−1 , S n ) a decomposition into g i : S 2n−1 → S n , with 1 ≤ i ≤ k, which have a Lipschitz constant controlled by C 1 sinh ρ i , with k i=1 ρ i ≤ C 2 λ. It follows then by Rivière's bound (1.7) and by convexity that which is quite far from the estimate proposed in open problem 6.