Topological singular set of vector-valued maps, II: $\Gamma$-convergence for Ginzburg-Landau type functionals

We prove a $\Gamma$-convergence result for a class of Ginzburg-Landau type functionals with $\mathcal{N}$-well potentials, where $\mathcal{N}$ is a closed and $(k-2)$-connected submanifold of $\mathbb{R}^m$, in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $\pi_{k-1}(\mathcal{N})$) which solves the Plateau problem in codimension $k$. The analysis relies crucially on the set of topological singularities, that is, the operator $\mathbf{S}$ we introduced in the companion paper arXiv:1712.10203.


Introduction
Let n ≥ 0, k ≥ 2, m ≥ 2 be integers, and let Ω ⊆ R n+k be a bounded, smooth domain. Let ε > 0 be a small parameter. For u ∈ W 1,k (Ω, R m ), we define the functional .
Here, f : R m → R is a non-negative, continuous potential, whose zero-set N := f −1 (0) is assumed to be a smooth, compact, (k − 2)-connected manifold without boundary. The aim of this paper is to understand the asymptotic behaviour of the functionals E ε in the limit as ε → 0, by a Γ-convergence approach. Our analysis builds upon the results obtained in a companion paper, [17]. Functionals of the form (1), which describe a kind of penalised k-harmonic map problem (see e.g. [19,42]), arise naturally in different contexts. A well-known example is the Ginzburg-Landau functional, which corresponds to the case k = m = 2 and f (u) := (|u| 2 − 1) 2 , so that the In case k = 2, we also assume that π 1 (N ) is Abelian.
(H 3 ) There exists a positive constant λ 0 such that f (y) ≥ λ 0 dist 2 (y, N ) for any y ∈ R m .
The assumption (H 2 ) is consistent with the setting of [17] and is satisfied, for instance, when k = 2 and N = S 1 (the Ginzburg-Landau case) or k = 2 and N = RP 2 (the Landau-de Gennes case).
The assumption (H 3 ) is both a non-degeneracy condition around the minimising set N and a growth condition.
Remark 1. We do not expect the assumption (H 3 ) to be sharp. In fact, (H 3 ) may probably be relaxed so as to include potentials that behave as dist s (·, N ), for some s > 2, in a neighbourhood of N .
We consider minimisers u ε,min of (1), subject to the boundary condition u = v on ∂Ω. On the boundary datum v, we assume Under the assumptions (H 1 )-(H 4 ), the rescaled energy densities µ ε,min : dx Ω |log ε| have uniformly bounded mass (see e.g. Remark 3.4 below; here, dx Ω denotes the Lebesgue measure restricted to Ω). Up to extraction of a subsequence, we may assume that µ ε,min converges weakly * (as measures in R n+k ) to a non-negative measure µ min , as ε → 0. We provide a variational characterisation of µ min in terms of flat chains with coefficients in (π k−1 (N ), | · | * ), where |·| * is a suitable norm, defined in Section 2 below. (For instance, in case k = 2 and N = S 1 , |d| * = π |d| for any d ∈ π 1 (S 1 ) Z.) We denote the mass of such a flat chain S by M(S), and the restriction of S to a set E by S E. We have Theorem A. Under the assumptions (H 1 )-(H 4 ), there exists a finite-mass n-chain S min , with coefficients in (π k−1 (N ), | · | * ) and support in Ω, such that µ min (E) = M(S min E) for any Borel set E ⊆ R n+k . Moreover, S min minimises the mass in its homology class -that is, for any (n + 1)-chain R with coefficients in (π k−1 (N ), | · | * ) and support in Ω, we have M(S min ) ≤ M(S min + ∂R).
In other words, in the limit as ε → 0 the energy of minimisers concentrates, to leading order, on the support of a flat chain S min that solves a homological Plateau problem. The homology class of S min is uniquely determined by the domain Ω and the boundary datum v (that is, S min belongs to the class C (Ω, v) defined by (2.6) below). We stress that Theorem A does not require any topological assumption, such as simply connectedness, on the domain Ω. However, the homology class of S min does depend on the topology of the domain and it can be described more easily if Ω has a simple topology (see the examples in Section 2 below). On the other hand, the topological assumption (H 2 ) on the manifold N is essential. An analogue of Theorem A in case k = 2 and the fundamental group of N is non-Abelian would already be of interest in terms of the applications; manifolds with non-Abelian fundamental group arise quite naturally, for instance, in materials science (e.g., as a model for biaxial liquid crystals). Unfortunately, the very statement of Theorem A does not make sense in the non-Abelian setting, because homology requires the coefficient group to be Abelian. Convergence results in case n = 0, k = 2 (see e.g. [15,44]) suggest that the energy concentration set may inherit some minimality properties, even if π 1 (N ) is non-Abelian. However, a general convergence result in the non-Abelian setting, along the lines of Theorem A, would presumably require some 'ad-hoc' tools from Geometric Measure Theory.
In some cases, the next-to-leading order term can be characterised, too. For instance, when n = 0, k = 2, the energy concentrates on a finite number of points and the next-to-leading order term in the energy expansion is a 'renormalised energy' which describes the interaction among the singular points. The renormalised energy was introduced, in the Ginzburg-Landau setting, by Bethuel, Brezis and Hélein [8] and it was extended very recently by Monteil, Rodiac and Van Schaftingen [44,45] to more general functionals. This raises the question as to whether a renormalised energy may be derived in case n = 0, k > 2. A higher-order energy expansion for the three-dimensional Ginzburg-Landau functional (n = 1, k = 2, N = S 1 ) was obtained by Contreras and Jerrard [21], in a setting where the energy concentrates on a cluster of 'nearly parallel' vortex filaments.
We deduce Theorem A from our Γ-convergence result, Theorem C in Section 2. The proof of the Γ-lower bound is based on the same strategy as in [2]. However, the construction of a recovery sequence is rather different from [2]. The main building block, Proposition 3.1 in Section 3.2, is inspired by the "dipole construction" [13,6,7]. Here, dipoles are suitably inserted into a non-constant and, in fact, singular background.
As an auxiliary result, we prove the following lower energy bound, which may be of independent interest.
Proposition B. Suppose that (H 1 )-(H 4 ) hold. Let Ω ⊆ R k be a bounded, Lipschitz domain that is homeomorphic to a ball. Then, for any u ∈ W 1,k (Ω, R m ) such that u = v on ∂Ω, there holds where σ ∈ π k−1 (N ) is the homotopy class of v and C is a positive constant that depends only on Ω, v.
If Ω ⊆ R k is homeomorphic to a ball and v ∈ W 1−1/k,k (∂Ω, N ), the homotopy class of v can be defined as in [14]. In the Ginzburg-Landau case, this inequality was proved by Sandier [50] (with k = 2) and Jerrard [38]; for the Landau-de Gennes functional, see e.g. [5,16]. The proof of Proposition B in contained in Appendix C (in fact, a slightly stronger statement is given there).
Remark 3. In case σ = 0, Proposition B does not provide any information. However, there could be critical points of the functional E ε whose energy diverges logaritmically even if the boundary datum is homotopically trivial. In other words, energy concentration may happen not only because of global topological contraints, but also for other reasons, such as symmetry. See, for instance, [37] for an analysis of two-dimensional Landau-de Gennes solutions (n = 0, k = 2, N = RP 2 ).
The paper is organised as follows. In Section 2 we recall some notation from [17] and we state the main Γ-convergence result, Theorem C. We prove the Γ-upper bound first, in Section 3, and give the proof of the Γ-lower bound in Section 4. Theorem A is deduced from Theorem C in Section 5. A series of appendices, with proofs of technical results, completes the paper.

Setting of the problem and statement of the Γ-convergence result
Throughout the paper, we will write A B as a shorthand for A ≤ CB, where C is a positive constant that only depends on n, k, f , N , and Ω. If F ⊆ R n+k is a rectifiable set of dimension d and u ∈ W 1,k loc (R n+k , R m ) we will write Additional notation will be set later on. Throughout the paper, we assume that (H 1 )-(H 4 ) are satisfied.
Remark 2.1. We do not require that |nσ| = n|σ| for any n ∈ N, σ ∈ π k−1 (N ); this is consistent with the theory of flat chains as developed in [26,55].
The proof of this result will be given in Appendix A. In case N = S k−1 , the group π k−1 (S k−1 ) is isomorphic to Z, S = {−1, 0, 1}, and for any d ∈ Z we have where L k (B k 1 ) is the Lebesgue measure of the unit ball in R k and |d| is the standard absolute value of d (see Example A.1).
Remark 2.2. When k = 2, the infimum in (2.2) is achieved by a minimising geodesic in the homotopy class σ, parametrised by multiples of arc-length. As a consequence, E min (σ) isup to a multiplicative constant -the length squared of a minimising geodesic in the class σ, and E 1/2 min is a norm on π 1 (N ). However, E 1/2 min may not coincide with | · | * , not even up to a multiplicative constant. For instance, when N is the flat torus, N = R 2 /(2πZ) 2 = S 1 × S 1 , we have π 1 (N ) Z × Z, for any (d 1 , d 2 ) ∈ Z × Z. We did not investigate whether, for arbitrary k > 2 and N , E 1/k min is a norm on π k−1 (N ).

Notation for flat chains
We follow the notation adopted in [17,Section 2]. In particular, we denote by F q (R n+k ; π k−1 (N )) the space of flat q-dimensional chains in R n+k with coefficients in the normed group (π k−1 (N ), |· | * ). We denote the flat norm by F, and the mass by M. The support of a flat chain S is denoted by spt S. The restriction of S to a Borel set E ⊆ R n+k is denoted S E. Given f ∈ C 1 (R n+k , R n+k ), we write f * S for the push-forward of S through f . (The reader is referred e.g. to [26,55] for the definitions of these objects.) Given a domain Ω ⊆ R n+k , we define F q (Ω; π k−1 (N )) as the set of flat chains such that spt S ⊆ Ω. We also define M q (Ω; π k−1 (N )) as the set of flat chains S ∈ F q (Ω; π k−1 (N )) such that M(S) < +∞. We will say that two chains S 1 , S 2 ∈ M q (Ω; π k−1 (N )) are cobordant in Ω if and only if there exists a finite-mass chain R ∈ M q+1 (Ω; π k−1 (N )) such that In this case, we write S 1 ∼ Ω S 2 . The cobordism in Ω defines an equivalence relation on the space of finite-mass chains, M q (Ω; π k−1 (N )). Moreover, due to the isoperimetric inequality (see e.g. [25, 7.6]), cobordism classes are closed with respect to the F-norm.
The group of flat q-chains relative to a domain Ω ⊆ R n+k is defined as the quotient To avoid notation, the equivalence class of a chain S ∈ F q (R n+k ; π k−1 (N )) will still be denoted by S. The quotient norm may equivalently be rewritten as For any S ∈ F n (Ω; π k−1 (N )) and R ∈ F k (R n+k ; Z) such that M(R) + M(∂R) < +∞, spt R ⊆ Ω, and spt(∂S) ∩ spt R = spt S ∩ spt(∂R) = ∅, we denote the intersection index of S and R (as defined in [17, Section 2.1]) by I(S, R) ∈ π k−1 (N ). For instance, if S is carried by a n-polyhedron with constant multiplicity σ ∈ π k−1 (N ), R is carried by a k-polyhedron with unit multiplicity and (the supports of) S, R intersect transversally, then I(S, R) = ±σ, where the sign depends on the relative orientation of S and R. The intersection index I is a bilinear pairing and satisfies suitable continuity properties (see e.g. [17,Lemma 8]).

The topological singular set
In [17], we constructed the topological singular set, S y (u), for u ∈ (L ∞ ∩W 1,k−1 )(Ω, R m ) and y ∈ R m . Here, we introduce a variant of that construction and define S y (u) in case u ∈ W 1,k (Ω, R m ), without assuming that u ∈ L ∞ (Ω, R m ). In both cases, the operator S y (u) generalises the Jacobian determinant of u -and indeed, the Jacobian of u : The starting point of the construction is the following topological property.

Proposition 2.2 ([30]
). Under the assumption (H 2 ), there exist a compact, polyhedral complex X ⊆ R m of dimension m − k and a smooth map : R m \ X → N such that (z) = z for any z ∈ N , and for any z ∈ R m \ X and some constant C = C(N , m, X ) > 0. This result, or variants thereof, was proved in [30, Lemma 6.1], [12,Proposition 2.1], [33,Lemma 4.5]. While in our previous paper [17] we required X to be a smooth complex, in this paper we require X to be polyhedral, because this will simplify some technical points in the proofs.
Let us fix once and for all a polyhedral complex X and a map , as in Proposition 2.2. Let δ * ∈ (0, dist(N , X )) be fixed, and let be the set of Lebesgue-measurable maps S : B * → F n (Ω; π k−1 (N )), respectively S : B * → F n (Ω; π k−1 (N )) (we use the notation y ∈ B * → S y in both cases), such that The sets Y , Y are complete normed moduli, with the norms · Y , · Y respectively. The space F n (Ω; π k−1 (N )), respectively F n (Ω; π k−1 (N )), embeds canonically into Y , respectively Y . If need be, we will identify a chain S ∈ F n (Ω; π k−1 (N )) with an element of Y , i.e. the constant map y → S. By [17, Theorem 3.1], there exists a unique operator S : in Y ) and satisfies (P 0 ) for any smooth u, a.e. y ∈ B * and any R ∈ F k (R n+k ; Z) such that M(R) + M(∂R) < +∞, spt(R) ⊆ Ω, spt(∂R) ⊆ Ω \ spt S y (u), there holds I(S y (u), R) = homotopy class of • (u − y) on ∂R.
We recall that I denotes the intersection index, defined as in [17, Section 2.1].
(P 2 ) For any u ∈ W 1,k (Ω, R m ) and any Borel subset E ⊆ Ω, there holdŝ The proof of Proposition 2.3 will be given in Apprendix B. Taking account of (P 1 ), we abuse of notation and write S instead of S from now on. As a consequence of (P 3 ), for any boundary datum v ∈ W 1−1/k,k (∂Ω, N ) there exists a unique cobordism class C (Ω, v) ⊆ M n (Ω; π k−1 (N )) such that (2.6) S y (u) ∈ C (Ω, v) for any u ∈ W 1,k (Ω, R m ) with trace v on ∂Ω and a.e. y ∈ B * .

The Γ-convergence result
The main result of this paper is a generalisation of [2, Theorem 5.5]. We let W 1,k v (Ω, R m ) denote the set of maps u ∈ W 1,k (Ω, R m ) such that u = v on ∂Ω (in the sense of traces).
Theorem C. Suppose that the assumptions (H 1 )-(H 4 ) are satisfied. Then, the following properties hold.
(i) Compactness and lower bound.
Then, there exists a (non relabelled) countable subsequence and a finite-mass chain S ∈ C (Ω, v) such that S(u ε ) → S in Y and, for any open sub- Theorem A follows almost immediately from Theorem C, combined with general properties of the Γ-convergence and standard facts in measure theory. There is a variant of Theorem C for the problem with no boundary conditions, which is analogue to [2, Theorem 1.1]. We will say that a chain S is a finite-mass, n-dimensional relative boundary if it has form S = (∂R) Ω, where R ∈ M n+1 (R n+k ; π k−1 (N )) is such that M(∂R) < +∞.
Proposition D. Suppose that the assumptions (H 1 )-(H 3 ) are satisfied. Then, the following properties hold.
(i) Compactness and lower bound. Let (u ε ) ε>0 be a sequence in W 1,k (Ω, R m ) that satisfies sup ε>0 |log ε| −1 E ε (u ε ) < +∞. Then, there exists a (non relabelled) countable subsequence and a finite-mass, n-dimensional relative boundary S such that S(u ε ) → S in Y and, for any open subset A ⊆ Ω, (ii) Upper bound. For any finite-mass, n-dimensional relative boundary S, there exists a se- Proposition D is not quite informative as it stands, because minimisers of the functional (1) under no boundary conditions are constant. However, since Γ-convergence is stable with respect to continuous perturbations, Proposition D can be extended to non-trivial minimisation problems with lower-order terms or under integral constraints, as long as these are compatible with the topology of Γ-convergence.

A few examples
We illustrate our results by means of a few simple examples. If A ⊆ R n+k is an n-dimensional polyhedral (or smooth) set, with a given orientation, the unit-multiplicity chain carried by A will be denoted A ∈ M n (R n+k ; Z).
Example 2.1. First, we suppose the domain is the unit ball in the critical dimension, i.e. n = 0 and Ω = B k , and consider the target N = S k−1 ⊆ R k . We need to identify the class C (Ω, v) defined by (2.6). For simplicity, suppose that the boundary datum v : could also be considered, by appealing to Brezis and Nirenberg's theory of the degree in VMO, [14]). Let u : B k → R k be any smooth extension of v. Let y ∈ R k be a regular value for u (i.e., det ∇u(x) = 0 for any x ∈ u −1 (y)) such that |y| < 1. Then, the inverse image u −1 (y) consists of a finite number points. Let r > 0 be a sufficiently small radius. By definition of S, we have consists of all and only the chains that differ from S y (u) by a boundary. It is not difficult to characterise C (Ω, v) using the following topological property, which holds true for any (normed, Abelian) coefficient group G and any connected, open set D ⊆ R d .
Then, there exists R ∈ M 1 (D; G) such that ∂R = T if and only if q j=1 σ j = 0. For a proof of this fact, see e.g. [31,Proposition 2.7]. Now, Brouwer's theory of the degree (or Property (P 0 ) above) implies that In agreement with the Ginzburg-Landau theory, mass-minimising chains in C (Ω, v) consist of exactly |d| points, with multiplicities equal to 1 or −1 according to the sign of d. This argument extends to more general manifolds N , with no essentially change; we obtain where σ ∈ π k−1 (N ) is the homotopy class of the boundary datum v : ∂B k → N . Massminimising chains in C (Ω, v) have the form q j=1 σ j z j , where the multiplicities σ j belong to the set S defined in (2.4) and satisfy q j=1 E min (σ j ) = |σ| * .
Example 2.2. Next, we discuss the case n = 1, Ω = B k+1 . Suppose that the boundary datum v : ∂B k+1 → N is smooth, except for finitely many isolated singularities at the points x 1 , . . . , x p . Let D 1 , . . . , D p be pairwise-disjoint closed geodesic disks in ∂B k+1 , centred at the points x 1 , . . . , x p . Each D i is given the orientation induced by the outward-pointing unit normal to B k+1 . Using orientation-preserving coordinate charts, we may identify v |∂D i : ∂D i → N with a map S k−1 → N ; the homotopy class of the latter is an element of π k−1 (N ), which we denote σ i . The coefficents σ i must satisfy the topological constraint Indeed, let D + ⊆ ∂B k+1 be a small geodesic disk that does not contain any singular point x i , and let D − := ∂B k+1 \ D + . Topologically, D − is a disk which contains all the singular points of v; therefore, the homotopy class of v restricted to ∂D − is the sum of all the σ i 's above. However, the homotopy class of v on ∂D + must be trivial, because v is smooth in D + . Thus, (2.7) follows. We consider the chain Thanks to (2.7), S bd (v) is the boundary of some 1-chain supported inB k+1 . More precisely, let u ∈ W 1,k (B k+1 , R m ) be any extension of v. The results of [17] (see, in particular, Proposition 1, Proposition 3 and Lemma 18) imply that for a.e. y ∈ R m of norm small enough. Chains in the same homology class have the same boundary; therefore, for any chain T ∈ C (Ω, v), there holds ∂T = S bd (v). Conversely, two chains inB k+1 that have the same boundary belong to same homology class (relative toB k+1 ), because the domainB k+1 is contractible. As a consequence, we have In particular, mass-minimising chains in C (Ω, v) will be carried by a finite union of segments, connecting the singularities of the boundary datum according to their multiplicities. In case N = S k−1 , such union of segments realises a 'minimising connection', in the sense of Brezis, Coron and Lieb [13]. For k = 2 and N = RP 2 , the condition (2.7) implies that v has an even number of non-orientable singularities; mass-minimising chains connect the non-orientable singularities in pairs. The characterisation (2.8) extends to general data v ∈ W 1−1/k,k (∂B k+1 , N ), provided that we define S bd (v) in a suitable way (see [17,Section 3]). It also extend to more general domains Ω ⊆ R n+k , so long as the n-th homology group H n (Ω; π k−1 (N )) is trivial. Example 2.3. If the domain has a non-trivial topology, then C (Ω, v) may contain non-trivial chains even if the boundary datum is smooth. For instance, take n = 1, k = 2, N = S 1 .
Let Ω ⊆ R 3 be a solid torus of revolution, defined as the image of the map Ψ : We consider the smooth map u : Ω → R 2 given by u(Ψ(x, θ)) := x for (x, θ) ∈ B 2 × R. The trace of u at the boudary, v, takes its values in S 1 and its restriction on each meridian curve of the torus ∂Ω has degree 1. Therefore,

Upper bounds 3.1 Notations and sketch of the construction
We say that a map u : Ω → R m is locally piecewise affine if u is continuous in Ω and, for any polyhedral set K ⊂⊂ Ω, the restriction u |K is piecewise affine. A set P ⊆ Ω is called locally n-polyhedral if, for any compact set K ⊆ Ω, there exists a finite union Q of convex, compact, n-dimensional polyhedra such that P ∩ K = Q ∩ K. In a similar way, we say that a finite-mass chain S ∈ M n (Ω; π k−1 (N )) is locally polyhedral if, for any compact set K ⊆ Ω, there exists a polyhedral chain T such that (S − T ) K = 0. If M is a polyhedral complex and j ≥ 0 is an integer, we denote by M j the j-skeleton of M , i.e. the union of all its faces of dimension less than or equal to j. We set M −1 := ∅.
Maps with nice and η-minimal singularities. To construct a recovery sequence, we will work with N -valued maps with well-behaved singularities, in a sense that is made precise by the definition below. Let M , S be polyhedral sets in R n+k of dimension n, n − 1 respectively, and let u : We say that u has a nice singularity at (M, S) if u is locally Lipschitz on Ω \ (M ∪ S) and, for any p > 1, there is a constant C p such that We say that u has a locally nice singularity at M (respectively, at (M, S)) if, for any open subset W ⊂⊂ Ω, the restriction u |W has a nice singularity at M (respectively, at (M, S)).  Throughout Section 3, we will work with maps with nice (or locally nice) singularities. However, in order to obtain sharp energy estimates, we will need to impose a further restriction on the behaviour of our maps near the singularities. Let u : Ω → N be a map with nice singularity at (M, S), where M , S are polyhedral sets of dimension n, n − 1 respectively. We triangulate M , i.e. we write M as a finite union of closed simplices such that, if K , K are simplices with K = K , K ∩ K = ∅, then K ∩ K is a boundary face of both K and K . Let K ⊆ M be a n-dimensional simplex of the triangulation, and let K ⊥ be the k-plane orthogonal to K through the origin. Given positive parameters δ, γ, we define the set Figure 1). We will identify each x ∈ U (K, δ, γ) with a pair x = (x , x ), where x , x are as in (3.1). By choosing δ, γ small enough (uniformly in K), we can make sure that the sets U (K, δ, γ) have pairwise disjoint interiors. We say that u is η-minimal if there exist positive numbers δ, γ, a triangulation of M and, for any n-simplex K of the triangulation, a Lipschitz map φ K : S k−1 → N that satisfy the following properties.
(ii) For any n-dimensional simplex K ⊆ M and a.e. x = (x , x ) ∈ U (K, δ, γ), we have (iii) For any n-dimensional simplex K ⊆ M and any map ζ ∈ W 1,k (S k−1 , N ) that is homotopic to φ K , we haveˆS The operator ∇ is the tangential gradient on S k−1 , i.e. the restriction of the Euclidean gradient ∇ to the tangent plane to the sphere. N ). Therefore, for any η > 0 and any homotopy class σ ∈ π k−1 (N ), there exists a smooth map φ : S k−1 → N in the homotopy class σ that satisfies Remark 3.3. It is possible to find C 1 -maps that satisfy a stronger version of (3.2), with η = 0. Indeed, the compact Sobolev emebedding W 1,k (S k−1 , N ) → C(S k−1 , N ) implies that homotopy classes of maps S k−1 → N are sequentially closed with respect to the weak W 1,k -convergence. Then, for each homotopy class σ ∈ π k−1 (N ), there exists a map φ σ the minimises the L k -norm of the gradient in σ. The map φ σ solves the k-harmonic map equation and, by Sobolev embedding, is continuous. Then, regularity results for k-harmonic maps (e.g. [24,Proposition 5.4]) imply that φ σ ∈ C 1,α (S k−1 , N ). However, the weaker condition (3.2) is enough for our purposes.
Construction of a recovery sequence: a sketch. In most of this section, we focus on the proof of Theorem C.(ii), i.e. we study the problem in the presence of boundary conditions; only at the end of section, we present the proof of Proposition D.(ii). As in [2], in order to define a recovery sequence, we first construct a map w : Ω → N with (locally) nice singularity and prescribed singular set S(w) = S. However, w must also satisfy the boundary condition, w = v on ∂Ω, where v ∈ W 1−1/k,k (∂Ω, N ) is a datum. This boundary condition makes the construction of w substantially harder. For such a w to exists, we need a topological assumption on S, namely, that S belongs to the homology class (2.6) determined by Ω and v. Our approach is rather different from that of [2,Theorem 5.3]. In [2], the authors first construct w inside Ω, then interpolate near ∂Ω, using the symmetries of the target S k−1 , so as to match the boundary datum. On the contrary, we start from a map that satifies the boundary conditions and we modify it inside Ω so to obtain S(w) = S. Before giving the details, we sketch the main steps of our construction.
First, we consider a locally piecewise affine extension u * ∈ (L ∞ ∩ W 1,k )(Ω, R m ) of v. Since we have assumed that X is polyhedral, the singular set S y (u * ) will be locally polyhedral, for a.e. y. By projecting u * onto N (using Hardt, Kinderlehrer and Lin's trick [29], see Section 3.3), we define a map w * : Ω → N such that w * = v on ∂Ω, S(w * ) = S y (u * ) (for a well-chosen y) is locally polyhedral, and w * has a locally nice singularity at spt S(w * ). We cannot make sure that the singularity is nice up to the boundary of Ω, because the boundary datum is not regular enough. Let S be a finite-mass n-chain in the homology class C (Ω, v) defined by (2.6). Thanks Figure 2: Sketch of the construction of a recovery sequence. Inside W S , the chain S (in red) takes multiplicities in the set S ⊆ π k−1 (N ). Outside W , the original map w * and the modified map w coincide. to (P 3 ), we know that S(w * ) = S y (u * ) ∈ C (Ω, v) and hence, S(w * ) and S differ by a boundary. By approximation (see Section 3.4.2), we reduce to the case where R is a polyhedral (n + 1)-chain with compact support in Ω. Actually, we can make a further assumption on S. Let W S ⊂⊂ Ω be an open set, with polyhedral boundary, whose closure contains the support of R (see Figure 2). Up to a density argument (Proposition 3.7), we can assume that S W S takes its multiplicities in the set S ⊆ π k−1 (N ) defined by (2.4). Roughly speaking, we replace each polyhedron K of S W S with a finite number of polyhedra, very close to each other, whose multiplicities add up to the multiplicity of K. This is possible, because S generates π k−1 (N ) by Proposition 2.1. The assumption on the multiplicity of S W S turns out to be essential to obtain sharp energy bounds for our recovery sequence.
Let W be another open set, with polyhedral boundary, such that W S ⊂⊂ W ⊂⊂ Ω (see Figure 2). In particular, W contains the support of R. We aim to modify w * inside W , so to obtain a new map w : Ω → N with locally nice singularities and S(w) = S(w * ) + ∂R = S. In other words, we need to "move" the singularities of w * along the boundary of R. This is the key step in the construction. We achieve this goal by a suitable generalisation of the so-called "insertion of dipoles", Proposition 3.1 in Section 3.2. For any (n + 1)-polyhedron T of R, we modify w * in a neighbourhood of T by inserting an N -valued map that depends only on the k −1 coordinates in the orthogonal directions to T . To define w near ∂T , we use radial projections repeatedly, first onto the n-skeleton of T , then onto its (n − 1)-skeleton, and so on. Eventually, we obtain a map w : Ω → N that agrees with w * out of a neighbourhood of spt R (in particular, it matches the boundary datum), has locally nice singularities at S and satisfies S(w) = S. By local surgery ([2, Lemma 9.3], stated below as Lemma 3.9), we can also make sure that w |W is η-minimal.
The map w does not belong to the energy space W 1,k (Ω, R m ), unless S = 0, because it has a singularity of codimension k. Therefore, we must regularise w to construct a recovery sequence. For x ∈ W , we define Since w is η-minimal in W , a fairly explicit computation allows us to estimate the energy of u ε on W , in terms of the area of spt S and the maps φ K given by Definition 3.2. Moreover, for any simplex K of S W S , the multiplicity σ K of S at K belongs to S and hence,

Insertion of dipoles along a simplex
Our next result, Proposition 3.1, is the main building block in the construction of the recovery sequence. Perhaps it is worth commenting on the assumptions of Proposition 3.1. In terms of regularity of N , we do not need to work with smooth manifolds: a compact, connected Lipschitz neighbourhood retract would do. The assumption that N is (k − 2)-connected could also be relaxed. (k − 2)-connectedness is used in [47,17] to construct S(u) for arbitrary u ∈ W 1,k−1 (Ω, N ); however, if u has nice singularities and π k−1 (N ) is Abelian, then S(u) can be defined in a straightforward way. On the other hand, we must assume that N is (k − 1)-free (that is, the fundamental group of N acts trivially on π k−1 (N )). Should N not be (k − 1)-free, we could not identify free homotopy classes of maps S k−1 → N with elements of π k−1 (N ). In this case, the product of free homotopy classes S k−1 → N is multi-valued and hence, the equality S(ũ) = S(u) + σ∂ T may fail.
The proof of Proposition 3.1 (see Figure 3) is based on a construction known as "insertion of dipoles". Several variants of this construction are available in the literature (see e.g. [13,6,7,27,47]), but all of them rely of the following fact: a map B k−1 → N that takes a constant value on ∂B k−1 may be identified with a map S k−1 → N , by collapsing the boundary of the disk to a The initial map u is plotted in (a); the values of u are represented by the colour code. We aim to insert singularities of degrees 1, −1 at the points x + , x − . First, we reparametrise u, creating a 'slit' along the segment of endpoints x + and x − (b). Then, we fill the slit by inserting a map that winds around the circle exactly once, as we move in the direction orthogonal to the segment of endpoints x + , x − (c). Finally, we defineũ in the disks V + , V − in such a way thatũ is homogeneous inside each disk (d). The new mapũ behaves as required. For instance, there are exactly three yellow points on ∂V + ; as we move anticlockwise around ∂V + , two of them carry the orientation 'from red to blue' and the other one carries the opposite orientation 'from blue to red'. If we orient the target S 1 'from red to yellow to blue', then the degree ofũ on ∂V + is 1.
point. As a consequence, if a continuous map φ : B k−1 → N is constant on ∂B k−1 , then we may define the homotopy class of φ as an element of π k−1 (N ). (In principle, we should distinguish between free or based homotopy, according to whether the boundary value of φ is allowed to vary during the homotopy or not; however, the assumption (H 2 ) guarantees that these two notions are equivalent.) and, for any σ ∈ π k−1 (N ), the homotopy class of u(x , ·) is σ.
The proof of Lemma 3.2 is completely standard, but we provide it for the sake of convenience.

Proof of Lemma 3.2.
We choose a point x 0 ∈ K and consider the map ψ : The map u is Lipschitz and satisfies (3.3); moreover, For any x ∈ K, the map u(x , ·) is (freely) homotopic to σ, via a reparametrisation and a change of base-point. Therefore, the homotopy class of u(x , ·) is σ.
Proof of Proposition 3.1. We triangulate Σ ∪ T , that is, we write Σ ∪ T as a finite union of closed simplices in such a way that, for any simplices K, K with K = K , K ∩ K is either empty or a boundary face of both K and K . We denote by T n the n-skeleton of this triangulation (i.e., the union of all simplices of dimension n or less). We will construct a sequence of maps u n+1 , u n , . . . , u 1 , u 0 by modifying the given map u first along the simplices of dimension n + 1 that are contained in T , then along those of dimension n, and so on. In order to do so, we first need to construct a suitable covering of T .
Step 1 (Construction of a covering of T ). Let K ⊆ T be a simplex of dimension j > 0. Let K ⊥ be the orthogonal (n + k − j)-plane to K through the origin. We fix positive numbers δ K , γ K and defineK Figure 4: The set V (K, δ, γ), in case n = 1, k = 2 (left) and n = 2, k = 1 (right). In both cases, the polyhedron K is in pink andK is in red.
(see Figure 4). If K is a 0-dimensional simplex, i.e. a point, we define V K := B n+k (K, δ K ) and Γ K := ∂V K . By choosing δ K , γ K in a suitable way, we can make sure that the following properties are satisfied: Property (b) implies that the V K 's do cover T . To construct a covering that satisfies (a)-(e), we first cover the 0-skeleton of T by pairwise disjoint balls that are compactly contained in D. Then, we cover each 1-dimensional simplex in T by a "thin cylinder", whose bases are contained in the balls we have chosen before. Next, we cover each 2-dimensional simplex by a "thin shell", and so on, as illustrated in Figure 5. At each step, we can make sure that the properties (a)-(e) are satisfied, because the simplices have pairwise disjoint interiors and only intersect along their boundaries. As a consequence of (d), for any simplex K ⊆ T there holds For any integer j ∈ {0, 1, . . . , n + 1}, we define and V <0 := ∅. Step 2 (Construction of u n+1 ). Let K ⊆ T be a (n + 1)-simplex of the triangulation, with the orientation induced by T . We identify V K withK × B k−1 (0, δ K ), whereK is given by (3.4). We construct a Lipschitz map u n+1 K : V K → N as follows. First, we let only depends on the variable x , we may apply Lemma 3.2 and define u n+1 in such a way that, for any x ∈K, The sign (−1) n+1 will be useful to compensate for orientation effects, later on in the proof. We define a map This definition is consistent. Indeed, the sets V K are pairwise disjoint, due to (c). Moreover, if a point x belongs both to V K and to D \ V <n+1 , then x ∈ Γ V because of (b), so u n+1 K (x) = u(x) by (i). Therefore, the map u n+1 is well-defined and locally Lipschitz out of Σ, with nice singularity at Σ.
Step 3 (Construction of u n ). Let K ⊆ T be a n-simplex. We identify V K withK ×B k (0, δ K ). The map u n+1 is Lipschitz continuous on Γ K , due to (3.7). Let σ K ∈ π k−1 (N ) be the homotopy class of u n+1 on an arbitrary slice of Γ K , of the form {x } × ∂B k (0, δ K ). If σ K = 0 then, by adapting the arguments of Lemma 3.2, we can construct a Lipschitz continuous map u n K : In both cases, by a straightforward computation, we obtain where the proportionality constant at the right-hand side depends on δ K and u n+1 . We define (c) and (3.10), we can argue as in Step 2 and check that u n is locally Lipschitz out of Σ ∪ T n , with nice singularity at Σ ∪ T n .
Step 4 (Construction of u j for j < n). We proceed by induction. Let j ∈ {0, 1, . . . , n − 1}. Suppose we have constructed a map The map u j K is locally Lipschitz out of the set By Property (d), the only simplices of Σ ∪ T n that intersect V K are those that contain K. Therefore, if H 1 , H 2 , . . . , H p denote the n-dimensional (closed) simplices of Σ∪T n that contain K, then Moreover, Property (d) and the convexity of H i imply that , λx ∈H i for any x ∈H i and any λ ≥ 0). As a consequence, where the proportionality constant at the right-hand side may depend on δ K . Given By the induction hypothesis, u j+1 has a nice singularity at Σ ∪ T n . Therefore, an explicit computation gives for a.e. x ∈ V K . By (3.11) and (3.12), the set Σ ∪ T n agrees withK × ∪ iHi in V K , and ∪ iHi is a cone. Then, by a geometric argument (see Figure 6), we have By combining (3.14) and (3.15), (3.13) follows. Finally, we define (c) and (3.13), the map u j is well-defined, locally Lipschitz out of Σ ∪ T n and has a nice singularity at Σ ∪ T n .
Step 5 (Conclusion). By induction, we have constructed a sequence of maps u n+1 , u n , . . . , u 1 , u 0 . Letũ := u 0 : D → N . By construction, the mapũ has a nice singularity at Σ ∪ T n and agrees with u out of V <n+1 ∪ V =n+1 . In particular,ũ = u in a neighbourhood of ∂D, because of (a).
It only remains to compute S(ũ). Let K be an n-simplex of T . By Property (e), K is not entirely contained in V <n ; we take a point x ∈ K \ V <n . Let K ⊥ be the orthogonal k-plane to K at x, and let F := V K ∩ K ⊥ . By Property (d), the only (n + 1)-simplices that intersect F are those that contain K; we call them H 1 , . . . , H p . We consider the restriction ofũ to the (k − 1)sphere ∂F . By construction (see (3.8) and (3.9) in Step 2),ũ |∂F consists (up to homotopy) of a reparametrisation of u |∂F , with the insertion of 'bubbles' around the points ∂F ∩ H i . Each bubble carries the homotopy class σ or −σ, depending on the orientation of H i (which, we recall, is the one induced by T ). The net topological contribution of all the bubbles may vanish or not, depending on whether the point x belongs to the boundary of T or not. As a result, we have (homotopy class ofũ |∂F : The sign of the second term in the right-hand side depends on the choice of the sign we made in Equation (3.9) (see, for instance, Property (iv) in Lemma 8 of [17]). Then, by Remark 3.1, S(ũ) = S(u) + σ∂ T .

Projection of a W 1,k -map onto N
Before we pass to the construction of a recovery sequence, we gather some useful results, based on earlier work by Hardt For any y ∈ R m , we consider the map˜ y : z → (z − y) which is well defined for z ∈ R m \ (X + y). This is not a retraction onto N , in general, because it does not restrict to the identity on N . However, for sufficiently small |y| -say, y ∈ B m σ with σ > 0 small enough -the restriction˜ y|N is a small perturbation of the identity and, in particular, it is a diffeomorphism. For y ∈ B m σ and z ∈ R m \ (X + y), let us define This map is indeed a smooth retraction of R m \ (X + y) onto N . We also define a function ψ : R m → R by The function ψ is Lipschitz and ψ = 1 on N . By Proposition 2.2 and (3.17), we have for any y ∈ B m σ and z ∈ R m \ (X + y). The proportionality constants here depend on σ, but σ = σ(N , X , ) is fixed once and for all. Finally, let ξ ε (t) := min(t/ε, 1) for t ≥ 0.
Then, the following properties hold.
(ii) For a.e. y ∈ B m σ and sufficiently small ε, where C Λ is a positive constant that only depends on N , k, X and Λ.
Proof of Lemma 3.3. Throughout the proof, we denote by C Λ a generic positive constant that only depends on N , k, X and Λ (and may change from one occurence to the other).
Step 1 (Proof of (i)). For a.e. y, we have • (u − y) ∈ W 1,k−1 (Ω, N ) (see e.g. [17, Lemma 14] for a proof of this claim). Moreover, by [17, Lemma 17] we know that Now, w y is obtained from • (u − y) by composition with a map, (˜ y|N ) −1 , which is homotopic to the identity on N . Therefore, from the identity above we obtain This can be first checked when u is smooth, using [17, Lemma 18], and remains true for a general u by a density argument, using the continuity of S and e.g. [17, Lemma 14].
Step 2 (Proof of (ii), (iii)). It is immediate to see that e. y, and by the chain rule, we have the pointwise bound for a.e. x ∈ Ω. Thanks to (3.18), we deduce that (where, as usual, 1 A denotes the characteristic function of a set A). On the other hand, the L ∞ -norm of w ε,y is uniformly bounded in terms of N only, and hence there holds Together, (3.20) and (3.21) imply that We integrate the previous inequality for y ∈ B m σ , apply Fubini theorem and make the change of variable z = u(x) − y: Since X is a finite union of simplices of codimension k or higher, for ε sufficiently small there holdsˆB m σ+Λ Lemma 8.3]). As a consequence, we obtain (iii).

Construction of an N -valued map with nice singularity at a locally polyhedral set
In this section, we give the construction of a recovery sequence. We first construct a map Ω → N that matches the Dirichlet boundary datum and has nice singularities along a locally polyhedral set.

Lemma 3.4. Any boundary datum
(c) the chain S y (u * ) takes its multiplicities in a finite subset of π k−1 (N ), which depends only on N , , X ; (d) there exists a locally (n − 1)-polyhedral set P y such that • (u * − y) has a locally nice singularity at spt S y (u * ) ∪ P y .
The proof of Lemma 3.4 relies on the following fact.
We give a proof of Lemma 3.5, for the convenience of the reader only.
Proof of Lemma 3.5. Arguing component-wise, we reduce to the case m = 1. Let u ∈ W 1,k v (Ω) be an extension of v. By a truncation argument, we can make sure that v ∈ L ∞ (Ω). Let Γ 1 := {x ∈ Ω : dist(x, ∂Ω) > 1/2} and, for any integer j ≥ 2, let Γ j := {x ∈ Ω : Using a partition of unity, we construct a sequence of smooth functions ϕ j ∈ C ∞ c (Γ j ) such that j≥1 ϕ j = 1. Thanks to e.g. [53, Theorem 1], for any j there exists a triangulation T j of R n+k such that the piecewise affine interpolant u j of ϕ j u along T j is well-defined (that is, all the vertices of T j are Lebesgue points of ϕ j u) and there holds Moreover, the proof of [53,Theorem 1] shows that for any r > 0, we can choose T j such that all the simplices of T j have diameter ≤ r. In particular, we can make sure that u j is still supported in Γ j . Now, we define u * := j≥1 u j . Since the support of u j intersects the support of u i only for finitely many i, the function u * is locally piecewise affine. Moreover, u * ∈ L ∞ (Ω) because, by construction, u j L ∞ (Ω) ≤ u L ∞ (Ω) for any j, and u ∈ W 1,k (Ω) due to (3.22).
Proof of Lemma 3.4. Let u * be the locally piecewise extension of v given by Lemma 3.5. Statement (a) follows from (P 2 ) in Proposition 2.3, because u * ∈ W 1,k (Ω, R m ). Let K ⊆ Ω be a (closed) (n + k)-simplex such that u * |K is affine. Since we have assumed that X is polyhedral, for any y ∈ R m the inverse image (u * − y) −1 (X ) ∩ K is polyhedral too. Take y ∈ R m such that (u * − y) |K is transverse to each cell of X . By [17, Corollary 1], we have S y (u * ) K = S y (u * |K ) and by definition (see [17,Section 3.2] and Section B below), the latter is a polyhedral chain supported on (u * − y) −1 (X ) ∩ K. Thus, S y (u * ) is locally polyhedral. Moreover, S y (u * ) take its multiplicities in the set which is a finite subset of π k−1 (N ), because X is a finite union of polyhedra. Finally, let us prove Statement (d). Take an open set W ⊂⊂ Ω, and take y ∈ R m such that u * |W is transverse to each cell of X . Let K be a (n + k)-simplex such that K ∩ W = ∅ and u * |K is affine. By transversality, we see that for any x ∈ K, where C K > 0 is a constant that depends on the (constant) gradient of u * on K and on y.
Since W is covered by finitely many simplices, we have where C W,y := min K : K∩W =∅ C K,y > 0. Then, by applying the chain rule and Proposition 2.2, we conclude that • (u * − y) |W has a nice singularity at (spt S y (u * ) ∪ P y ) ∩ W , where P y := (u * − y) −1 (X m−k−1 ).

Reduction of the problem
Throughout the rest of Section 3, we fix the boundary datum v ∈ W 1−1/k,k (∂Ω, N ) and let u * be the map given by Lemma 3.4. We also fix y * ∈ R m , with |y * | sufficiently small, in such a way that Statements (a)-(d) in Lemma 3.4 are satisfied. Let w * := y * • u * , where y * is defined by (3.16). By Lemma 3.4, the map w * has a locally nice singularity at spt S y * (u * ) ∪ P y * , where P y * is a locally polyhedral set of dimension n − 1. By Lemma 3.3, we can choose y * so to have w * ∈ W 1,k−1 (Ω, N ) and S(w * ) = S y * (u * ) as well.
Let S be a finite-mass n-chain, supported in Ω, that is cobordant to S(w * ). By definition of C (Ω, v), Equation (2.6), S and S(w * ) differ by a boundary. By an approximation argument, we will reduce to the case S has a special form. Proposition 3.6. Let S ∈ C (Ω, v) be a finite mass chain. Then, there exists a sequence of polyhedral (n + 1)-chains R j , with compact support in Ω, such that S(w * ) + ∂R j → S (with respect to the F-norm) and M(S(w * ) + ∂R j ) → M(S) as j → +∞.
The proof of Proposition 3.6 is left to Appendix D.1. Thanks to Proposition 3.6, and a diagonal argument, we can assume with no loss of generality that S has the form where R is a polyhedral (n+1)-chain, compactly supported in Ω. There is one further assumption we can make. Let W S ⊂⊂ Ω be an open set, with polyhedral boundary, such that ∂W S is transverse to spt S (more precisely, there exist triangulations of ∂W S and spt S such that any simplex of the triangulation of ∂W S is transverse to any simplex of the triangulation of spt S) and The condition S ∂W S = 0 is satisfied because, by transversality, spt S ∩∂W S has dimension (n− 1) or less and hence, it cannot support a non-trivial polyhedral n-chain.

Proposition 3.7.
There exists a sequence of polyhedral (n + 1)-chains R j , supported in W S , such that the following hold: (i) S + ∂R j → S, with respect to the F-norm, as j → +∞; (ii) M(S + ∂R j ) → M(S) as j → +∞; (iii) for any j, (S + ∂R j ) ∂W S = 0; (iv) for any j, the chain (S + ∂R j ) W S takes multiplicities in the set S ⊆ π k−1 (N ) defined by (2.4).
The proof of Proposition 3.7 will be given in Appendix D.1. Thanks to Proposition 3.7, it is not restrictive to assume that (3.25) S W S takes its multiplicities in S in addition to (3.23), (3.24). Indeed, if (3.24) does not hold, we replace S with with a chain of the form S + ∂R j as given by Proposition 3.7, we replace R with R + R j , then we use a diagonal argument to pass to the limit as j → +∞.

Construction of an N -valued map with prescribed singular set
Our next task is to construct a map w : Ω → N , with locally nice singularities, in such a way that S(w) = S. To do so, we fix an open set W ⊂⊂ Ω such that W S ⊂⊂ W and ∂W is transverse to spt S (i.e., there exist triangulations of ∂W S and spt S such that any simplex of the triangulation of ∂W S is transverse to any simplex of the triangulation of spt S). We also fix a small parameter η > 0.
Lemma 3.8. For any W as above and any η > 0, there exists a map w ∈ W 1,k−1 (Ω, N ) that satisfies the following properties: (ii) w has a locally nice singularity at (spt S, Q * ), where Q * ⊇ (spt S) n−1 is a locally (n − 1)polyhedral set; (iii) S(w) = S; (iv) w |W is η-minimal. (ii)ũ has a nice singularity at (K, ∂K); (iii) S(ũ) = S(u); In [2], this result is proved in the particular case N = S k−1 . However, the same proof applies to a general target N : the mapũ is constructed by a suitable reparametrisation of the domain U (K, δ, γ), and the arguments do not rely on properties of the target N other than (Lipschitz) path-connectedness. Property (iii) follows from Remark 3.1 and (ii), (iv).
Proof of Lemma 3.8. By (3.24), we have spt R ⊆ W S ⊆ W . By triangulating, we can write R in the form where the coefficients σ i belong to π k−1 (N ) and each T i ⊂⊂ W is a convex (n + 1)-simplex. We apply Proposition 3.1, so to modify w * in a neighbourhood of T 1 . We obtain a new map w 1 ∈ W 1,k (Ω, N ) that has a locally nice singularity at spt S(w * ) ∪ (T 1 ) n ∪ P y * (with (T 1 ) n is the nskeleton of a suitable triangulation of T ), satisfies w 1 = w * on Ω\W and S(w 1 ) = S(w * )+σ 1 ∂ T 1 . Now, we use Proposition 3.1 to modify w 1 in a neighbourhood of T 2 , and so on. By applying iteratively Proposition 3.1, we construct a sequence of maps w 1 , w 2 , . . . , w q . The map w q has a locally nice singularity at spt S(w * ) ∪ (spt R) n ∪ P y * , satisfies w q = w * on Ω \ W and S(w q ) = S(w * ) + ∂R = S.
To complete the proof, it only remains to modify w q so as to satisfy (iv). Since W ⊂⊂ Ω has polyhedral boundary, the restriction S W is a polyhedral chain. Let K be a n-face of spt S(w * ) ∪ (spt R) n . The interior of K is contained in W and hence, for sufficiently small parameters δ > 0, γ > 0, the interior of U (K, δ, γ) is contained in W . Let σ K ∈ π k−1 (N ) be the homotopy class of w q around K. By Remark 3.2, there exists a smooth map φ K : If σ K = 0, we choose φ K to be constant. We apply Lemma 3.9 to u = w q and φ = φ K . By doing so for each K, we obtain a map w : Ω → N that agrees with w * on Ω \ W and is η-minimal on W . By Remark 3.1, S(w) = S(w q ) = S. Moreover, since φ K is constant if σ K = 0, w has a locally nice singularity at (spt S, Q * ) where Q * := (spt S(w * )) n−1 ∪ (spt R) n−1 ∪ P y * . Therefore, w has all the desired properties.

ε-regularisation
The map w : Ω → N given by Lemma 3.8 has a singularity of codimension k at spt S, so w / ∈ W 1,k (Ω, N ) unless S = 0. Therefore, in order to define a recovery sequence, we need to regularise w around spt S. We do so by defining the maps for any x ∈ Ω.
where the constant C w,D depends on the map w and on dist(D, ∂Ω).
where C is a constant that depends only on N , X , and k.
Step 1 (Proof of (i)). Let D ⊂⊂ Ω be an open set. We choose a number p, with 1 < p < (k + 1)/(k − 1). Since w has a locally nice singularity at (spt S, Q * ), at a.e. point of D we have where C w,D is a constant that depends on w, dist(D, ∂Ω) and p, but not on ε. Therefore, By our choice of p, we have p − kp > −(k + 1). Since Q * has codimension k + 1, [2, Lemma 8.3] implies that the function dist p−kp (·, Q * ) is integrable and that As a consequence, we havê as ε → 0, and (i) follows.
Step 2 (Proof of (ii)). Let D ⊂⊂ Ω and 1 < p < 1 + 1/k. From (3.27), we deduce The second and third term at the right-hand side are uniformly bounded with respect to ε → 0, due to [2, Lemma 8.3] and (3.28). Since spt S ∩ D is contained in a finite union of polyhedra of codimension k or higher and D has polyhedral boundary, a computation based on Fubini theorem gives lim sup On the other hand, H n (spt S ∩ D) M(S D) because the coefficient group (π k−1 (N ), | · | * ) is discrete (Proposition 2.1). Thus, (ii) follows (and in particular, w ε ∈ W 1,k loc (Ω, R m )).
Step 3 (Proof of (iii)). The inequality (3.28) implies (3.29) lim sup |log ε| ε k = 0, so we only need to estimate the gradient terms. By Lemma 3.8, w |W is η-minimal, with nice singularity at ((spt S)∩W, Q * ∩W ). Therefore, there exist positive numbers δ, γ, a triangulation of (spt S) ∩ W and, for any n-simplex K of the triangulation, a Lipschitz map φ K : S k−1 → N that satisfy the conditions (i)-(iii) in Definition 3.2. By taking smaller δ, γ if necessary, we can also assume that the interior of U (K, δ, γ) is contained in W , for any n-simplex K of the triangulation. Let F := W \ ∪ K U (K, δ, γ), where the union is taken over all n-simplices K of the triangulation. We estimate separately the energy on F and on each U (K, δ, γ). Let us estimate the energy on F first. Since Q * ⊇ (spt S) n−1 , the definition (3.1) of U (K, δ, γ) implies that (3.30) dist(x, Q * ) dist(x, spt S) for any x ∈ F.
(The proportionality constant at the right-hand side depends on δ, γ.) Let us choose a number p with 1 < p < 1 + 1/k. Since w has a locally nice singularity at (spt S, Q * ), we obtain for a.e. x ∈ F and some constant C w,W that depends on w, W , p, δ and γ. This implies The right-hand side is uniformly bounded with respect to ε, due to [2, Lemma 8.3] and (3.28), so Next, we estimate the energy on U (K, δ, γ), with K an n-dimensional simplex in the triangulation of (spt S) ∩ W . We write U := U (K, δ, γ) for brevity, and let x = (x , x ) denote the variable in U , as in (3.1). Using Condition (ii) in Definition (3.2), we can compute explicitly the gradient of w ε , and we obtain for a.e. x ∈ U , where ∇ denotes the tangential gradient on S k−1 . (In the second inequality, we use that φ K is Lipschitz.) We raise to the power k both sides of this inequality, integrate over U , apply Fubini theorem and pass to polar coordinates for the integral with respect to x : Using Condition (iii) in Definition 3.2, we deduce lim sup where σ K ∈ π k−1 (N ) is the homotopy class of φ K and E min (σ K ) is defined by (2.2). We need to distinguish two cases, depending on whether the interior of K is contained W S or not. If the interior of K is contained in W S , then σ K ∈ S because of (3.25), and (3.32) becomes lim sup for some constant C that depends only on N . (Here again, we have used that M(S K) H n (K), due to Proposition (2.1).) Suppose now that the interior of K is not contained in W S . The intersection between the interior of K and ∂W S has dimension n − 1 at most, because we have taken ∂W S to be transverse to spt S. Therefore, up to refining the triangulation, we may assume that the interior of K is contained in W \ W S . Then, thanks to (3.23) and (3.24), S agrees with S(w * ) in the interior of K. The chain S(w * ) takes its multiplicity in a finite set that depends only on N , X , (by Lemma 3.4) and hence, E min (σ K ) ≤ C. Thus, (3.32) becomes lim sup

Proof of Theorem C.(ii) and Proposition D.(ii)
Proof of Theorem C.(ii). Let S ∈ C (Ω, v) be a finite-mass chain, and let η > 0 be a small number. Given a countable sequence ε → 0, we aim to construct u ε ∈ (L ∞ ∩ W 1,k v )(Ω, R m ), where ε ranges in a non-relabelled subsequence, in such a way that lim ε→0ˆBm (0, dist(N , X )) where C is a constant that does not depend on η. If we do so, the theorem will follow, by a diagonal argument. As we have seen, thanks to Proposition 3.6, Proposition 3.7 and a diagonal argument, it is not restrictive to assume that S satisfies (3.23), (3.24), (3.25). Moreover, we have Step 1 (Definition of u ε ). To define the recovery sequence near the boundary of Ω, we apply Lemma 3.3 to u * and y * , and consider the map (with ξ ε , ψ as in Lemma 3.3). Thanks to Lemma 3.3 and an averaging argument, by possibly modifying the value of y * we have Our recovery sequence will coincide with w ε given by (ii) in W , where W S ⊂⊂ W ⊂⊂ Ω is the open set introduced in Section 3.4.3. We need to interpolate between w ε and w ε,y * near W . To this end, we take a small parameter θ > 0, and we let D θ : We have u ε ∈ (L ∞ ∩ W 1,k v )(Ω, R m ) and sup ε u ε L ∞ (Ω) < +∞.
Step 2 (Bounds on E ε (u ε )). The energy of u ε on Ω \ (W ∪ D θ ) is bounded from above by (3.39). The energy of u ε is bounded from above by Lemma   It remains to estimate the energy of u ε on D θ . We first note that |∇t θ | = θ −1 and hence, By Lemma 3.8, w = w * a.e. in Ω \ W and in particular, w = w * a.e. in D θ . Therefore, for a.e. x ∈ D θ such that w ε (x) = w(x) and w ε,y * (x) = w * (x), we have u ε (x) = w * (x) ∈ N . Since the maps u ε are uniformly bounded, we deduce that
Step 4 (Proof of (3.35)). Let us take a larger, bounded domain Ω ⊃⊃ Ω and a map V ∈ (L ∞ ∩ W 1,k )(Ω \ Ω, R m ) with trace v on ∂Ω. We definẽ Since the traces of u ε , w agree with that of V on ∂Ω, F Ω (S y (ũ ε ) − S y (w)) dy → 0 as ε → 0.  Proof of Proposition D. Let S be an n-dimensional relative boundary of finite mass -that is, S has the form S = (∂R) Ω, where R is an (n + 1)-chain of finite mass such that M(∂R) < +∞. By a density argument, we can assume without loss of generality that R is polyhedral. By Proposition 3.7, we can also assume that ∂R takes its multiplicities in the set S ⊆ π k−1 (N ) defined by (2.4). Finally, by translating the support of R and applying Thom's transversality theorem, we can assume that (3.49) (∂R) ∂Ω = 0.
Let w * ∈ N be a constant, and let η > 0 be a small parameter. We repeat the same arguments of Lemma 3.8 and modify the constant map w * in a neighbourhood of spt R. We obtain a new map w : R n+k → N that (i) has a nice singularity at (spt(∂R), (spt(∂R)) n−1 ); (ii) satisfies S(w) = S(w * ) + ∂R = ∂R; By the same computations as in Lemma 3.10, we obtain that w ε → w strongly in W 1,k−1 (R n+k ) and that where Ω ⊃⊃ Ω is any open set, with polyhedral boundary, such that ∂Ω is transverse to spt(∂R).

A local compactness result
The aim of this section is to prove Statement (i) of Theorem C. As an intermediate step, we will prove the following result, which is a local version of Theorem C.(i). We recall that we have fixed a number δ * ∈ (0, dist(N , X )) and that B * := B m (0, δ * ) ⊆ R m .

Then, there exist a (non-relabelled) subsequence and a finite-mass chain
(F U is the relative flat norm, see (2.5)).
Throughout this section, we fix bounded domains U ⊂⊂ U ⊆ R n+k and a countable sequence (u ε ) in W 1,k (U , R m ) that satisfies (4.1). By an approximation argument, using the continuity of S (Proposition 2.3 and [17, Theorem 3.1]), we can assume without loss of generality that the maps u ε are smooth and bounded. For any ε > 0 and y ∈ B * , we define the measure Thanks to (P 2 ) (Proposition 2.3), µ ε,y is a bounded Radon measure for a.e. y.

Choice of a grid
As in [2], we define a grid G of size h > 0 as a collection of closed cubes of the form for some a ∈ R n+k . For j ∈ N, 0 ≤ j ≤ n+k, we denote by G j the collection of the (closed) j-cells of G , and we define the j-skeleton of G , R j := ∪ K∈G j K. We letR k be the union of all the cells K ∈ G k that are parallel to the k-plane spanned by {e n+1 , . . . , e n+k }. Given an open set V ⊆ U , we denote by R k (V ) the union of the k-cells , h), the grid will be called the dual grid of G . We will denote by G k the collections of k-cells of G and by R k its k-skeleton. For each K ∈ G k there exists a unique K ∈ G n , called the dual cell of K, such that K ∩ K = ∅. We are now going to construct a sequence of grids G ε with suitable properties. The construction is analogous to [2, Lemma 3.11]. Let us take a function h : (0, 1) → R + such that (4.6) ε α h(ε) |log ε| −1 for any α > 0, as ε → 0.

Lemma 4.2.
For any fixed parameter δ > 0 and any ε < 1 there exists a grid G ε of size h(ε) that satisfies the following properties: Here µ ε,y is the measure defined by (4.4).
Proof. We take a grid G ε := G (a, h(ε)) of the form (4.5). We claim that it is possible to choose a ∈ (0, h(ε)) n+k in such a way that (4.7)-(4.10) are satisfied. For (4.7)-(4.9), we can repeat verbatim the arguments in [2]. As for (4.10), let us call R ε k−1 (a) the (k − 1)-skeleton of the grid G (a, h(ε)). Thanks to (P 2 ) in Section 2, µ ε,y is a finite, non-negative Radon measure for a.e. y ∈ B * . By applying [25, Lemma 5.2], together with a scaling argument, we obtain for a.e y ∈ B * . By integrating the previous inequality with respect to y and applying (P 2 ), we obtain Now the lemma follows by an averaging argument, see e.g. [2, Lemma 8.4].
Throughout the rest of this section, we suppose that (4.6) is satisfied, we fix δ ∈ (0, 1) and we consider the sequence of grids G ε given by Lemma 4.2. Without loss of generality, we will also assume that is the union of the closed cubes K ∈ G ε such that K ∩ U = ∅). Lemma 4.3. For any α ∈ (0, k/(k 2 − k + 2)), there holds where C(δ, α) is a positive constant that only depends on N , k, f , n, δ and α.
Proof. We repeat the arguments of [2, Lemma 3.4]. Let d ε := dist(u ε , N ), let λ ∈ (0, 1/k) be a parameter, and let G : R + → R + be defined by G(t) := t 2λ/(k−kλ)+1 . Thanks to (H 3 ), we have d 2 ε f (u ε ). Therefore, by (4.9) and (4.11), we obtain (4.12) The Young inequality and the chain rule imply Since we have assumed that λ < 1/k, we have k − kλ > k − 1 and hence, for any (k − 1)cell K ⊆ R ε k−1 (U ), we can bound the oscillation of G • d ε on K by Sobolev embedding: The inverse G −1 of G is well-defined and Hölder continuous of exponent On the other hand, we can bound the integral average of d ε on K thanks to (4.12): Combining (4.14) with (4.15), and letting λ 1/k, the lemma follows.

A polyhedral approximation of S y (u ε )
Let y ∈ B * be fixed in such a way that S y (u ε ) has finite mass for any ε. (Thanks to (P 2 ), the set of y such that this property is not satisfied is negligible, because the sequence (u ε ) is assumed to be countable.) We are going to construct a polyhedral approximation of S y (u ε ), supported on the dual n-skeleton (R ε ) n of the grid. Thanks to Lemma 4.3, there exists ε 0 > 0 (depending on δ * , but not on y) such that for any x ∈ R ε k−1 (U ) and any ε ∈ (0, ε 0 ]. As a consequence, the projection (u ε − y) is welldefined and smooth on R ε k−1 (U ) for ε ∈ (0, ε 0 ]. For any K ∈ G ε k , let γ ε (K) ∈ π k−1 (N ) be the homotopy class of (u ε −y) on ∂K. The quantity γ ε (K) does not depend on the choice of y ∈ B * , because (u ε − y) |∂K and (u ε ) |∂K are homotopic to each other, due to (4.16); a homotopy is defined by (x, t) ∈ ∂K × [0, 1] → (u ε (x) − ty). We define the polyhedral chain where K ∈ (G ε ) n is the dual cell to K. The chain T ε depends on the choice of the grid, but not on y.
Proof. Essentially, this lemma is a particular instance of the Deformation Theorem for flat chains [26,Theorem 7.3] (see also [2,Lemma 3.8] for a statement which is specifically tailored for application to Ginzburg-Landau functionals). Nevertheless, we provide details for the convenience of the reader. Let ε ∈ (0, ε 0 ] be fixed. By [2, Lemma 3.8.(i)] there exists a locally Lipschitz retraction ζ ε : R n+k \ R ε k−1 → (R ε ) n , which maps each cube of G ε into itself and satisfies By (4.16), we have u ε (x) − y / ∈ X for any x ∈ R ε k−1 (U ). By construction (see [17,Section 3]), and let I be the 1-chain, with integer multiplicity, carried by the interval [0, 1] with positive orientation. We remark that (4.11). This implies .

(4.22)
To conclude the proof of (4.18), it suffices to show that ζ ε * (S y (u ε )) agrees with T ε inside U . By [26,Lemma 7.2], ζ ε * (S y (u ε )) U is a n-polyhedral chain of the grid (G ε ) ; in particular, its multiplicity is constant on every n-cell of (G ε ) . We want to compute such multiplicities. Let us take K ∈ G ε k and its dual cell K ∈ (G ε ) n , and let x be the unique element of K ∩ K . By construction of ζ ε (see [2, Lemma 3.8 and Figure 3.2]), we have (ζ ε ) −1 (x) = K \ ∂K. By Thom's parametric transversality theorem, we can assume with no loss of generality that K intersects transversally the support of S y (u ε ). Then, by definition of push-forward, we have multiplicity of ζ ε * (S y (u ε )) at x = I(S y (u ε ), (ζ ε ) −1 (x) ) = I(S y (u ε ), K ) and hence Now, (4.18) follows from (4.22) and (4.23). Moreover, (4.23) implies To bound the mass of T ε , we will use the following result.

Lemma 4.5. There exist positive numbers
Then, The proof of Lemma 4.5 will be given in Appendix C.
Proof. We first remark that Let K ∈ G ε k be a k-cell such that K ∩ U = ∅. We claim that (4.27) |γ Indeed, thanks to (P 0 ) and the definition of I (see e.g. [17, Section 2.1]), we have for any y ∈ B * . By averaging both sides with respect to y ∈ B * , and by applying (P 2 ) from Proposition 2.3, we obtain We can bound the right-hand side from above with the help of (4.8), so the claim (4.27) follows. From (4.1), (4.6) and (4.27), we deduce and this fact, together with (4.16), shows that the assumptions of Lemma 4.5 are satisfied for ε small enough. By applying Lemma 4.5, (4.6) and (4.27), we obtain the following bound: We multiply both sides by h(ε) n |log ε| −1 and sum over K. Thanks to (4.26), we obtain The right-hand side can now be bounded from above with the help of Lemma 4.2,so (4.24) follows. The proof of (4.25) is analougous; in this case, we sum over the cells K that are parallel to the k-plane spanned by {e n+1 , . . . , e n+k } and use (4.7).

Proof of Proposition 4.1
By combining the results in the previous section, we prove the following lemma, which is analougous to [2, Proposition 3.1]. For any n-plane L ⊆ R n+k , we denote by π L : R n+k → L the orthogonal projection onto L.

Lemma 4.7.
Let U ⊂⊂ U be bounded domains in R n+k . Let (u ε ) ε be a countable sequence of smooth, bounded maps that satisfy (4.1). Let L ⊆ R n+k be a n-plane. Then, there exist a (non-relabelled) subsequence and a finite-mass chain S ∈ M n (U ; π k−1 (N )) such that |log ε| . (4.29) Proof. Up to rotations we can assume without loss of generality that L is the k-plane spanned by {e n+1 , . . . , e n+k }. By Lemma 4.4 and Lemma 4.6, we know that ∂T ε U = 0 and M(T ε U ) is uniformly bounded with respect to ε. Then, by applying compactness results for the flat norm (see e.g. [17, Lemma 5 and 6] for a statement in terms of the relative flat norm), we find a (non-relabelled) subsequence and a finite-mass chain S ∈ M n (U ; π k−1 (N )) such that The triangle inequality and Lemma 4.4 implŷ and the right-hand side tends to zero as ε → 0, due to (4.1) and (4.30). Thus, (4.28) follows. By passing to the limit in (4.31) first as r → 0, then as δ → 0, we obtain (4.29). Now, Proposition 4.1 can be deduced from Lemma 4.7 by a localisation argument, with the help of the following lemma.
where the supremum is taken over all sequences of pairwise disjoint open sets U i and n-planes The proof will be given in Appendix D.

Compactness and lower bounds for the boundary value problem
The aim of this section is to complete the proof of Theorem C.(i). We will deduce Theorem C.(i) from its local counterpart, i.e. Proposition 4.1, with the help of the extension result, Lemma 3.3.
This condition, combined with (4.34), implies that S y (ũ) (Ω \ Ω) =S (Ω \ Ω) and hence, the chain S :=S − S y (ũ) (Ω \ Ω) =S Ω is supported in Ω. At the same time, we have S y (u ε ) = S y (ũ ε ) Ω for a.e. y ∈ B * . For chains supported in a compact subset of Ω , the relative flat norm F Ω is equivalent to F (see e.g. [17, Remark 2.2]) and hence, (4.35) implieŝ ) for any ε and a.e. y ∈ B * . The set C (Ω, v) is closed with respect to the F-norm (this follows from the isoperimetric inequality, see e.g. [26, Statement (7.6)]). Therefore, S ∈ C (Ω, v). It only remain to prove the upper bound on the mass of S. Let A ⊆ R n+k be an open set. We extract a (non-relabelled) subsequence, in such a way that lim inf ε→0 |log ε| −1 E ε (u ε , A ∩ Ω) is achieved as a limit. For any integer j ≥ 1, let A j := {x ∈ A : dist(x, ∂A) ≥ 1/j}. By applying Proposition 4.1 and a diagonal argument, we find a subsequence such that |log ε| for any j ≥ 1.
By construction, S is supported in Ω, so S (A j ∩ Ω ) = S A j . Then, by applying Lemma 3.3, we obtain for some constant C that does not depend on ε, j, Ω . Letting j → +∞, Ω Ω, we conclude that |log ε| and the proof is complete. Statement (i) in Proposition D also follows by Proposition 4.1, in a similar way.

Proof of Theorem A
Let u ε,min be a minimiser of the functional E ε subject to the boundary condition u = v on ∂Ω, and let µ ε,min : dx Ω |log ε| .
We have sup ε µ ε,min (R n+k ) < +∞ by Remark 3.4 and hence, up to a subsequence, µ ε,min converges weakly * to a limit µ min , in the sense of measures on R n+k . By applying Theorem C.(i), we find a chain S min ∈ C (Ω, v) such that for any open set A ⊆ R n+k .
Theorem C.(ii) implies that S min is mass-minimising in C (Ω, v). Moreover, by the properties of weak * convergence, from (5.1) we obtain for any open set A ⊆ R n+k such that µ min (∂A) = 0.
Let E ⊆ R n+k be a Borel set, let U ⊆ R n+k be an open set and let K ⊆ R n+k be a compact set such that K ⊆ E ⊆ U . For any t ∈ (0, dist(K, ∂U )), let U t := {x ∈ U : dist(x, ∂U ) > t} ⊇ K. Since µ min is a finite measure, we have µ min (∂U t ) = 0 for all but countably many t ∈ (0, dist(K, ∂U )). Therefore, there holds Letting U K, K E, we conclude that M(S min E) ≤ µ min (E). (The measure M(S min ·) is Radon, because by construction, it is the weak * limit of a sequence of Radon measures, associated with polyhedral approximations of S min ; see [26,Section 4].) As a consequence, µ min −M(S min ·) is a non-negative measure. However, Theorem C.(ii) implies that µ min (R n+k ) = lim ε→0 µ ε,min (R n+k ) ≤ M(S min ), so µ min = M(S min ·). The aim of this section is to prove Proposition 2.1. In Section 2, we have defined for any σ ∈ π k−1 (N ), with ∇ the tangential gradient on S k−1 (i.e. the restriction of ∇ to the tangent plane to S k−1 ). The compact Sobolev embedding W 1,k (S k−1 , N ) → C(S k−1 , N ) implies that W 1,k (S k−1 , N ) ∩ σ is sequentially W 1,k -weakly closed, so the infimum at the righthand side is achieved. We must have for otherwise there would exist a sequence of non-null-homotopic maps v j ∈ W 1,k (S k−1 , N ) that converge W 1,k -strongly, and hence uniformly, to a constant. Moreover, there holds Indeed, for any v ∈ W 1,k (S k−1 , N ) ∩ σ and any x ∈ S k−1 , definev(x) := v(−x 1 , x 2 , . . . , x k ). The map that sends v →v is a bijection W 1,k (S k−1 , N ) ∩ σ → W 1,k (S k−1 , N ) ∩ (−σ) that preserves the L k -norm of the gradient, and hence (A.3) follows. Our candidate norm | · | * on π k−1 (N ), which was also introduced in Section 2, is defined for any σ ∈ π k−1 (N ) by Proposition A.1. The function | · | * is a norm on π k−1 (N ) that satisfies The infimum in (A.5) is achieved, for any σ ∈ π k−1 (N ). Moreover, the set is finite, and for any σ ∈ π k−1 (N ) there exists a decomposition σ = q i=1 σ i such that |σ| * = q i=1 |σ i | * and σ i ∈ S for any i.
Proof. The function | · | * is certainly non-negative, and its definition (A.4) immediately implies the triangle inequality, |σ 1 + σ 2 | * ≤ |σ 1 | * + |σ 2 | * . The property |σ| * = | − σ| * follows by (A.3), while (A.2) yields (A.5) (in particular, |σ| * = 0 only if σ = 0). The property (A.6) is immediate from the definition of | · | * . We check now that the set S is finite. Under the assumption (H 2 ), Hurewicz theorem (see e.g. [31, Theorem 4.37 p. 371]) implies that π k−1 (N ) is isomorphic to the homology group H k−1 (N ). The latter is Abelian and finitely generated, because the manifold N is compact and hence, homotopically equivalent to a finite cell complex. Therefore, we have where p ≥ 0 is an integer and T is a finite group. Let (g i ) p i=1 be a basis for the torsion-free part of H k−1 (N ) (i.e., the quotient H k−1 (N )/T Z p ). By de Rham theorem, there exist closed, smooth (k − 1)-forms ω 1 , . . . , ω p that satisfŷ where c i is a smooth (k − 1)-cycle in the homology class g i . Let σ ∈ π k−1 (N ). By abusing of notation, and identifying g i with its image under the Hurewicz isomorphism, we can write uniquely where d i ∈ Z and σ T ∈ T . Then, for any v ∈ W 1,k (S k−1 , N ) ∩ σ, we have where C k, N > 0 is a constant depending only on k and the ω i 's. This implies On the other hand, the definition of | · | * immediately gives the upper bound If σ ∈ S then, by comparing (A.8) and (A.9), we obtain i |d i | ≤ M for some constant M > 0 depending only on k, N . Therefore, S is a finite set. For any σ ∈ π k−1 (N ) there exists a finite decomposition σ = q i=1 σ i which achieves the infimum in the right-hand side of (A.4). Indeed, it suffices to minimise among the decompositions with q ≤ (inf g∈π k−1 (N )\{0} E min (g)) −1 E min (σ) and E min (σ i ) ≤ E min (σ) for any i, and there are only finitely many such decompositions because of (A.2), (A.8). Let σ = q i=1 σ i be a decomposition that achieves the minimum in (A.4). Then, the triangle inequality implies and, since |σ i | * ≤ E min (σ i ) for any i, we must have |σ i | * = E min (σ i ), i.e. σ i ∈ S, for any i.
Z and E min (d) = πd 2 for any d ∈ Z, since the infimum in (A.1) is achieved by a curve that parametrises the unit circle |d| times, with constant speed and orientation depending on the sign of d. Therefore, S = {−1, 0, 1} and |d| * = π |d| for any d ∈ Z.
More generally, when N = S k−1 the constant that appears in the lower bound (A.8) can be computed explicitely, and we have . On the other hand, by using the identity as a comparison map for (A.1), we see that Therefore, also in case N = S k−1 we have S = {−1, 0, 1}. By Proposition 2.1, we conclude that |d| * = β k |d| for any d ∈ π k−1 (S k−1 ) Z.

B The operator S: Proof of Proposition 2.3
The aim of this section is to prove Proposition 2.3, which we recall here for the convenience of the reader. We recall that δ * ∈ (0, dist(N , X )) is a fixed constant, B * := B m (0, δ * ) ⊆ R m , and Y := L 1 (B * , F n (Ω; π k−1 (N ))) is equipped with the norm (P 1 ) For any u ∈ (L ∞ ∩ W 1,k )(Ω, R m ) and a.e y ∈ B * , S y (u) = S y (u) (more precisely, the chain S y (u) belongs to the equivalence class S y (u) ∈ F n (Ω; π k−1 (N ))).
Proof of Proposition B.1. For the sake of clarity, we split the proof into steps.
Step 1 (Construction of S). First, we consider a smooth map u ∈ C ∞ c (R n+k , R m ) and the topological singular operator, S y (u), as defined in [17,Section 3.2,Eq. (3.4)]. By definition, we can write Here, the sum is taken over all (m − k)-dimensional polyhedra K in X . The coefficient γ(K) ∈ π k−1 (N ) is the homotopy class of restricted to a small (k − 1)-sphere Σ around K, |Σ : Σ We claim that, for any u, u 0 , u 1 ∈ C ∞ c (R n+k , R m ) and any open set U ⊆ R n+k , there holdŝ These inequalities differ from the corresponding ones in [17,Theorem 3.1] because the multiplicative constants in front of the right-hand sides do not depend on the L ∞ -norm of u, u 0 , u 1 . We postpone the proof of (B.3)-(B.4). As a consequence of (B.4), by a density argument we can extend S to a continuous operator W 1,k (R n+k , R m ) → L 1 (B * , F(R n+k , π k−1 (N ))), still denoted S for simplicity. The property (B.3) is preserved for any u ∈ W 1,k (R n+k , R m ), by the lower semi-continuity of M (see e.g. [17,Lemma 3 and Lemma 5]).
Since the domain Ω ⊆ R n+k is bounded and smooth, by reflection about ∂Ω and multiplication with a cut-off function we can define a linear extension operator T : For any u ∈ W 1,k (Ω, R m ) and a.e. y ∈ B * , the chain S y (T u) has finite mass, due to (B.3). Therefore, the restriction S y (u) := S y (T u) Ω is well-defined and belongs to M n (Ω; π k−1 (N )).
Step 3 (Proof of (P 1 )). By construction, S y (u) = S y (u) for any u ∈ C ∞ c (R n+k , R m ) and a.e. y ∈ B * . By continuity of both S and S [17, Theorem 3.1], we deduce that S = S on (L ∞ ∩ W 1,k )(Ω, R m ).
Step 4 (Proof of (P 2 )). Let E ⊆ Ω be a Borel set and U ⊇ E be open. By (B.3), we havê and (P 2 ) follows by letting U E.
Step 6 (Proof of (B.3)). Let u ∈ C ∞ c (R n+k , R m ) and let E ⊆ R n+k be a Borel set. Since X contains finitely many (m − k)-cells K, there exists a constant C such that |γ(K)| * ≤ C for any K. Then, using the definition (B.2) of S y (u), we deduce where the sum is taken over all the (m − k)-dimensional polyhedra K in X . We fix K and assume, without loss of generality, that K ⊆ {y ∈ R m : y 1 = . . . = y k = 0} R m−k . Let ζ ⊥ be the orthogonal projection R m → {y ∈ R m : y m−k+1 = . . . = y m = 0} R k . Then, If we integrate this inequality over y ∈ B * , and use the variable y = (z, z ⊥ ) ∈ R m−k × R m , we obtain The right-hand side can be estimated by applying the coarea formula: Combining (B.8) and (B.9), (P 2 ) follows.

C Energy lower bounds when n = 0
The aim of this section is to prove energy lower bounds in the critical dimension, i.e. when n = 0. In the contest of the Ginzburg-Landau theory, i.e. when N = S k−1 , energy bounds of this type were proved by Jerrard [38] and, in case k = 2, by Sandier [50].
Then, we can define the homotopy class of u (or, more precisely, of • u) on ∂Ω as an element of π k−1 (N ). This is immediate in case u is continuous on ∂Ω and Ω is homeomorphic to a disk. If Ω has not the topology of a disk, this is still possible due to the Hurewicz isomorphism π k−1 (N ) H k−1 (N ), which holds true thanks to (H 2 ) (see e.g. [31, Theorem 4.37 p. 371] and (C.10) below). If u is not continuous we can define its homotopy class by approximating •u with smooth functions Ω → N , as in [14] (see also (C.10) below for more details).

Proposition C.1.
Let Ω ⊆ R k be a bounded, Lipschitz domain and let r > 0. There exist a number δ 0 > 0, depending only on N , and positive constants ε 0 , M , depending only on Ω, r, N , k and f , such that the following statement holds. Suppose that u ∈ W 1,k (Ω, R m ) satisfies (C.1), and let σ ∈ π k−1 (N ) be the homotopy class of u on ∂Ω. Let ε ∈ (0, 1/2) be such that ε |log ε| |σ| * ≤ ε 0 . Then, The aim of this section is to prove Proposition C.1. Once Proposition C.1 is proved, Proposition B follows by an extension argument in a neighbourhood of ∂Ω (see e.g. [10, Theorem 2]). Lemma 4.5 also follows from Proposition C.1, by exactly the same arguments as in [2, Lemma 3.10].

C.1 Reduction to the cone-valued case
For the purposes of this section, it will be convenient to consider the nearest-point projection onto N . If z ∈ R m is sufficiently close to N , there exists a unique π(z) ∈ N such that |z − π(z)| ≤ |z − w| for any w ∈ N . Moreover, the map z → π(z) is a smooth in a neighbourhood of N . Throughout the rest of the section, we fix a small parameter θ 0 and assume that π is well-defined and smooth in a θ 0 -neighbourhood of N .
Lemma C.2. If u : Ω → R m is a smooth map that satisfies dist(u(x), N ) < θ 0 for any x ∈ Ω and if d := dist(u, N ), then there holds where C 1 , C 2 are positive constants that only depend on N .
Proof. Let x 0 ∈ Ω be arbitrarily fixed. Let ν 1 , . . . , ν p be a smooth orthonormal frame for the normal space to N , locally defined in a neighbourhood of (π • u)(x 0 ). Then, for each x in a neighbourhood of x 0 there exist numbers α 1 (x), . . . , α p (x) such that The functions α i are as regular as u. By differentiating this equation, raising both sides to the square, using the fact that ∇(π • u) is tangent to N and that (∇ν i )ν i = 0, we obtain Since N is smooth and compact, we have |∇ν i | ≤ C for some constant C that only depends on N and not on u. Therefore, setting d := dist(u, N ) = ( i α 2 i ) 1/2 , we obtain On the other hand, by differentiating the identity d = ( i α 2 i ) 1/2 , we see that By combining (C.2) and (C.3), the lemma follows. (ii) φ(y) = 0 if dist(y, N ) ≥ θ 0 , and in particular π(y) is well-defined for any y ∈ R m such that φ(y) > 0; (iii) for any u ∈ W 1,k (Ω, R m ), there holds pointwise a.e. on Ω.

C.2 Proof of Proposition C.1
Throughout this section, we fix a bounded, smooth map u : Ω → R m and we let s := φ • u, v := π • u, where φ is the function given by Lemma C.3 and π is the nearest-point projection onto N . Thanks to Lemma C.3, in order to provide lower bounds for E ε (u) it suffices to bound from below the functional To this end, we adapt Jerrard's approach in [38]. We explain here the main steps of the construction and point out the differences, referring the reader to [38] for more details.
Let us fix a small number η 0 , such that Let V ⊆ Ω be an open set such that s > 0 on ∂V . By Lemma C.3, we have dist(u(x), N ) < θ 0 for any x ∈ ∂V . Therefore, by (C.8), we have spt(S y (u))∩∂V = ∅ for a.e. y ∈ R m such that |y| ≤ η 0 . In fact, S y (u) is a 0-chain of finite mass, and hence we can write where σ i ∈ π k−1 (N ) and x i ∈ V . The quantity I(S y (u), V ) := q i=1 σ i ∈ π k−1 (N ) plays the rôle of the topological degree and indeed, it coincides with the homotopy class of π • u on ∂V because of Proposition B.1.(P 0 ) and (C.8) (see [17,Section 2.4 and Theorem 3.1] for the details on the case u is not smooth). In particular, I(S y (u), V ) is independent of the choice of y.
As in [38], we define an "approximate homotopy class", which allows us to disregard sets where s is small but u does not carry topological obstruction. We let S := {x ∈ Ω : s(x) ≤ 1/2}. The componentsS of S are closed sets and it is possible to define I(S y (u), S ) as above. We define the "essential part" of S: where C 0 > 0, C 1 > 0 and N > 1 are parameters that we will need to choose, depending on k, α and β. It can be shown (see [38, Theorem 2.1, proof of (2. 2)]) that where C > 0 only depends on k, C 0 and ν := 1/(N − 1) > 0. As a consequence, λ ε (ρ) ≥ C 1 /ε if ρ ≥ C 2 ε, for some constant C 2 > 0 depending on k, C 0 , C 1 and N . Therefore, after integration we obtain that Proof. First of all, given ρ > 0 and a map v ∈ W 1,k (∂B k ρ , N ) in the homotopy class σ ∈ π k−1 (N ), there holds where E min (σ) is defined by (A.1). This inequality follows immediately from the definition of E min (σ), combined with a scaling argument. Now, for any ρ ≥ ε > 0 and any smooth u : ∂B k ρ → R m such that µ := min ∂B k ρ φ • u > 1/2, there holds Here G ε is defined by (C.7), s := φ • u, v := π • u, and σ denotes the homotopy class of v on ∂B k ρ . The constants C 0 , N are suitably chosen at this stage. The proof of this claim follow by repeating, almost word by word, the arguments in [38, Theorem 2.1]; the only difference is that we need to apply (C.12) instead of [38, Lemma 2.4]. Due to (A.6), we obtain On the other hand, in case 0 ≤ µ ≤ 1/2, [38, Lemma 2.3] implies that (C.14) G ε (s, v; ∂B k ρ ) ≥ C 1 ε for some C 1 > 0 that depends on k, C 0 and N . Therefore, by integrating the inequalities (C.13)-(C.14) with respect to ρ, we deduce that and, thanks to Lemma C.3, the lemma follows.
We also have an analogue of [38,Proposition 3.3].
Lemma C.5. Suppose that u ∈ W 1,k (Ω, R m ) is smooth and that S E ⊂⊂ Ω. Then, there exists a finite collection of closed, pairwise disjoint balls Proof. We claim that, ifS is a connected component of S E such thatS ⊂⊂ Ω, then This inequality parallels [38, Lemma 3.2]; once (C.15) is established, the rest of the proof follows exactly as in [38]. Lemma C.5 and the definition of Λ ε imply that The last step in the proof of Proposition C.1 is the so-called "ball construction" [38, Proposition 4.1]. If u satisfies (C.1) for some r > 0 then, by choosing δ 0 = δ 0 (N ) < θ 0 sufficiently small, we obtain as a consequence (C. 18) |s(x)| ≥ 1 2 for any x ∈ Ω such that dist(x, ∂Ω) < r.
Moreover, we can assume without loss of generality that u satisfies for some ε-independent constant C, for otherwise Proposition C.1 holds trivially.

D.1 Approximation results for flat chains
We give the proof of the approximation results, Proposition 3.6 and 3.7, we have used in Section 3.4.2. For convenience, we recall the statements here. Let S ⊆ G be a set of generators for G. We assume that, for any g ∈ G, there exist g 1 , . . . , g p ∈ S such that The set defined by (A.7) satisfies this assumption, by Proposition 2.1. (i) S + ∂R j → S, with respect to the F-norm, as j → +∞; (ii) M(S + ∂R j ) → M(S) as j → +∞; (iii) for any j, (S + ∂R j ) ∂W S = 0; (iv) for any j, the chain (S + ∂R j ) W S takes multiplicities in the set S ⊆ π k−1 (N ) defined by (2.4).
Proof. Since ∂W S is transverse to spt S, the intersection spt S ∩ ∂W S has dimension n − 1 at most and hence, S ∂W S = 0. By triangulating S, we can write S W S as a finite sum where σ K ∈ π k−1 (N ) and the K's are closed n-simplices, whose interiors are contained in W S and pairwise disjoint. We fix positive parameters δ, γ and, for any n-simplex K of S W S , we consider the set U (K, δ, γ) defined by (3.1). We choose δ, γ small enough, so that the interiors of the U (K, δ, γ)'s are pairwise disjoint and contained in W S . By assumption (D.2), we can Take distinct vectors y K,1 , . . . , y K,p ∈ R n+k that are orthogonal to K and satisfy |y K,1 | = . . .  The chain R j , in case n = 1, k = 2 and S consists of a segment only, S = σ K K . The chain S is in black, R j is in gray, and S + ∂R j is in red.
For any integer j ≥ 1, we define Figure 7). The chain R j is polyhedral, because the h K,i 's are piecewise affine, and supported in W S . The support of R j may intersect ∂W S only along its (n−1)-skeleton, so (∂R j ) ∂W S = 0. We compute the mass of R j . Since the maps h K,i are Lipschitz, and their Lipschitz constant only depends on γ, which is fixed, the area formula implies M(R j ) Thus, (i) follows. Now, we compute the boundary of R j . For each simplex K and each i, we have h K,i (t, x ) = x if x ∈ ∂K and h K,i (0, x ) = x for any x ∈ K. As a consequence, By multiplying this identity by σ K,i , and taking the sum over i, K, we obtain In particular, S W S + ∂R j = (S + ∂R j ) W S takes multiplicities in S. Finally, by applying the area formula to (D.4), we deduce M(S W S + ∂R j ) →  (iii) for any j, we can write S j = S 0 + ∂R j for some finite-mass (n + 1)-chain R j with compact support in Ω.
Proof. By assumption, there exists R ∈ M n (Ω; G) such that S = S 0 +∂R. Thanks to Lemma D.4 and a diagonal argument, we can assume without loss of generality that R is compactly supported in Ω. For any positive t, let Ω t := {x ∈ Ω : dist(x, ∂Ω) > t}. We take a positive number t 0 such that spt R ⊆ Ω 2t 0 , and an open set U , with polyhedral boundary, such that Ω 2t 0 ⊂⊂ U ⊂⊂ Ω t 0 . Because S and S 0 differ by a boundary, we have ∂(S U ) + ∂(S (R n+k \ U )) = ∂S = ∂S 0 = ∂(S 0 U ) + ∂(S 0 (R n+k \ U )).
However, S and S 0 agree out of U , so ∂(S U ) = ∂(S 0 U ). In particular, since S 0 is locally polyhedral in Ω and U is polyhedral, ∂(S U ) is a polyhedral chain. Thanks to, e.g., [26,Theorem 5.6 and 7.7], there exists a sequence of polyhedral n-chains T j that F-converges to S U , satisfies spt T j ⊆ Ω t 0 for any j and We do not know a priori whether the chains P j , Q j are supported in Ω, so we perform a truncation argument. Define P j,t := (∂P j ) Ω t − ∂(P j Ω t ) for t ∈ (0, t 0 ) and j ∈ N. By applying Fatou's lemma and [26,Theorem 5.7], we obtain that = 0.
Moreover, by taking the boundary of both sides of (D.8), we deduce that The right hand side converges to zero as j → +∞, due to (D.5) and (D.11). Thus, we deduce that lim sup j→+∞ M(S j ) ≤ M(S), and hence M(S j ) → M(S) as j → +∞. Finally, we define R j := R − P j − C j . Then, (D.13) gives S j − S 0 = ∂R j and the lemma follows.
Proof of Proposition D.2. Let S 0 , S be given, as in the statement. By applying Lemma D.5, we find a sequence of locally polyhedral chains S j ∈ M n (Ω; G) and a sequence of finite-mass (n + 1)-chainsR j , compactly supported in Ω, such that S j → S in the F-norm, M(S j ) → M(S) and S j = S 0 + ∂R j for any j. Since S 0 , S j are locally polyhedral in Ω, ∂R j is polyhedral. We apply Lemma D.3 to eachR j . We find polyhedral (n + 1)-chains R j , compactly supported in Ω, and (n + 2)-chains C j of finite mass, such that Then, S j = S 0 + ∂(R j + ∂C j ) = S 0 + ∂R j , and the proposition follows.

D.2 A characterisation of the mass of a rectifiable chain
For any linear subspace L ⊆ R n+k , we let π L : R n+k → L be the orthogonal projection onto L. A n-chain of class C 1 is a chain S that can be written in the form S = f * P , with f a map of class C 1 and P a polyhedral chain. The set of rectifiable n-chains is defined as the closure of n-chains of class C 1 with respect to the M-norm.
Lemma D.6. Let S ∈ M n (R n+k ; G) be a rectifiable n-chain. Then, where the supremum is taken over all sequences of pairwise disjoint open sets U i and n-planes L i ⊆ R n+k .
Proof. Let (U i ) i∈N be a sequence of pairwise disjoint open sets, and let (L i ) i∈N be a sequence of n-planes in R n+k . For any i, the projection π L i is a 1-Lipschitz map and hence M(π L i , * (S U i )) ≤ M(S U ) (see e.g. [26, Eq. (5.1)]). Since the U i 's are assumed to be pairwise disjoint, we obtain (D.14) This proves one of the inequalities. To prove the opposite inequality, we first suppose that S is a C 1 -polyhedron, then a C 1 -chain, and finally we extend the result to an arbitrary rectifiable chain. We denote by int A the interior of a set A ⊆ R n+k , and by diam A its diameter.
Step 1 (S is a C 1 -polyhedron). We suppose that S = f * (σ K ), where σ ∈ G, K is a convex, compact n-polyhedra, and f : R n+k → R n+k is a C 1 -diffeomorphism. Let η > 0 be arbitrarily fixed. Since f is C 1 and K is compact, there exists ρ > 0 such that (D. 15) ∇f (x) − ∇f (y) ≤ η if (x, y) ∈ K × K and |x − y| ≤ ρ, where · denotes the operator norm on the space of real (n + k) × (n + k)-matrices. Let (T i ) q i=1 be a collection of n-simplices that triangulate K, such that Let V i := int U (T i , ρ/2, ρ/2) where U (T i , ρ/2, ρ/2) is defined as in (3.1), and U i := f (V i ). The U i 's are pairwise disjoint open sets, because f is a diffeomorpism. Let L be the n-plane passing through the origin that is parallel to K. For any i, we choose a point x i ∈ int T i and we define L i := ∇f (x i )(L). The L i 's are indeed n-planes, because ∇f (x i ) is an invertible linear map. For any x ∈ K ∩ T i and any y ∈ L, we have Therefore, by applying the area formula we obtain for some constant C depending only on n, k. This implies where η is arbitrarily small. To complete the proof in this case, it only remains to notice that M(S) = M(S ∪ i U i ), because H n (K \ ∪ i V i ) = 0 and H n (S \ ∪ i U i ) = 0 by the area formula.
Step 2 (S is a C 1 -chain). We suppose that S = f * P , where P is a polyhedral n-chain and f : R n+k → R n+k is a C 1 -diffeomorphism. This case follows easily from the previous one, by additivity. Indeed, let us write S = p j=1 σ j K j with σ j ∈ G, K j a convex, compact npolyhedra. Given positive parameters δ, γ, let W j := f (int U (K j , δ, γ)). For δ, γ small enough, the W j have pairwise disjoint interiors. Let η > 0 be fixed. By applying Step 1, for any j we find a sequence (V j i ) i∈N of pairwise disjoint open sets and a sequence (L j i ) i∈N of n-planes such that (D.17) +∞ i=0 M(π L j i , * (f * (σ j K j ) U j i )) ≥ M(f * (σ j K j )) − η.
We define U j i := V j i ∩ W j . The U j i 's are pairwise disjoint open sets. We have H n (K j \ int U (K j , δ, γ)) = 0 and hence, by the area formula, M(f * (σ j K j ) (R n+k \W j )) = 0. Therefore, we obtain 17) ≥ M(S) − pη and the lemma is proved also in this case, because η may be taken arbitrarily small.
Step 3 (S is a rectifiable chain). Since S is rectifiable, for any η > 0 there exist a polyhedral n-chain P and a C 1 -diffeomorphism f : R n+k → R n+k such that  Let us set Q := S − f * P . Then, using the linearity of π L i , * , · U i , and the triangle inequality for M, we obtain