Pull-back of metric currents and homological boundedness of BLD-elliptic spaces

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping $f\colon X\to Y$ between oriented cohomology manifolds $X$ and $Y$ induces a pull-back operator $f^\ast \colon M_{k,loc}(Y) \to M_{k,loc}(X)$ between the spaces of metric $k$-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward $f_\ast \colon M_{k,loc}(X)\to M_{k,loc}(Y)$. As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology $n$-manifolds $X$ admitting a BLD-mapping $\mathbb{R}^n \to X$.


Introduction
In [4], Ambrosio and Kirchheim extended the Federer-Fleming theory of currents to general metric spaces by viewing currents as multilinear functionals acting on tuples of Lipschitz functions instead of on di erential forms; see also Lang [17] for a localized theory on locally compact spaces. Under this formalism, a locally Lipschitz map f : X → Y between metric spaces gives rise to a natural push-forward operator f * : M k (X) → M k (Y), given by f * T(π , . . . , π k ) = T(π • f , . . . , π k • f ), (π , . . . , π k ) ∈ LIP∞(X) k+ , (1.1) between the spaces of nite mass k-currents M k (X) and M k (Y), respectively. In this article we develop a pull-back of metric currents for a special class of Lipschitz maps, called BLD-maps, between locally geodesic oriented cohomology manifolds. A continuous, open and discrete map f : X → Y between metric spaces is an L-BLD map, for L ≥ , if it satis es the bounded length distortion estimate for each path γ in X, where (·) is the length of a path. We call a map f : X → Y simply a BLD-map if it is L-BLD for some L ≥ . Condition (1.2) may be regarded as a locally non-injective variant of the bi-Lipschitz condition. Indeed, every bi-Lipschitz bijection is a BLD-map. Note that, since the spaces we consider are locally geodesic, Condition (1.2) is not vacuous. Heuristically, the pull-back is a local left inverse of the push-forward; see Theorem 1.1 below. We construct it as an adjoint of a push-forward of polylipschitz forms introduced in [24], which give a pre-dual for metric currents. The construction relies on two key properties of BLD-maps between oriented cohomology manifolds.
A comment on the assumptions in Theorem 1.5 is in order. We assume that the target space X is locally Lipschitz contractible, that is, we assume that for every point x ∈ X and a neighborhood U of x there is a neighborhood V ⊂ U of x and a Lipschitz map h : V × [ , ] → U for which h is the inclusion V → U and h is a constant map. Clearly, Riemannian manifolds in the theorem of Bonk and Heinonen are locally Lipschitz contractible. In the proof of Theorem 1.5 this assumption yields an a priori nite dimensionality for the current homology H * (X), which in turn allows us to obtain lling inequality (Proposition 6.3) for normal currents on X.
A more commonly used assumption is local linear contractibility, cf. [13]. Local Lipschitz contractibility does not imply local linear contractibility, nor is it implied by it. However, if X satis es the hypotheses of Theorem 1.5 and is locally linearly contractible, it is a generalized manifold of type A in the terminology of Heinonen and Rickman; see [13,De nition 5.1]. Indeed, local Ahlfors-regularity follows from the work of Heinonen-Rickman [13] and is discussed in Section 5.1 (see Remark 5.1). The remaining condition, local bi-Lipschitz embeddability into some Euclidean space, follows from the work of Almgren, see [3] and De Lellis-Spadaro [7]. We give the details in Appendix A; see Theorem A.1.
This article is organized as follows. In Sections 2 and 3, we discuss preliminaries on metric currents and polylipschitz forms, and BLD-mappings, respectively. Section 4 is devoted to the pull-back of metric currents under BLD-maps and we prove Theorems 1.1 and 1.2 in this section. In Section 5 we prove equidistribution of the pull-back currents under BLD-maps and in Section 6 we discuss current homology and prove Theorem 1.5. The article is concluded with an appendix on local bilipschitz embeddability of BLD-elliptic spaces into Euclidean spaces.

Metric currents and polylipschitz forms
In this section, we recall rst basic notions from the Ambrosio-Kirchheim theory of metric currents [4] and then brie y discuss the construction of polylipschitz forms introduced in [24].

. Metric currents
Let X be a locally compact metric space. A function f : We denote by LIP∞(X) and LIPc(X) the vector spaces of bounded Lipschitz functions and Lipschitz functions with compact support, respectively. We equip LIP∞(X) and LIPc(X) with locally convex vector topologies such that (1) fn → f in LIP∞(X) if fn → f pointwise and sup n LIP(fn) < ∞; and (2) fn → f in LIPc(X) if there is a compact set K ⊂ X for which spt(fn) ⊂ K for all n ∈ N, and fn → f in LIP∞(X).
The vector space of metric k-currents is denoted D k (X).

Mass
A k-current T ∈ D k (X) is said to have locally nite mass, if there is a Radon measure µ on X satisfying |T(π , . . . , π k )| ≤ Lip(π ) · · · Lip(π k ) X |π |dµ, (π , . . . , π k ) ∈ D k (X) (2.1) for every (π , . . . , π k ) ∈ D k (X). If T ∈ D k (X) has locally nite mass, it admits a mass measure, denoted T , a Radon measure on X that is minimal with respect to satisfying (2.1). If T (X) < ∞, we say that T has nite mass. The space of k-currents of locally nite mass is denoted by M k,loc (X), and the space of k-currents of nite mass M k (X).

Normal currents
A k-current T ∈ D k (X) is called locally normal, if T ∈ M k,loc (X) and ∂T ∈ M k− ,loc (X), and normal if T ∈ M k (X) and ∂T ∈ M k− (X). The normal mass N : is a norm on N k (X) and the normed space (N k (X), N) is a Banach space; see [17,Proposition 4.2]. For k = , the norm N is the total variation norm.

Flat norm
Let E ⊂ X be a Borel set and let F E : N k,loc (X) → [ , ∞] be the function For each non-empty Borel set E, F E is a seminorm and F := F X is a norm on N k (X), called the at norm of N k (X). We recall that, for each T ∈ N k,loc (X) and a Borel set E ⊂ X, We record standard properties of the at norm of a restriction of a current as a lemma.
Lemma 2.1. Let T ∈ N k,loc (X) and E ⊂ X a Borel set. Then for all Lipschitz functions η ∈ LIP∞(X). Moreover, if η ∈ LIP∞(X) satis es η| E ≡ , then Proof. Let A ∈ N k,loc (X) and B ∈ N k+ ,loc (X) satisfy T = A + ∂B. Then, for any η ∈ LIPc(X) we have Thus, by (2.1), we have The rst claim follows. For the second claim, note that and the second claim follows.
The following compactness result for the at norm provides a crucial tool in the proof of homological boundedness. This result is used for currents in R n and it is an immediate consequence of [9, Corollary 7.3] and the weak compactness of normal currents [17,Theorem 5.4]. For an analogous compactness result in compact metric spaces, see [8].
Theorem 2.2. Let A ⊂ R n be a compact subset and λ ≥ . Then the set . Polylipschitz forms

Polylipschitz functions
Let k ∈ N and X be a metric space. Given functions f , . . . , f k : Note that if each f j is Lipschitz and bounded, then f ⊗ · · · ⊗ f k is Lipschitz and bounded on X k+ (here we endow X k+ with the Euclidean product metric). The norm L(f ) on LIP∞(X), given by makes LIP∞(X) into a Banach space.
Consider the algebraic tensor product LIP∞(X) ⊗(k+ ) . The projective tensor norm on LIP∞(X) ⊗(k+ ) is given by The completion of LIP∞(X) ⊗(k+ ) with respect to the projective tensor norm is called the (completed) projective tensor product and denoted LIP∞(X)⊗ π (k+ ) . The projective tensor product has the following universal property which characterizes it up to isometric isomorphism in the category of Banach spaces: Let B be a Banach space and A : LIP∞(X) k+ → B a continuous (k + )-linear map. Then there exists a unique continuous linear map A : LIP∞(X)⊗ π (k+ ) → B satisfying where ȷ : LIP∞(X) k+ → LIP∞(X)⊗ π (k+ ) is the continuous (k + )-linear map (π , . . . , π k ) → π ⊗ · · · ⊗ π k . In particular, the map extends to a continuous linear mapĀ we identify the projective tensor product with the image of this map in LIP∞(X k+ ).

De nition 2.3. Let X be a metric space and k
In other words a polylipschitz function is an element of the completed projective tensor product under the identi cation explained above.

Polylipschitz forms
Fix a metric space X. Given open sets U ⊂ V ⊂ X we denote by the restriction map. The collection ranging over all open sets U ⊂ V ⊂ X is known as the (polylipschitz) presheave over X. Given x ∈ X and two polylipschitz functions π ∈ Poly k (U), π ∈ Poly k (U ) de ned on open neighbourhoods U and U of x, respectively, we say that π and π are equivalent, denoted π ∼ π , if there is a neighbourhood W ⊂ U ∩ U such that ρ W ,U (π) = ρ W ,U (π ).
The equivalence class [π]x of a polylipschitz π ∈ Poly k (U) de ned on a neighbourhood U of x is called the germ of π on x.
The étalé space Poly k (X) consists over all such equivalence classes. There is a natural projection map For each x ∈ X, the set is called the stalk of Poly k (X) at x, and it is a real vector space.
A k-polylipschitz section on X is a section of Poly k (X), i.e. a map ω : X → Poly k (X) satisfying q•ω = id X . We denote the space of k-polylipschitz sections on X by G k (X). The support of a k-polylipschitz section ω ∈ G k (X) is the set spt ω = cl{x ∈ X : ω(x) ≠ }.
The space Poly k (X) can be equipped with the étalé topology which makes q into a local homeomorphism. See [28,Section 5.6] for the details. Note that Poly k (X) is usually a rather pathological space; for example it is rarely Hausdor . Instead of describing the topology, we describe what continuity of sections means: a section ω is continuous if there is there is a locally nite open cover U of X and a collection {π U } U∈U , where π U ∈ Poly k (U), such that [π U ]x = ω(x) for all U ∈ U and x ∈ U, and the collection {π U } U∈U satis es the overlap condition Conversely, any collection {π U } U∈U satisfying (2.5) de nes a continuous section ω of Poly k (X) by setting x , whenever U ∈ U and x ∈ U.

De nition 2.4. Let X be a metric space, and k ∈ N. A k-polylipschitz form on X is a continuous section of
Poly k (X).
The space of k-polylipschitz forms on X is denoted by Γ k (X), and Γ k c (X) denotes the set of polylipschitz forms whose support is compact.

Piecewise continuous polylipschitz forms
Given any set B ⊂ X, the restriction operators ρ U∩B,V : Poly k (V) → Poly k (U ∩ B), for U ⊂ V ⊂ X, form a presheaf homomorphism, giving rise to a restriction homomorphism ρ B : G k (X) → G k (B), where B is considered as a metric space with the restricted metric from X. We denote ρ B (ω) =: ω| B for ω ∈ G k (X).

De nition 2.5.
A polylipschitz section is partition-continuous if it is E-continuous for some countable Borel partition E of X.
We denote by Γ k pc (X) the space of partition-continuous polylipschitz sections, and by Γ k pc,c (X) those elements of Γ k pc (X) which have compact support. Clearly Γ k c (X) ⊂ Γ k pc,c (X).

Exterior derivative and cup-product
We refer to [24,Section 4.5] for further details. Following the construction of Alexander-Spanier cohomology we introduce the linear map d = d k X : Poly k (X) → Poly k+ (X) by for π ∈ Poly k (X) and x , . . . , x k+ ∈ X. This map satis es d • d = . The presheaf homomorphism {d k The cup-product is a bilinear map : Γ k pc,c (X) × Γ m pc,c (X) → Γ k+m pc,c (X), de ned in the same manner starting from the bilinear map : Poly k (X) × Poly m (X) → Poly k+m (X) for α ∈ Poly k (X), β ∈ Poly m (X) and x , . . . , x k+m ∈ X. Note that the cup product restricts to a bilinear map :

. Duality of metric currents and polylipschitz forms
We refer to [ x for (π , . . . , π k ) ∈ D k (X) and x ∈ X. We slightly abuse notation by using the symbol ı also for the embedding D k (X) → Γ k pc,c (X).
Extensions of currents of nite mass also satisfy natural integrability bounds. Given π ∈ Poly k (X) and V ⊂ X, de ne a variant of the projective norm L k (·) as follows: De ne the pointwise norm ω x of ω ∈ Γ k (X) at x ∈ X, by for any π such that [π]x = ω(x

Remark 2.8.
In the forthcoming sections we do not distinguish a metric current T ∈ M k,loc (X) from the extension T provided by Theorem 2.6. We will consider metric currents as acting on D k (X), Poly k c (X), Γ k c (X), or Γ k pc,c (X) interchangeably and without mentioning it explicitly.

Preliminaries on BLD-maps . Branched covers
A continuous mapping f : X → Y between metric spaces is a branched cover if f is discrete and open; recall In what follows, all mappings between metric spaces are continuous unless otherwise stated.
We recall that, given a branched cover f : X → Y and a normal domain U ⊂ X of f , the restriction f | U : U → fU is a proper map; see e.g. Rickman [26] and Väisälä [27].
Let f : X → Y be a branched cover between locally compact spaces. For x ∈ X and r > , we denote by When the map f is clear from the context we omit the subscript and write U(x, r) in place of U f (x, r). The following lemma is extensively used throughout the paper. It follows from [ (a) For every x ∈ X, there exists a radius rx > , for which U(x, r) is a normal domain of x for every r < rx.
Furthermore, given a compact set K ⊂ X and y ∈ f (K), there exists ry > so that U(x, r) is a normal neighborhood for x, for every x ∈ f − (y) ∩ K and r < ry.

Remark 3.2.
It follows that, if f : X → Y is a proper branched cover, then, for every x ∈ X, there is a radius r > for which

. Oriented cohomology manifolds
Following [13] we say that a separable and locally compact space X is an oriented cohomology n-manifold if (a) X has nite covering dimension, is a surjection for any neighborhood W of x contained in V.
The notation H * c (−; Z) above refers to the compactly supported Alexander-Spanier cohomology with integer coe cients. We refer to [13, De nition 1.1] and the ensuing discussion for more details. Here we only mention that a more widely used notion of cohomology manifolds requires all local cohomology groups of dimension < k < n to vanish, see e.g. [6, De nition 6.17].

. Global and local degree
Let X and Y be oriented cohomology manifolds of the same dimension n ∈ N and x orientations c X and c Y of X and Y, i.e. generators c X and c Y of H n c (X; Z) and H n c (Y; Z), respectively. For open sets U ⊂ X and V ⊂ Y we have local orientations given by c U = ι * UX c X and c V = ι * VY c Y , where ι UX : U → X and ι VY : V → Y are inclusions. As described in [13,26,27], continuous maps X → Y admit a local degree in the following sense.
Here we follow the presentation in [13].
Given a precompact domain U ⊂ X, the local degree µ f (U, y) ∈ Z with respect to a point y ∈ Y \ f (∂U) and domain U is , and otherwise 2. the unique integer λ ∈ Z for which the pull-back homomorphism Then A standard property of the local degree is that, for precompact domains V ⊂ U and a point y ∈ Y satisfying This immediately yields a summation formula for pairwise disjoint domains U , . . . , U N contained in U and satisfying As a consequence we obtain that, for a branched cover f : X → Y, the local degree function i f : X → Z, de ned by x where U is any normal neighborhood of x, is well-de ned. For branched covers, we may express the summation formula (3.1) in terms of the local index. Indeed, let f : X → Y be a branched cover between oriented cohomology manifolds of the same dimension and suppose U ⊂ X is a normal domain for f . Then for any y ∈ f (X); see [26] and [13]. The local index satis es a chain rule analogous to the chain rule for derivatives. More precisely, given branched covers f : X → Y and g : Y → Z between oriented cohomology manifolds, we have that . It is known that branched cover between oriented cohomology manifolds is either sense preserving or sense reversing [27]. Thus we may always choose the orientations c X and c Y of X and Y, respectively, so that a given branched cover f is sense preserving. In particular we may assume i f ≥ everywhere.

Branch set
Local homeomorphisms are always branched covers. However the converse fails, that is, a branched cover f : X → Y between oriented cohomology manifolds need not be a local homeomorphism. We de ne the set B f to be the set of points x ∈ X for which f is not a local homeomorphism at x. The branch set is easily seen to be a closed set.
It is known that the branch set B f as well as its image fB f of a branched cover between oriented cohomology n-manifolds has topological dimension at most n − ; see [27]. In particular B f and fB f do not locally separate X and Y, respectively, that is, [13, 3.1].
An orientation preserving proper branched cover f : X → Y is (deg f )-to-one in the sense that, for any y ∈ Y \ fB f , the preimage f − (y) contains exactly deg f points.

. BLD-maps and path-lifting
A BLD-map f : X → Y between metric spaces X and Y is a branched cover satisfying the bounded length distortion inequality (1.2) for some L ≥ . BLD-maps rst appeared in [19] as a subclass of quasiregular maps between Euclidean spaces, and in [13] in the present metric context. We refer to [18] for alternative characterizations of BLD-maps between metric spaces.
A path-lifting yields a bijection between preimages of points not in the image of the branch set of the map. In what follows, we use the following version of [18,Lemma 4.4]. We omit the details. Lemma 3.3. Let f : X → Y be an L-BLD map between two oriented cohomology manifolds. Suppose there exists a geodesic joining p, q ∉ fB f . Let K ⊂ X is a compact set. Then there is a bijection ψ : for every x ∈ f − (p) ∩ K.

The pull-back of metric currents by BLD-maps
Given a branched cover f : Recall that, by Lemma 3.1, for a compact set K ⊂ X, su ciently small balls Br(y), for y ∈ Y and r > , are spread neighborhoods with respect to K. By Remark 3.2, su ciently small balls Br(y), for y ∈ Y and r > , are spread neighborhoods for proper BLD-maps.
We say that a metric space X is locally geodesic if any point x ∈ X has a neighborhood U ⊂ X with the property that, for any two points p, q ∈ U, there is a geodesic joining them, i.e. a curve γ : We call such neighborhoods geodesic neighborhoods. Note, however, that the geodesic γ is not required to lie inside the neighborhood U. We also say that a ball Br(y) ⊂ Y is a geodesic spread neighborhood with respect to a set E ⊂ X if it is both a geodesic neighborhood, and a spread neighborhood with respect to E. Similarly, a geodesic spread neighborhood is a spread neighborhood that is also a geodesic neighborhood. We use the notation γ : x y to denote a curve γ : [a, b] → X joining two points x, y ∈ X.
In what follows, we consider only locally geodesic oriented cohomology manifolds.

Push-forward of functions by BLD-maps
Recall that the push-forward of a compactly supported Borel function g : It is not di cult to see that the push-forward f g is a Borel function.

locally Lipschitz and satis es the bound
Proof. The second estimate follows by a direct computation. Indeed, for any p ∈ Y, we have by the summation formula (3.2) for the local index. We now prove the rst estimate. Let p ∈ Y and take a geodesic spread neighborhood Br(p) of p. The preimage Ux is a mutually disjoint union of normal neighborhoods Ux of preimage points x. For any q ∈ Br(p) we have and further By substituting the local summation formula (3.2) into (4.1) we have the estimate where γ : x y is a lift of a geodesic γ : p q. Thus Suppose that y ∈ Y and Br(y) is a spread neighborhood. Then, for any p, q ∈ Br(y), choosing a geodesic γ connecting them, we have This proves that f # η is locally Lipschitz and satis es the rst estimate in the claim.
The following lemma shows that the push-forward is natural with respect to composition.
and, by (3.4), Thus for every z ∈ Z.

Push-forward of polylipschitz functions by BLD-maps
To simplify notation, we denote by For example, for the local index i f : Let f : X → Y be a BLD-map between locally geodesic, oriented cohomology manifolds. Let U ⊂ X be a normal domain for f . Given a normal domain U ⊂ X for f , consider the continuous (k + )-linear linear map

De nition 4.3. Let f : X → Y be a BLD-map between locally geodesic, oriented cohomology manifolds, and let U ⊂ X be a normal domain for f . The push-forward
is the unique continuous linear extension of A U f for which (2.2) holds.
The claim follows from this immediately.
For the next three lemmas, we assume that f : X → Y is an L-BLD map between geodesic, oriented cohomology manifolds, U ⊂ X is a normal domain for f , and that k ≥ is a xed integer. We show that the push-forward commutes with the cup product and the exterior derivative.
Lemma 4.6. Given π ∈ Poly k (U) and σ ∈ Poly m (fU) we have Proof. We observe rst that, given functions g, h : U → R and p ∈ fU, we have . Now let π = π ⊗ · · · ⊗ π k ∈ Poly k (U) and σ = σ ⊗ · · · ⊗ σm ∈ Poly m (fU) be polylipschitz functions. Then Since the cup product is bi-linear and the pull-back is linear we have, by (4.2), that the claim holds for all π ∈ Poly k (U) and σ ∈ Poly m (V).

Lemma 4.7.
For each π ∈ Poly k (U), we have Proof. As before, it su ces to consider the case π = π ⊗ · · · ⊗ π k ∈ Poly k (U). Then The following lemma shows that the push-forward is sequentially continuous.
Finally, we show that the push-forward is natural in the sense that the composition of push-forwards is the push-forward of compositions Lemma 4.9. Let f : X → Y and g : Y → Z be BLD-maps between locally geodesic oriented cohomology manifolds. Let U ⊂ X be a normal domain for f and V ⊂ f (U) a normal domain for g. Set W = g(V) and Then for every π ∈ Poly k (U ).

. Push-forward of polylipschitz forms
Let f : X → Y be a BLD-map between locally geodesic oriented cohomology manifolds X and Y. We show that the push-forwards f U# : Poly k (X) → Poly k (Y). x. Then Proof. We may assume U ⊂ U. Since [π]x = [π ]x, there exists ρ > , for which U(x, ρ) ⊂ U and Since U(x, ρ) is a normal neighborhood of x we have, by the summation formula of the local index (3.2) that, for every q ∈ Bρ(p), Thus The claim follows.
De nition 4.11. Let f : X → Y be a BLD-map between locally geodesic oriented cohomology manifolds. The local averaging map A f : Poly k (X) → Poly k (Y) is the map where, for each x ∈ X, Ux is a normal neighborhood of x.
Note that, since ω ∈ G k c (X) has compact support, the sum in De nition 4.13 has only nitely many nonzero summands.
Let ω ∈ G k c (X) and y ∈ Y. The value of the push-forward f # ω at y can be given as follows. Let r > be a radius with the property that Br(y) is a geodesic spread neighborhood with respect to spt ω; cf. Lemma 3.1. For Indeed, it su ces to note that We use this fact in the sequel. The next proposition lists the basic properties of the push-forward.
Proof. Linearity is straighforward to check (see Remark 4.12). Let ω ∈ G k c (X) and p ∈ Y, p ∉ f (spt ω). Then spt ω ∩ f − (p) = ∅ and therefore all the terms in the sum de ning f # ω(p) are zero. This proves (1).
Let Br(p) be a geodesic spread neighborhood with respect to spt ω. By Corollary 4.4 we have Similarly r)).
We prove Proposition 4.15 at the end of Section 4.2. For the proof, we brie y recall the monodromy representation of a proper branched covers. Let f : X → Y be a proper branched cover. Then there is a locally compact geodesic space X f , a nite group G = G f , called the monodromy group of f , acting on X f by homeomorphisms, and a subgroup H ≤ G satisfying The quotient maps and φ : X f → X, x → Hx, are branched covers for which the diagram commutes. When f is a BLD-map, the group G acts on X f by bilipschitz maps, and f and φ are BLD-maps. See [1] and the references therein for details on monodromy representations. The following multiplicity formula is a counterpart of (3.4). We refer to [1] for similar multiplicity formulas.
Proof. Let w ∈ X f and let W ⊂ X f be a normal domain forf . Then φ(W) ⊂ X is a normal neighborhood of φ(w) with respect to f . We denote g = (f )| W : W →f (W). The stabilizers Gw and Hw act on W and the restrictions g and φ| W are orbit maps with respect to the action. Thus, the commuting diagram is a monodromy representation of f | φ(W) , with monodromy group Gw, and φ| W is the orbit map for Hw. Since gBg and fB f are nowhere dense, there exists p ∈f \ gBg ∪ fB f . Since g is the orbit map for Gw, we have that On the other hand, (f | φ(W) ) − (p ) ∩ (φ| W )B φ| W = ∅. We conclude that Since deg(f | φ(W) ) = i f (w), the claim follows.

Fiber equivalence
Throughout this subsection we x a proper BLD-map f : X → Y. We introduce the ber equivalence on Y using the monodromy representation  (k , . . . , km) is Borel, set The sets Em and E(k) are clearly Borel. Observe that whence the Borel measurability of the equivalence classes follows. Proof. Let z ∈f − (p) and z ∈f − (q). Then For second claim let Hw ≤ Gw be the stabilizer subgroups of w ∈ X f in H and G, respectively. Since φ is an orbit map of the action H X f , we have that |Hw| = |φ − φ(w)|. Hence Thus, by Lemma 4.16, we have, for w = z j ∈ φ − (x j ) and w = z j = φ − (y j ), that for each j = , . . . , m.

Lemma 4.19. Let p ∼ f q. Let Br(p) and Bs(q) be spread neighborhoods for f andf . Then
Since The claim now follows from the identity Proof of Proposition 4. 15. Suppose rst that spt π ⊂ U where U ⊂ X is a normal domain for f . Then f # (ω)(p) = whenever p ∈ Y \ f (U).
De ne the partition E on f (U) as the collection of equivalence classes of the ber equivalence relation Let p, q ∈ f (U) be ber equivalent, that is p ∼ q, and let Br(p) and Br(q) be geodesic spread neighborhoods for f | U and f | U . Then, by Lemma 4.19, for j = , . . . , m and z ∈ Br(p) ∩ Bs(q).
For each l = , . . . , M + , spt(φ l π ) is contained in a normal domain for f . Thus f # (φ l ω) is E-continuous. We conclude that the nite sum is E-continuous.

. Pull-back of currents of locally nite mass by BLD-maps
To de ne the pull-back of a k-current T ∈ M k,loc (X) as T • f # (see the discussion in the introduction) it remains to show that the resulting functional is weakly continuous.  Em(µ , . . . , µm) We have Since σ n → in Poly k U (X), and hence the restrictions converge in Poly k (U(x l , r i )), it follows from Lemma 4.8 . The claim follows.
We now de ne the pull-back of currents of locally nite mass.

Proposition 4.22.
Let f : X → Y be an L-BLD-map between locally geodesic, oriented cohomology manifolds, and let T ∈ M k,loc (Y). Then f * T ∈ M k,loc (X) and Proof. By Lemma 4.20 and Theorem 2.6, f * T is sequentially continuous. Let (π , . . . , π k ) ∈ D k (X) and π = π ⊗ · · · ⊗ π k . Suppose E , . . . , E N is a Borel partition of Y for which for each i = , . . . , N. By [24, Proposition 6.7] and Lemma 4.14, we may estimate This proves f * T is a k-current of locally nite mass and provides the desired estimate. and f | Em∩U(x, Lr) is injective.
The fact that f | Em∩U(x,r) is L-Lipschitz is clear. Moreover the proof of injectivity shows that U(x, r))).
Let z , w ∈ f (Em ∩ U(x, r)), and let z, w ∈ Em ∩ U(x, r) satisfy z = f (z) and w = f (w). Suppose γ is a geodesic joining z and w in B r (f (x)). Since f − (w ) ∩ U(x, r) = {w}, we have that a lift γ in U(x, Lr) of γ starting at z ends at w. Thus This nishes the proof of the claim.
Suppose now that x ∈ B f . Let r > be a radius for which U(x, Lr) is a normal neighborhood of x and for Then . By the same argument as above Thus We have proven that, for each x ∈ X, there exists a radius r > such that T Br(x) = .
Proof of Theorem 1.2. By Theorem 1.1 (1) and the assumption on S, we have for every precompact Borel set E ⊂ X. Proposition 4.23 implies that completing the proof.
We now prove the naturality of the pull-back. The rst auxiliary result is the naturality of the push-forward of polylipschitz forms.
Denote ω = f # π ∈ G k c (Y), and spt π = K. Let q ∈ Z and x r > for which Br(q) is a geodesic spread neighborhood for g • f with respect to K, and Ug(y, r) is a geodesic spread neighborhood of y, for each y ∈ g − (q) ∩ f (K), with respect to K. Since we have that, for each y ∈ g − (q) ∩ f (K), the set f − (Ug(y, r)) is a pairwise disjoint union f − (Ug(y, r) r).
see the discussion after De nition 4.13. Then ig(y) k g Ug(y,r)# σy .
For each y ∈ f − (q) ∩ f (K) we have, by Lemma 4.9, that Thus we have on Br(q) k+ . From this we conclude that for all q ∈ Z.  Note that, if T ∈ M k,loc (Z), then, by de nition and Proposition 4.25, we have for π ∈ D k (X). Unfortunately, since f # π ∈ G k c (Y) is not necessarily in Γ k c (Y), we cannot conclude that g * T(f # π) (strictly speaking, g * T(f # π)) is given by T(g # f # π).

. Pull-back of proper BLD maps
Throughout this subsection, f : X → Y is a proper L-BLD map between geodesic, oriented cohomology manifolds X and Y. Recall that a proper branched cover is (deg f )-to-one; see the discussion on the branch set in Section 6.3.
In this subsection we prove Corollary 1.4, that is, we prove that the pull-back f * : M k,loc (Y) → M k,loc (X) satis es the following properties: Let (K j ) be an increasing sequence of compact sets in X for which j∈N K j = X. Then This proves (3). By (4.4) and Theorem 1.1(1), we have On the other hand weakly in M k,loc (X) as j → ∞. Thus (1) in Corollary 1.4 is proven. By Theorem 1.1 (2) and (3), f # maps N k (Y) to N k (X).

Equidistribution estimates for pull-back currents . BLD-maps from R n into metric spaces
Let X be a metric space and f : R n → X a Lipschitz map. We will use the metric Jacobian Jf of f , de ned by Kirhchheim [16]: for almost every x ∈ R n the limit exists for all v ∈ R n and de nes a seminorm. The metric di erential of f at such a point x is the seminorm given by (5.1) and zero otherwise. This induces the metric Jacobian Jf : R n → R, a Borel function de ned for any point where the limit (5.1) exists, by Here σ n− is the normalized surface measure on the unit sphere S n− of R n . The metric Jacobian plays a prominent role in the co-area formula We refer to [16] for details.
Remark 5.1. By [18,Lemma 2.4] we obtain that, if the limit in (5.1) exists for x ∈ R n , then Throughout the rest of this section X is a compact geodesic oriented cohomology n-manifold, and f : R n → X an L-BLD map. We denote |X| = H n (X) and D = diam(X).
By Remark 5.1 and the discussion after it the space X is Ahlfors n-regular under the present assumptions. In particular |X| ≤ CD n , where C > is the Ahlfors regularity constant.

. Equidistribution
We turn our attention to the value distribution of BLD-maps. The following theorem will be used in the next subsection to obtain estimates on the mass of pullbacks of currents. For the theorem, let for every p ∈ X and R ≥ LD.
Theorem 5.2 gives a quantitative equidistribution estimate with constants depending only on n and L. We refer to [20] and [23] for similar results for quasiregular maps. We begin with an observation which we record as a lemma.
for almost every y ∈ R n . The rest follows directly from the change of variables formula (5.2).
Proof of Theorem 5.2. Let R > LD, and let χ B(R−δ) ≤ η ≤ χ B(R) be a Lipschitz function. By Lemma 3.3, for points p, q ∈ X \ fB f , there is a bijection for all x ∈ f − (q) ∩ B(R). Thus We have |η(ψ(x)) − η(x)| ≤ for all x. Moreover, and we obtain Integrating with respect to q we obtain In similar fashion we may obtain the estimate Fixing q and integrating with respect to p yields Since f # η is continuous, estimates (5.4) and (5.5) hold for all p ∈ Y. Letting δ → we obtain This implies the claim.

. Mass and flat norm estimates
We apply the equidistribution Theorem 5.2 to prove estimates for the mass and at norm of pull-backs of locally normal currents. Proof. Denote χ B(R) =: χ R . By Theorem 5.2 we have establishing the rst estimate.
To estimate the at norm, let A ∈ N k+ (X). Then, by Proposition 4.22, we have that .
For the opposite inequality, let η : R n → [ , ∞), η(x) = ( − dist(B R− , x))+ be a Lipschitz function. By the proof of Proposition 1.1 (2) we have where c(n, L) is a constant depending only on n and L.
Proof of Claim. We observe rst that Thus it su ces to estimate f # (χ R − χ R− )/A f (R). By Theorem 5.2 we have that for R > CD.
By Lemmas 2.1 and 4.1, we have If R > C(n, L)(D + ), then (5.6) yields the estimate from which the remaining inequality readily follows.

Homology of normal metric currents
In this section we assume that X is a compact oriented cohomology manifold and, in addition, that X is locally Lipschitz contractible. Recall that X is locally Lipschitz contractible if every neighborhood U of every point x ∈ X contains a neighborhood V ⊂ U of x so that there is a Lipschitz map so that h (y) = y for every y ∈ V and h is constant. We remark that this is similar to the notion of γ-Lipschitz contractibility in [30,Section 3.2]. For compact spaces it is not di cult to see that the two notions coincide in the sense that a locally Lipschitz contractible is γ-Lipschitz contractible for some γ, and a γ-Lipschitz contractible space is locally Lipschitz contractible.

. Current homology and oriented cohomology manifolds
The boundary map satis es ∂ k− ∂ k = , which can be readily seen from the de nition of metric currents; see also [17,Section 3]. Thus the boundary map induces a chain complex As is customary we omit the subscripts from ∂.
We study the homology of the chain complex (6.1) for a BLD-elliptic oriented cohomology manifold X and we denote the homology groups of (6.1) by H k (X) := ker ∂ k / im ∂ k+ , (6.2) for k ≥ . It is known that H * (·) de nes a homology theory satisfying the Eilenberg-Steenrod axioms; see [21] and also [30] for integral currents, and [8] for the homology of normal chains and cohomology of charges. For us, homology always refers to the homology (6.2) of (6.1).

. Filling inequalities
We say that a locally compact metric space X admits a lling inequality for N k (X) if there is a constant C > such that each T ∈ N k+ (X) satis es FillVol(∂T) ≤ CM(∂T).
Recall that the lling volume of a current A ∈ N k (X) is de ned to be FillVol(A) = inf{M(B) : ∂B = A}, the in mum over the empty set being understood as in nity. This means in particular that, if S ∈ N k (X) and S = ∂T for some T ∈ N k+ (X), then there exists T ∈ N k+ (X) satisfying S = ∂T and There is a related notion of cone type inequalities introduced by Wenger [29]. A space X is said to support cone type inequalities for N k (X) if there exists a constant C > with the property that, if S ∈ ker ∂ k , then there exists T ∈ N k+ (X) satisfying ∂T = S and

M(T) ≤ C diam(spt S)M(S).
A space X supporting a cone type inequality for N k (X) necessarily has trivial current homology H k (X), whereas spaces admitting lling inequalities only require (6.3) for currents S a priori known to have a lling. Remark 6.2. In [8], De Pauw, Hardt, and Pfe er introduce the notion of locally acyclic spaces, see [8,De nition 16.10]. Locally Lipschitz contractible spaces are locally acyclic spaces, but the connection between lling inequalities and local acyclicity is not clear to us.
In this subsection we prove that compact BLD-elliptic spaces as in Theorem 1.5 support lling inequalities. Proposition 6.3. Let f : R n → X be an L-BLD map into a compact, geodesic, oriented and locally Lipschitz contractible cohomology n-manifold X, and let ≤ k ≤ n. Then there exists a constant C > having the property that, for every T ∈ im ∂ k+ there exists S ∈ N k+ (X) satisfying ∂S = T and

M(S) ≤ CM(T).
Filling inequalities are equivalent to the closedness of the range of ∂. We show this using nite dimensionality of the homology. Proof. By [14, Theorem V.7.1] the space X is an Euclidean neighborhood retract and by [11,Corollary A.8] it has the homotopy type of a nite CW-complex. By [21, Corollary 1.6] the normal current homology groups are isomorphic to the singular homology groups (with real coe cients), and thus nite dimensional. Lemma 6.4 immediately yields the desired nite dimensionality as a corollary. Corollary 6.5. Let X be a compact, locally geodesic, orientable, and locally Lipschitz contractible cohomology n-manifold. Then the normal current homology groups H k (X) are nite dimensional for all k ∈ N. Lemma 6.6. Let X be a compact, locally geodesic, orientable, and locally Lipschitz contractible cohomology n-manifold, and k ≥ . Then the boundary operator has closed range.
We are now ready for the proof of the lling inequality.
Proof of Proposition 6.3. Let k ≥ and consider the operator ∂ = ∂ k+ . By Lemma 6.6, (im ∂, N) is a Banach space. The canonical operator ∂ : N k+ (X)/ ker ∂ → im ∂ is injective and onto. By the open mapping theorem, there is a constant < c < ∞ for which for every T ∈ N k (X). Let A ∈ ker k+ ∂. Then This implies and consequently the lling inequality for N k (X).

. Homological boundedness
We use the lling inequality to establish the existence of mass minimal elements in homology classes of H * (X). By passing to a subsequence we may assume that the sequence (Am) converges weakly to a normal current A ∈ N k+ (X). By the lower semicontinuity of the mass, Taking in mum over B proves the last claim.

Proof of a non-smooth Bonk-Heinonen theorem
To prove Theorem 1.5 we introduce a norm | · | : H k (X) → [ , ∞) on the homology group H k (X) by c → inf{M(T) : T ∈ c}.
By Lemma 6.7 each homology class c ∈ H k (X) contains an element of minimal norm, and in particular |c| > if and only if c ≠ .

A Local Euclidean bilipschitz embeddability of BLD-elliptic spaces
In this appendix we prove the following embeddability theorem mentioned in the introduction.
Theorem A.1. Let X be a locally geodesic, orientable cohomology manifold admitting a BLD-map f : R n → X. Let x ∈ X. For every radius r > , for which there exists y ∈ f − (x) such that U(y, r) is a normal neighborhood of y, Br(x) is bilipschitz equivalent to a subset of a Euclidean space.
In the proof we use Almgren's theory of Q-valued maps. We refer to [7] for a recent exposition. Denote by A Q (R n ) the space of unordered Q-tuples of points in R n . For the purpose of introducing a metric, we formally de ne where δx is the Dirac mass at x ∈ R n . Given T , T ∈ A Q (R n ), suppose for each x ∈ f − (p) ∩ U. Thus where [x i , y σ(i) ] denotes the geodesic line segment from x i to y σ(i) . Thus d Q (g f (p), g f (q)) ≥ Qd(p, q)/L.
We have established the bilipschitz condition for points p, q in the dense set Br(x)\ fB f , whence it follows for all p, q ∈ Br(x).
Proof of Theorem A. 1. Let x ∈ X and let r > be a radius with the property that there exists y ∈ f − (x) for which U = U(y, r) is a normal neighborhood of y. Set Q = i f (y) and consider the map g f : Br(x) → A Q (R n ). Then the map ξ • g : Br(x) → R N ,