The metric fundamental class of non-orientable manifolds and manifolds with boundary

THE METRIC FUND AMENT AL CLASS OF NON-ORIENT ABLE MANIF OLDS AND MANIF OLDS WITH BOUND AR Y DENIS MAR TI Abstract. W e introduce the metric fundamen tal class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possi- bly empty) boundary . This is an integer rectiﬁable current that provides an analytic representation of the topological fundamen tal class of the space. Un- der certain w eak geometric conditions, we show the existence of suc h a curren t, extending earlier results for orien table, closed manifolds obtained in collabora- tion with Basso and W enger. As an application, we present new rectiﬁabilit y results. 1. Introduction In an earlier w ork jointl y with Basso and W enger, the author show ed that, under certain geometric conditions, a metric space that is homeomorphic to a closed, ori- en table smo oth manifold admits a non-trivial integral cycle in the sense of Am bro- sio–Kirc hheim [3]. This integral cycle serves as a metric analogue of the topological fundamen tal class of a smo oth manifold and is referred to as the metric fundamen tal class of the manifold. Deﬁnition 1.1. L et X b e a metric sp ac e with ﬁnite Hausdorﬀ n -me asur e that is home omorphic to a close d, oriente d, Riemannian n -manifold M . A metric funda- mental class of X is an inte gr al curr ent T ∈ I n ( X ) with ∂ T = 0 such that (a) φ # T = deg( φ ) · J M K for every Lipschitz map φ : X → M ; (b) ther e exists C > 0 such that ∥ T ∥ ≤ C · H n . Here, M is equipp ed with an y Riemannian metric and J M K denotes its fundamen tal class given b y integration of diﬀerential n -forms, and ∥ T ∥ is the mass measure of T . Informally speaking, the conditions (a) and (b) ensure that T is compatible with the top ological and metric structure of X , resp ectiv ely . W e note that a metric fundamen tal class, when it exists, is unique and generates the n th homology group via integral curren ts; c.f. [5, Proposition 5.5]. A metric space homeomorphic to a smooth manifold is called a metric manifold. Suc h spaces and the relationship b et w een their geometric and analytic prop erties pla y a central role in metric ge- ometry; see e.g. [11, 12, 24, 41, 46]. In this vein, the metric fundamen tal class oﬀers a p ow erful tool for studying metric manifolds. F or instance, a deep theorem of Semmes [46] guarantees the v alidit y of a Poincar ´ e inequality in Ahlfors regular and linearly lo cally contractible metric spaces that are homeomorphic to a closed, orien ted, smooth manifold. In [5], a new pro of of this result was provided using the metric fundamental class; see also [5, 22, 25, 40] for more applications. The goal of this article is to extend these ideas and results to manifolds that are non-orien table and to manifolds with b oundary . In the non-orien table case, every top-dimensional integral curren t necessarily has non-zero b oundary and the notion 2020 Mathematics Subje ct Classiﬁc ation. Primary 53C23; Secondary 49Q15, 28A75. Key wor ds and phrases. Metric geometry , Metric manifolds, Integral curren ts, Nagata dimen- sion, degree theory . D.M. was supp orted by Swiss National Science F oundation grant 212867. 1 2 DENIS MAR TI of degree is not w ell-deﬁned. Therefore, w e adapt the deﬁnition of a metric fun- damen tal class. W e require the curren t to ha v e no boundary mo dulo 2 as deﬁned in [2, 4]. Moreov er, in Deﬁnition 1.1(a) we use the degree deﬁned via singular ho- mology with co eﬃcients in Z / 2 Z . More precisely , an integer rectiﬁable current T satisﬁes ∂ T = 0 mod 2 if F 2 ( ∂ T ) = 0. Here, F 2 ( ∂ T ) denotes the ﬂat norm mo dulo 2 of ∂ T . This distance is deﬁned as the inﬁm um of M ( U ) + M ( V ) ov er in teger rectiﬁable curren ts U and V of dimensions n − 1 and n , resp ectiv ely , such that ∂ T = U + ∂ V + 2 Q for some ﬂat current Q . W e refer to Section 3.4 for a detailed o verview of ﬂat currents modulo 2. There also exists a w ell-dev elop ed theory of ﬂat chains with co eﬃcients in any normed ab elian group [15] that generalizes the ﬂat chains considered in [51] to metric spaces. This theory also pro vides a natural framew ork to deﬁne a metric fundamental class for non-orientable manifolds. Nev- ertheless, we hav e chosen to work with Ambrosio–Kirc hheim curren ts b ecause of a direct connection with the results in [5] for orientable manifolds. 1.1. Statemen t of main results. A metric space X is called a metric n -manifold if it is homeomorphic to a compact smo oth n -manifold M . W e sa y that X is ori- en table, non-orientable, or closed, dep ending on whether M has the corresp onding prop erties. In case M has non-empty b oundary ∂ M , we denote b y ∂ X the image of ∂ M under the homeomorphism from X to M . Ev ery manifold considered in this article is assumed to be connected. W e write I n ( X ) for the space of in teger rectiﬁable currents in X ; see Section 3 for the relev an t deﬁnitions. The ﬁrst result pro vides the existence of a metric fundamental class in orien table metric manifolds that are linearly lo cally contractible and extends [5, Theorem 1.1] to manifolds with boundary . A metric space is said to b e linearly lo cally contractible if there exists λ > 0 such that ev ery ball of radius 0 < r < diam X/λ is contractible within the ball with the same center and radius λr . This prop ert y often app ears in v arious contexts in metric geometry; see e.g. [11, 12, 20, 21, 30, 47]. Theorem 1.2. L et X b e a c omp act, orientable metric n -manifold with ﬁnite Haus- dorﬀ n -me asur e and (p ossibly empty) b oundary. Supp ose that X is line arly lo c al ly c ontr actible and H n ( ∂ X ) = 0 . Then, X has a metric fundamental class T ∈ I n ( X ) with (1.1) C − 1 H n ≤ ∥ T ∥ ≤ C H n , and whenever S ∈ I n ( X ) satisﬁes spt( ∂ S ) ⊂ ∂ X , then ther e exists k ∈ Z such that S = k · T . Here, C depends only on n . If X is closed, then T is as in Deﬁnition 1.1 and the second prop ert y holds for all S ∈ I n ( X ) with ∂ S = 0. An in teger rectiﬁable curren t T ∈ I n ( X ) in a metric manifold X with non-empt y b oundary is said to b e a metric fundamen tal class of X if T satisﬁes Deﬁnition 1.1(b) and Deﬁnition 1.1(a) for ev ery Lipsc hitz map φ : X → M satisfying φ ( ∂ X ) ⊂ ∂ ( M ). Next, we consider non-orientable manifolds. As men tioned earlier, the deﬁnition of the metric fundamental class needs to b e adapted for these manifolds. Deﬁnition 1.3. L et X b e a metric sp ac e with ﬁnite Hausdorﬀ n -me asur e that is home omorphic to a close d, non-orientable, smo oth n -manifold M . A metric funda- mental class mo dulo 2 of X is an inte ger r e ctiﬁable curr ent T ∈ I n ( X ) with ∂ T = 0 mo d 2 such that (1.2) φ # T = deg( φ, Z / 2 Z ) · J M K for every Lipsc hitz map φ : X → M . If X has b oundary , then (1.2) is only true for Lipschitz maps φ : X → M satisfying φ ( ∂ X ) ⊂ ∂ M . The manifold M can b e equipped with an y Riemannian metric and J M K denotes the corresponding fundamen tal class modulo 2. Unlike in the orien table case, we do not require the second prop ert y since we alw a ys hav e ∥ T ∥ 2 ≤ THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 3 C H n , where C dep ends only on n . The degree deg ( φ, Z / 2 Z ) is deﬁned in terms of singular homology groups with co eﬃcien ts in Z / 2 Z and is interpreted as either 0 or 1, dep ending on whether it is trivial or not. The next result is the equiv alen t of Theorem 1.2 for non-orientable manifolds. Theorem 1.4. L et X b e a c omp act, non-orientable metric n -manifold with ﬁnite Hausdorﬀ n -me asur e and (p ossibly empty) b oundary. Supp ose that X is line arly lo c al ly c ontr actible and H n ( ∂ X ) = 0 . Then, X has a metric fundamental class T ∈ I n ( X ) mo dulo 2 satisfying (1.3) C − 1 H n ≤ ∥ T ∥ 2 = ∥ T ∥ ≤ C H n , and whenever S ∈ I n ( X ) satisﬁes spt 2 ( ∂ S ) ⊂ ∂ X , then S = k · T mo d 2 for k e qual to either 1 or 0 . Here, C dep ends only on n , and ∥ T ∥ 2 is the 2-mass measure of T . The 2-supp ort spt 2 S of an in tegral curren t S ∈ I n ( X ) is deﬁned as the supp ort of its 2-mass measure ∥ S ∥ 2 . If X is closed, then the uniqueness prop ert y holds for all S ∈ I n ( X ) with ∂ S = 0 mo d 2. W e note that w e pro v e Theorem 1.2 and Theorem 1.4 using a weak er v ersion of linear lo cal contractibilit y , as in the standard deﬁnition explained ab o v e. Concretely , b oth results (and their corollary b elo w) hold for metric manifolds that are almost ev erywhere linearly lo cally con tractible; see Section 4. As the name indicates, this can b e though t of as a measure theoretic v ersion of the standard deﬁnition. W e obtain the following corollary . Corollary 1.5. L et X b e a c omp act metric n -manifold with ﬁnite Hausdorﬀ n - me asur e and (p ossibly empty) b oundary. Supp ose that X is line arly lo c al ly c on- tr actible and satisﬁes H n ( ∂ X ) = 0 . Then X is n -r e ctiﬁable. This extends the rectiﬁability result in [5] and is a direct consequence of the mass measure b ounds of the fundamental class in Theorem 1.2 and Theorem 1.4. F or non- orien table manifolds, we are also able to pro v e the existence of a metric fundamen tal class mo dulo 2 when the space is not linearly lo cally contractible. Theorem 1.6. L et X b e a c omp act, non-orientable metric n -manifold with ﬁnite Hausdorﬀ n -me asur e and (p ossibly empty) b oundary. Supp ose that X has ﬁnite Nagata dimension and H n ( ∂ X ) = 0 . Then, X has a metric fundamental class T ∈ I n ( X ) mo dulo 2 satisfying ∥ T ∥ 2 = ∥ T ∥ ≤ C H n and whenever S ∈ I n ( X ) satisﬁes spt 2 ( ∂ S ) ⊂ ∂ X , then S = k · T mo d 2 for k e qual to either 1 or 0 . Here, C dep ends only on n . W e note that a metric manifold with ﬁnite Nagata dimension does not need to b e rectiﬁable. Indeed, by [48, Theorem A.1] there exists a geo desic metric space Y with ﬁnite Hausdorﬀ 2-measure that is homeomorphic to the 2-sphere but is not 2-rectiﬁable. It follows from [27] that any geo desic metric surface, and in particular Y , has Nagata dimension 2. Therefore, in general, an in teger rectiﬁable current as in the previous result cannot satisfy a lo w er b ound of the form (1.3). Finally , for metric 2-manifolds, called metric surfaces, w e do not require any conditions. Theorem 1.7. L et X b e a c omp act, non-orientable metric surfac e with ﬁnite Haus- dorﬀ 2 -me asur e and (p ossibly empty) b oundary. Supp ose that H 2 ( ∂ X ) = 0 . Then, X has a metric fundamental class T ∈ I 2 ( X ) mo dulo 2 satisfying ∥ T ∥ 2 ≤ 2 H 2 and whenever S ∈ I 2 ( X ) satisﬁes spt 2 ( ∂ S ) ⊂ ∂ X , then S = k · T mo d 2 for k e qual to either 1 or 0 . F or closed, orientable surfaces, the result was already known [5, Theorem 1.3]. 4 DENIS MAR TI W e deﬁne the (lo w er) Minkowski c ontent of a subset E ⊂ X with resp ect to an op en subset U ⊂ X by M − ( E | U ) = lim inf r ↘ 0 H n ( E r ∩ U ) − H n ( E ∩ U ) r , where E r = { x ∈ X : d ( x, E ) < r } denotes the op en r -neigh borho o d of E in X . F or an op en ball B = B ( x, r ) and λ > 0, w e write λB for the op en ball B ( x, λr ) with the same center x and radius λr . Theorem 1.8. L et X b e a close d, non-orientable metric manifold of dimension n ≥ 2 . Supp ose that X is line arly lo c al ly c ontr actible and Ahlfors n -r e gular. Then ther e exist C , λ ≥ 1 such that (1.4) min  H n  E ∩ B  , H n  B \ E  ≤ C · M −  E | λB  n n − 1 for every Bor el subset E ⊂ X and every op en b al l B ⊂ X . Here, the constants C and λ dep end only on the data of X . The inequalit y (1.4) is called a relative isop erimetric inequalit y and is strongly related to the Poincar ´ e inequalit y . In particular, it follows that a metric manifold, as in Theorem 1.8, also satisﬁes a weak 1-Poincar ´ e inequality; see [10, 29, 31]. 1.2. Structure of the article. In Section 2, we introduce the notation and basic results we need throughout the article. W e review the theory of metric currents in Section 3 and give a detailed ov erview of ﬂat currents mo dulo p . There, we also pro ve a compactness and rectiﬁabilit y result for such curren ts. W e present the new measure theoretic version of linear local contractibilit y , which w e call almost everywhere linear local contractibilit y in Section 4. In addition, w e explain the construction of the orientable double co ver of a metric manifold and ho w one can glue tw o copies of the manifold along their b oundaries. F or b oth spaces, w e construct a metric using the metric of the original manifold. The main motiv ation b ehind the new version of linear lo cal con tractibility is that it in teracts w ell with these constructions. Indeed, if X is linearly lo cally con tractible, then the manifold ˆ X obtained by gluing t w o copies along their b oundaries does not need to b e linearly lo cally contractible. How ev er, ˆ X will b e almost everywhere linearly lo cally contractible in case the boundary of X has zero Hausdorﬀ n -measure. Finally , we discuss ho w to extend bi-Lipsc hitz maps from Euclidean space into metric manifolds in a controlled manner. These techniques allo w us to show that the degree of Lipsc hitz maps b eha v es nicely in metric manifolds that are almost ev erywhere linearly lo cally con tractible. Similar results w ere already obtained in [5]. In Section 5, we prov e the existence of a metric fundamen tal class for orientable manifolds that are almost ev erywhere linearly lo cally con tractible. W e ﬁrst consider only closed manifolds. In this case, the argument is very similar to the pro of of [5, Theorem 1.1]. Indeed, by using linear lo cal contractibilit y , we can equip the rectiﬁ- able part of the manifold with an orientation that is compatible with the top ological manifold. This ”metric orientation” induces an in teger rectiﬁable current T satis- fying the upper measure bound in (1.1). It follows from a deep theorem of Bate, Theorem 2.3, and degree theory , that T is a cycle and Deﬁnition 1.1(a) holds. The result for manifolds with b oundary can easily be reduced to the closed case with the constructions in Section 4. The pro ofs of the diﬀeren t existence results of the metric fundamen tal class mo dulo 2 are contained in Section 6. W e begin with the case when the manifold X is closed and almost ev erywhere linearly lo cally con tractible. By construction, the orien table double cov er ˜ X is orien table and lo cally isometric to the original manifold. As a THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 5 consequence, ˜ X is almost everywhere linearly lo cally con tractible as well and thus, has a metric fundamental class ˜ T . W e construct an integral current T in X from ˜ T that satisﬁes the measure b ound (1.3). T o show that T is a cycle modulo 2 and satisﬁes (1.2) we use degree theory and the techniques introduced in Section 4. In case X is closed and has ﬁnite Nagata dimension, w e use an approximation of X by a simplicial complex. This is a construction frequen tly used in the context of Nagata dimension; see e.g. [6, 34, 40]. More precisely , there exists a simplicial complex Σ with bounded Hausdorﬀ measure and Lipsc hitz maps φ : X → Σ, and ψ : Σ → E ( X ). F urthermore, the comp osition ( ψ ◦ φ ) : X → E ( X ) is arbitrarily close to the inclusion of X into its injectiv e h ull E ( X ). The space E ( X ) has similar prop erties as l ∞ but is compact; see Section 2.1. W e then approximate the composition ( φ ◦ ϱ ) b y a Lipschitz map, where ϱ denotes the homeomorphism from the smo oth manifold M into X . Using this approximation, w e can push the fundamental class mo dulo 2 of M in to Σ. In this wa y , we construct a sequence of in tegral cycles mo dulo 2 with uniformly b ounded 2-mass. It follo ws from the compactness and rectiﬁabilit y theorem for ﬂat currents modulo 2 established in Section 3 that X has a metric fundamen tal class mo dulo 2. F or non-orientable closed metric surfaces X , w e use a recent uniformization result [43]. This result implies that there exists a w eakly conformal map from a smo oth surface in to X . W e can approximate this map by Lipsc hitz maps with b ounded area. As b efore, w e can use a pushforward argument and ﬁnd a metric fundamental class mo dulo 2 in X . Finally , we show the existence of the metric fundamen tal class for non-orientable manifolds with b oundary by reducing the pro of to the corresp onding case for closed manifolds. 2. Preliminaries 2.1. Metric notions. Let ( X , d ) b e a metric space. Giv en x ∈ X and r > 0, we write B ( x, r ) for the open ball with center x ∈ X and radius r > 0. The inﬁmal distance b etw een tw o subsets A and B is deﬁned by d ( A, B ) = inf  d ( a, b ) : a ∈ A and b ∈ B  . Giv en A ⊂ X and r > 0, we denote the op en r -neigh b orhoo d of A b y N X r ( A ) = { x ∈ X : d ( A, x ) < r } . If the ambien t space X is clear from the context, we simply write N r ( A ). W e call a map f : X → Y betw een metric spaces L -Lipschitz if d ( f ( x ) , f ( y )) ≤ Ld ( x, y ) for all x, y ∈ X . The smallest constant that satisﬁes this inequality is called the Lipschitz c onstant of f and is denoted b y Lip( f ). In case f is injective and its inv erse f − 1 is Lipschitz as w ell, we say f is bi-Lipschitz . W e write LIP( X, Y ) for the set of all Lipsc hitz maps from X to Y . If Y = R we abbreviate LIP( X, R ) = LIP( X ). The uniform distance b et w een tw o maps f , g : X → Y is given b y d ( f , g ) = sup  d ( f ( x ) , g ( x )) : x ∈ X  . The following result is easy to verify; see e.g. [5, Lemma 2.1]. Lemma 2.1. L et f : X → Y b e a c ontinuous map fr om a c omp act metric sp ac e X to a sep ar able metric sp ac e Y that is an absolute Lipschitz neighb orho o d r etr act. Then, for every ε > 0 ther e exists a Lipschitz map g : X → Y with d ( f , g ) < ε . Here, Y is said to b e an absolute Lipsc hitz neighborho od retract if there exists C > 0 suc h that the following holds. Whenever Y is a subset of another metric space Z then there exists an op en neighborho od U of Y in Z and a C -Lipschitz retraction π : U → Y . Note that ev ery closed Riemannian n -manifold is an absolute Lipschitz neigh b orhoo d retract; see e.g. [26, Theorem 3.1]. W e also need the following result ab out Lipschitz neigh b orhoo d retracts. 6 DENIS MAR TI Lemma 2.2. L et Y b e a c omp act metric sp ac e that is an absolute Lipschitz neigh- b orho o d r etr act. Then ther e exist C > 0 and ε > 0 with the fol lowing pr op erty. If f , g : X → Y ar e Lipschitz maps fr om a c omp act metric sp ac e X into Y that satisfy d ( f , g ) < ε , then ther e exists a Lipschitz homotopy H : [0 , 1] × X → Y b etwe en f and g satisfying d ( H ( t, x ) , f ( x )) ≤ C d ( f , g ) for every x ∈ X and e ach t ∈ [0 , 1] . Pr o of. W e embed Y ⊂ l ∞ . Since Y is an absolute Lipsc hitz neighborho o d retract there exists a C -Lipschitz retraction π : N l ∞ ε ( Y ) → Y for some C > 0 and ε > 0. Let h : [0 , 1] × X → l ∞ b e the straight-line homotopy b et w een f and g . Notice that h ( t, x ) ∈ N ε ( Y ) for every x ∈ X and each t ∈ [0 , 1]. Therefore, H = π ◦ h is w ell-deﬁned and satisﬁes the desired prop erties. □ Finally , w e introduce injective metric spaces and the injective hull. A metric space X is said to b e injective if the following holds. Whenev er f : A → X is 1-Lipschitz and A is a subset of a metric space B , then there exists a 1-Lipschitz extension f : B → X of f . Imp ortant examples of injective metric spaces are R and l ∞ . Moreo ver, for ev ery metric space X , there exists a minimal injectiv e metric space, called the injectiv e h ull E ( X ) of X , such that X can b e embedded isometrically in to E ( X ). Here, minimal means that an y isometric embedding X → Y in to an injectiv e space Y admits an isometric extension E ( X ) → Y . If X is compact, then E ( X ) is compact as well and can b e realized as a subset of l ∞ . W e refer to [33] for more information. 2.2. Orien tation and degree. W e review the basic deﬁnitions and prop erties of top ological orientation and degree. F or a detailed exp osition of the theory , see [17]. Let X b e a top ological n -manifold without b oundary but not necessarily compact. Notice that throughout this article, all manifolds are assumed to b e connected. F or x ∈ X , we denote the n -th lo cal singular homology group with co eﬃcien ts in Z by H n ( X, X \ x ). It follo ws from the excision theorem that H n ( X, X \ x ) is inﬁnite cyclic. A lo c al orientation o x at x is a generator of H n ( X, X \ x ). F urthermore, an orien tation of X is a choice of lo cal orien tations o x for each x ∈ X that satisﬁes a certain contin uit y condition. In case an orien tation exists, X is said to b e orientable, and X together with an orien tation is called orien ted. It follows that if X is an orien ted manifold, then for each connected, compact subset K ⊂ X the group H n ( X, X \ K ) is inﬁnite cyclic. Moreov er, there exists a generator o K ∈ H n ( X, X \ K ) such that for each x ∈ K the homomorphism from H n ( X, X \ K ) to H n ( X, X \ x ) induced by the inclusion sends o K to o x . F or a compact oriented n -manifold X w e call o X the fundamental class of X and denote it b y [ X ]. No w, let f : X → Y be a con tinuous map betw een tw o orien ted n -manifolds without b oundary . Supp ose that K ⊂ Y is a compact connected subset such that f − 1 ( K ) is compact as well. Then f ∗ : H n ( X, X \ f − 1 ( K )) → H n ( Y , Y \ K ) is a well-deﬁned homomorphism and sends o f − 1 ( K ) to an integer m ultiple of o K . This integer is denoted by deg K ( f ) and called the de gr e e of f over K . If X is compact, then deg K ( f ) is the same num ber for each compact, connected subset K ⊂ X and we write deg ( f ). W e call deg ( f ) the de gr e e of f . It is a homotop y in v ariant. More precisely , if g : X → Y is another con tin uous map that is homotopic to f , then deg( f ) = deg ( g ). If Y is also compact, then the degree is characterized b y the following equality f ∗ [ X ] = deg( f ) · [ Y ]. Next, we deﬁne the lo cal degree. Let x ∈ X and let U ⊂ X be an op en neighborho o d of x . By excision, the homomorphism H n ( U, U \ x ) → H n ( X, X \ x ) is an isomorphism. If V ⊂ Y is an open neighborho o d of f ( x ) such that f ( U ) ⊂ V and f ( y )  = f ( x ) for all y ∈ U \ x , then f ∗ : H n ( U, U \ x ) → H n ( V , V \ f ( x )) is well-deﬁned. This homomorphism sends the local orien tation o x to an in teger m ultiple of the local orien tation o f ( x ) . W e call this num ber the lo c al de gr e e of f at x and denote it by deg( f , x ). Notice that the local degree do es not dep end on U and V . The lo cal degree satisﬁes the follo wing multiplicit y property; see [17, Chapter 8, Corollary THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 7 4.6]. If h : Y → Z is a contin uous map into an orien ted top ological n -manifold Z without b oundary and the lo cal degrees deg( f , x ) and deg( h, f ( x )) are well-deﬁned, then deg( h ◦ f , x ) = deg( h, f ( x )) · deg( f , x ) . W e hav e the following relation betw een the degree and the lo cal degree. Suppose that y ∈ Y is a p oint suc h that f − 1 ( y ) is ﬁnite. Then the lo cal degree of f at eac h x ∈ f − 1 ( y ) is well-deﬁned and we hav e deg( f ) = X x ∈ f − 1 ( y ) deg( f , x ) . W e refer to this property as the additivit y prop ert y of the degree; see [17, Chapter 8, Prop osition 4.7]. A top ological n -manifold with b oundary X is said to be orien table if its interior X \ ∂ X is orientable. In case X is compact and orien ted, it follows that H n ( X, ∂ X ) is inﬁnite cyclic and the orientation of X induces a generator [ X ] of H n ( X, ∂ X ). Again, w e call [ X ] the fundamen tal class of X . Let f : X → Y be a contin uous map b et w een tw o compact, oriented n -manifolds with b oundary such that f ( ∂ X ) ⊂ ∂ Y . Then, the de gr e e deg( f ) of f is the unique integer such that the homomorphism f ∗ : H n ( X, ∂ X ) → H n ( Y , ∂ Y ) satisﬁes f ∗ [ X ] = deg( f ) · [ Y ]. Finally , we note that it is p ossible to deﬁne orientabilit y and the degree using singular homology groups with co eﬃcien ts in Z / 2 Z instead of Z in the same manner as b efore; see [17, Chapter 8, Deﬁnition 4.1]. Ev ery manifold is orientable in this sense. F urthermore, the degree obtained in this wa y is an elemen t of Z / 2 Z and satisﬁes all prop erties analogous to those explained ab o ve. 2.3. Rectiﬁabilit y. Let X b e a complete metric space. W e denote b y H n the n - dimensional Hausdorﬀ measure on X , which is normalized such that H n is equal to the Leb esgue measure L n on R n . A H n -measurable subset E ⊂ X is called n - r e ctiﬁable if there exist compact sets K i ⊂ R n and bi-Lipschitz maps φ i : K i → X suc h that H n E \ [ i φ i ( K i ) ! = 0 . Notice that this deﬁnition is not the standard deﬁnition, but using [3, Lemma 4.1] it can easily b e sho wn that the deﬁnitions are equiv alent. W e say a H n -measurable subset P ⊂ X is pur ely n -unr e ctiﬁable if H n ( P ∩ E ) = 0 for ev ery n -rectiﬁable subset E ⊂ X . W e need the following deep result due to Bate. Theorem 2.3. ([9, Theorem 1.2]) L et X b e a c omplete metric sp ac e and let P ⊂ X b e pur ely n -unr e ctiﬁable with ﬁnite Hausdorﬀ n -me asur e. Then, for every m ∈ N the set { f ∈ LIP 1 ( X, R m ) : H n ( f ( P )) = 0 } is r esidual. Here, Lip 1 ( X, R m ) denotes the space of all Lipschitz maps X → R m with Lipsc hitz constan t at most 1. The theorem w as ﬁrst pro v en in [7] with the additional as- sumption that the lo w er densit y of P is p ositive at almost every point. How ev er, [9, Theorem 1.5] sho ws that this assumption is not necessary . The next result can b e seen as a strong conv erse of the previous result in Euclidean space. Theorem 2.4. ([8, Theorem 1.1]) L et E ⊂ R k b e n -r e ctiﬁable and m > n . Then the set { f ∈ LIP 1 ( R k , R m ) : H n ( f ( E )) = H n ( E ) } is r esidual. If X is a complete metric space, then LIP 1 ( X, R m ) equipp ed with the uniform distance is a complete metric space as w ell. In this case, the Baire category theorem implies that every residual subset of LIP 1 ( X, R m ) is dense. 8 DENIS MAR TI 2.4. Nagata dimension. The Nagata dimension can b e seen as a quantitativ e v ersion of the top ological dimension that is b etter adapted to Lipsc hitz analysis. It has recen tly gained atten tion in the study of metric spaces, most prominently in the context of Lipschitz extension problems; see e.g. [6, 13, 34, 35]. W e say a cov ering of a metric space X has s -multiplicity at most N if each subset of X with diameter less than s intersects at most N mem bers of the cov ering. Deﬁnition 2.5. A metric sp ac e has Nagata dimension dim N X ≤ N if ther e exists c > 0 such that for every s > 0 ther e exists a c overing { U i } i ∈ I of X with s - multiplicity at most N + 1 and diam U i < cs for every i ∈ I . The Nagata dimension is alwa ys at least the top ological dimension and at most the Assouad dimension of a space [36]. In particular, every doubling metric space has ﬁnite Nagata dimension. Let Σ b e a ﬁnite simplicial complex and denote by I the set of vertices of Σ. Then, up to rescaling, Σ can b e realized as a sub complex of Σ( I ) = ( x ∈ l 2 ( I ) : x i ≥ 0 and X i ∈ I x i = 1 ) . The l 2 -metric on Σ is the metric induced by the l 2 -norm on l 2 ( I ) and is denoted by | · | l 2 . F or more details on simplicial complexes, w e refer to [6]. W e need the following factorization theorem. The result is essentially [6, Proposition 6.1] com bined with a F ederer-Fleming deformation type theorem. Theorem 2.6. L et X b e a c omp act metric sp ac e with ﬁnite Hausdorﬀ n -me asur e and ﬁnite Nagata dimension. Then ther e exists a c onstant C > 0 , dep ending only on the data of X , such that for every ε > 0 ther e exist a ﬁnite simplicial c omplex Σ e quipp e d with the l 2 -metric and Lipschitz maps φ : X → Σ , ψ : Σ → E ( X ) with the fol lowing pr op erties (1) ψ is C -Lipschitz on every simplex in Σ ; (2) H n ( φ ( X )) ≤ C H n ( X ); (3) Σ has dimension ≤ n and every simplex in Σ is a Euclide an simplex of side length ε ; (4) Hull( φ ( X )) = Σ and d ( x, ψ ( φ ( x ))) ≤ C ε for al l x ∈ X . Here, Hull( A ) denotes the hul l of A in Σ and is the smallest sub complex of Σ con taining A . Pr o of. Let ε > 0. Since E ( X ) is an injective metric space, it follo ws directly from [6, Prop osition 6.1] and a simple rescaling argument that there exist a ﬁnite simplicial complex Σ ′ of dimension ≤ dim N X and ev ery simplex in Σ ′ is a Euclidean simplex of side length ε and there are Lipsc hitz maps φ ′ : X → Σ ′ , ψ : Σ ′ → E ( X ) such that φ ′ is C -Lipschitz, ψ is C -Lipschitz on each simplex of Σ ′ and d ( x, ψ ( φ ′ ( x ))) ≤ C ε for each x ∈ X . Here, Σ ′ is equipp ed with the l 2 -metric | · | l 2 and C dep ends only on the data of X . F urthermore, [40, Prop osition 5.3] implies that there exists a map p : Σ ′ → Σ ′ suc h that comp osition φ = p ◦ φ ′ is Lipschitz and has the follo wing prop erties; the image φ ( X ) is contained in the n -skeleton of Σ ′ , φ satisﬁes (2) and φ ( x ) , φ ′ ( x ) are con tained in a common simplex for all x ∈ X . Notice that the prop osition actually prov es a stronger version of (2). No w, let x ∈ X . Then, there exist t w o n -dimensional simplices ∆ 1 , ∆ 2 ⊂ Σ ′ with ∆ 1 ∩ ∆ 2  = ∅ and suc h that φ ′ ( x ) ∈ ∆ 1 and φ ( x ) ∈ ∆ 2 . It follows from Lemma [40, Lemma 2.3] that there exists z ∈ ∆ 1 ∩ ∆ 2 satisfying | φ ′ ( x ) − z | l 2 + | z − φ ( x ) | l 2 ≤ 4 √ n | φ ′ ( x ) − φ ( x ) | l 2 . THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 9 Since each simplex in Σ ′ has side-length ε and ψ is C -Lipschitz on each simplex, w e conclude that d ( ψ ( φ ′ ( x )) , ψ ( φ ( x ))) ≤ d ( ψ ( φ ′ ( x )) , ψ ( z )) + d ( ψ ( z ) , ψ ( φ ( x ))) ≤ 8 √ nC ε. Therefore, d ( x, ψ ( φ ( x ))) ≤ d ( x, ψ ( φ ′ ( x ))) + d ( ψ ( φ ′ ( x )) , ψ ( φ ( x ))) ≤ (1 + 8 √ n ) C ε, for all x ∈ X . Finally , we pick a point z i ∈ in t ∆ i \ φ ( X ) for eac h n -simplex ∆ i ⊂ Σ ′ whose in terior is not fully cov ered by φ . W e comp ose φ with a radial pro jection on eac h ∆ i with z i as the pro jection center (we denote the resulting map still by φ ). The comp osition is again Lipschitz and w e do not increase H n ( φ ( X )) b ecause the pro jections centers are not in the image of φ . Clearly , Hull( φ ( X )) is equal to the n -sk eleton of Σ ′ . The same argument as abov e, using [40, Lemma 2.3], sho ws that φ still satisﬁes the second part of (4). This completes the pro of with Σ equal to the n -sk eleton of Σ ′ . □ 3. Metric currents 3.1. Basic deﬁnitions. In this section w e present the theory of metric currents. F or more details we refer to [3] and [32]. Throughout this section let X b e a complete metric space. F or k ∈ N , w e let D k ( X ) = LIP b ( X ) × LIP( X ) k , where LIP b ( X ) denotes the set of b ounded Lipschitz functions X → R . Deﬁnition 3.1. A multiline ar map T : D k ( X ) → R is c al le d metric k -curr ent (of ﬁnite mass) if the fol lowing holds: (1) (c ontinuity) If π j i c onver ges p ointwise to π i for every i = 1 , . . . , k , and Lip π j i < C for some uniform c onstant C > 0 , then T ( f , π j 1 , . . . , π j k ) → T ( f , π 1 , . . . , π k ) as j → ∞ ; (2) (lo c ality) T ( f , π 1 , . . . , π k ) = 0 if for some i ∈ { 1 , . . . , k } the function π i is c onstant on { x ∈ X : f ( x )  = 0 } ; (3) (ﬁnite mass) Ther e exists a ﬁnite Bor el me asur e µ on X such that (3.1) | T ( f , π 1 , . . . , π k ) | ≤ k Y i =1 Lip( π i ) Z X | f ( x ) | dµ ( x ) for al l ( f , π 1 , . . . , π k ) ∈ D k ( X ) . The minimal measure µ satisfying (3.1) is called the mass me asur e of T and is denoted by ∥ T ∥ . W e deﬁne the supp ort of T as spt T =  x ∈ X : ∥ T ∥ ( B ( x, r )) > 0 for all r > 0  . W e write M k ( X ) for the vector space of all metric k -curren ts on X . Endo wed with the mass norm M ( T ) = ∥ T ∥ ( X ), it is a Banach space. Given a Borel set B ⊂ X , the r estriction of T to B is deﬁned by ( T B )( f , π 1 . . . , π k ) = T ( 1 B · f , π 1 , . . . , π k ) . This is a w ell-deﬁned metric k -curren t b ecause eac h metric k -current can b e ex- tended uniquely to L 1 ( ∥ T ∥ ) × LIP k ( X ). The restriction satisﬁes ∥ T B ∥ = ∥ T ∥ B . A sequence of metric k -currents T i ∈ M k ( X ) is said to con v erge weakly to T ∈ M k ( X ) if lim i →∞ T i ( f , π 1 , . . . , π k ) = T ( f , π 1 , . . . , π k ) for all ( f , π 1 , . . . , π k ) ∈ D k ( X ) and w e write T i ⇀ T . F or T ∈ M k ( X ), the b oundary of T is the multilinear map on D k − 1 ( X ) given by ∂ T ( f , π 1 , . . . , π k − 1 ) = T (1 , f , π 1 , . . . , π k − 1 ) 10 DENIS MAR TI for all ( f , π 1 , . . . , π k − 1 ) ∈ D k − 1 ( X ). If ∂ T satisﬁes (3.1), then it is a metric ( k − 1)- curren t, and we call T a normal k -curren t. The set of all normal k -curren ts in X is denoted by N k ( X ) and we deﬁne the normal mass by N ( T ) = M ( T ) + M ( ∂ T ). Again, N k ( X ) equipp ed with the normal mass is a Banach space. The pushforwar d of T ∈ M k ( X ) under a Lipschitz map φ : X → Y is the metric k -current in Y given b y φ # T ( f , π 1 , . . . , π k ) = T ( f ◦ φ, π 1 ◦ φ, . . . , π k ◦ φ ) for all ( f , π 1 , . . . , π k ) ∈ D k ( Y ). It is not diﬃcult to v erify that the pushforw ard satisﬁes the follo wing prop erties: ( φ # T ) B = φ # ( T φ − 1 ( B )), the supp ort of φ # T is contained in the closure of φ (spt T ) and M ( φ # T ) ≤ Lip( φ ) k M ( T ). 3.2. In teger rectiﬁable and in tegral currents. In Euclidean space, ev ery metric curren t is given b y integration with resp ect to a L 1 function as follows. F or θ ∈ L 1 ( R k ), the metric k -curren t J θ K ∈ M k ( R k ) is deﬁned by J θ K ( f , π 1 , . . . , π k ) = Z R k θ f det( D π ) d L k for all ( f , π ) = ( f , π 1 , . . . , π k ) ∈ D k ( R k ). Notice that this c haracterization of metric k -curren ts in R k is a v ery deep result; see [16]. An in teger rectiﬁable curren t is deﬁned in analogy to rectiﬁable sets, using the deﬁnition of currents in Euclidean space describ ed ab ov e. More precisely , a k -current T ∈ M k ( X ) is said to be integer rectiﬁable if there exist countably many compact sets K i ⊂ R k and functions θ i ∈ L 1 ( R k , Z ) with spt θ i ⊂ K i and bi-Lipschitz maps φ i : K i → X suc h that T = X i ∈ N φ i # J θ i K and M ( T ) = X i ∈ N M ( φ i # J θ i K ) . W e call a triple { φ i , K i , θ i } as abov e a p ar ametrization of T . The space of integer rectiﬁable k -curren ts on X is denoted by I k ( X ). F or T ∈ I k ( X ), the char acteristic set of T is deﬁned as (3.2) set T =  x ∈ X : lim inf r → 0 ∥ T ∥ ( B ( x, r )) r k > 0  . By [3, Theorem 4.6], the c haracteristic set of T is k -rectiﬁable and ∥ T ∥ is concen- trated on set T . No w, supp ose that there exists C > 0 such that T has a parametriza- tion { φ i , K i , θ i } satisfying | θ i ( x ) | ≤ C for almost all x ∈ K i and every i ∈ N . It follo ws from [3, Lemma 9.2 and Theorem 9.5] that C − 1 λ H k set T ≤ ∥ T ∥ ≤ C λ H k set T , where λ ∈ L 1 ( X ) satisﬁes k − k/ 2 ≤ λ ≤ k k/ 2 . In particular, w e ha v e M ( T ) ≤ C k k/ 2 H k (set T ). W e say an integer rectiﬁable current is an integral current if it is also a normal current and let I k ( X ) = I k ( X ) ∩ N k ( X ) be the space of integral k -curren ts on X . If an in tegral current T has zero b oundary ∂ T = 0, then we call T an integral cycle. The b oundary-rectiﬁabilit y theorem of Ambrosio and Kirchheim [3, Theorem 8.6] implies that if T ∈ I k ( X ), then ∂ T ∈ I k − 1 ( X ). Therefore, · · · ∂ k +1 − → I k ( X ) ∂ k − → I k − 1 ( X ) ∂ k − 1 − → · · · ∂ 1 − → I 0 ( X ) is a c hain complex. F or k ≥ 0, we deﬁne the k th homology group of this c hain com- plex by H IC k ( X ) = k er ∂ k / im ∂ k +1 . Finally , let M be a closed, orien ted Riemannian n -manifold and let J M K ( f , π ) = Z M f det( Dπ ) d H n for all ( f , π ) ∈ D n ( M ). This deﬁnes an integral n -curren t on M and by Stokes’ theorem T is a cycle. Moreo v er, this cycle satisﬁes ∥ J M K ∥ = H n and generates the n -th homology group via in tegral currents H IC n ( X ). W e th us refer to J M K as the fundamen tal class of M . THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 11 3.3. Isop erimetric inequality. Let T ∈ I k ( X ) b e an integral k -cycle. W e say U ∈ I k +1 ( X ) is a ﬁlling of T if ∂ U = T . Moreov er, if M ( U ) = inf { M ( V ) : V ∈ I k +1 ( X ) , ∂ V = T } then U is called a minimal ﬁlling of T . The isop erimetric inequalit y guaran tees the existence of ﬁllings that satisfy a certain mass b ound. More precisely , we sa y X satisﬁes a k -dimensional (Euclidean) isop erimetric inequalit y if there exists D k > 0 suc h that for every T ∈ I k ( X ) with ∂ T = 0, there exists a ﬁlling S ∈ I k +1 ( X ) of T satisfying M ( S ) ≤ D k M ( T ) k +1 k . In [50], W enger prov ed that metric spaces that satisfy a certain cone inequality also satisfy the isop erimetric inequality for in tegral curren ts. This includes, for example, all Banach spaces and injective metric spaces. W e refer to [5, Theorem 3.4] for the v ersion of the isoperimetric inequalit y presen ted here. Theorem 3.2. L et k ≥ 0 b e an inte ger. Then ther e exists a c onstant D = D k ≥ 1 such that the fol lowing holds. If Y is an inje ctive metric sp ac e and T ∈ I k ( Y ) a cycle, then ther e exists a minimal ﬁl ling U ∈ I k +1 ( Y ) of T and any such ﬁl ling satisﬁes (3.3) M ( U ) ≤ D M ( T ) k +1 k and ∥ U ∥ ( B ( y , r )) ≥ D − k r k +1 for e ach y ∈ spt U and al l r ∈ (0 , d ( y , spt T )) . In p articular, spt U is c ontaine d in the R -neighb orho o d of spt T for R = D M ( U ) 1 k +1 . If k = 0 then the ﬁrst inequality in (3.3) is understo od as an empt y statement. 3.4. Flat currents mo dulo p . W e follo w the theory of ﬂat currents mo dulo p in tro duced in [2] and [4]. F or k ≥ 0, we denote by F k ( X ) the space of ﬂat currents in X , that is, the curren ts that can b e written as T = R + ∂ S , where R ∈ I k ( X ) and S ∈ I k +1 ( X ). Clearly , this deﬁnes an additiv e group. If T is a ﬂat k -current of the form T = R + ∂ S with R ∈ I k ( X ) and S ∈ I k +1 ( X ), then its b oundary ∂ T is the ﬂat ( k − 1)-curren t given by ∂ T = ∂ R . Notice that a ﬂat current does not need to hav e ﬁnite mass as deﬁned in (3.1). Th us, the mass norm do es not provide a go o d notion of distance in this space. Instead, w e consider the ﬂat norm F ( T ) = inf { M ( R ) + M ( S ) : T = R + ∂ S, R ∈ I k ( X ) , S ∈ I k +1 ( X ) } . It follows from the subadditivit y of F and the completeness of I k ( X ) with respect to the mass norm that F deﬁnes a complete metric on F k ( X ). W e ha v e F ( ∂ T ) ≤ F ( T ) for all T ∈ F k ( X ). Let p > 1 be an in teger. W e deﬁne the ﬂat norm mo dulo p of T ∈ F k ( X ) as F p ( T ) = inf {F ( T − pQ ) : Q ∈ F k ( X ) } . F or T , T ′ ∈ F k ( X ), w e write T = T ′ mo d p whenever F p ( T − T ′ ) = 0. In particular, if there exists S ∈ F k ( X ) such that T = T ′ + pS , then T = T ′ mo d p . Giv en T ∈ F k ( X ), we deﬁne the (r elaxe d) p -mass of T by M p ( T ) = inf n lim inf i →∞ M ( T i ) : T i ∈ I k ( X ) , F p ( T − T i ) → 0 o . Clearly , M p ( T ) = M p ( T ′ ) if T = T ′ mo d p . A direct computation shows that the p -mass is subadditiv e and lo w er semi-con tin uous with resp ect to F p -con vergence. Moreo ver, we hav e F p ( T ) ≤ M p ( T ) ≤ M ( T ) for all T ∈ I k ( X ). Let φ : X → Y b e a Lipsc hitz map b et w een tw o complete metric spaces and T ∈ F k ( X ). Then the pushforw ard φ # T is a ﬂat current in Y satisfying F p ( φ # T ) ≤ Lip( φ ) k F p ( T ) and M p ( φ # T ) ≤ Lip( φ ) k M p ( T ) . No w, let E be a compact and con v ex subset of a Banac h space. F urthermore, let T ∈ F k ( E ) b e a ﬂat k -curren t with ﬁnite M p -mass and let π : E → R b e Lipschitz. 12 DENIS MAR TI It was shown in [2] that for almost all r ∈ R , the restriction T { π < r } is a well- deﬁned ﬂat k -curren t and there exists a ﬁnite, non-negativ e and σ -additiv e Borel measure ∥ T ∥ p suc h that M p ( T { π < r } ) = ∥ T ∥ p ( { π < r } ) for almost all r ∈ R . W e refer to ∥ T ∥ p as the p -mass measure of T . Exactly as in [4, Section 2.4], we can extend this equality to closed and op en subsets of E . Notice that in [4], it is assumed that the am bien t Banach space satisﬁes a strong ﬁnite-dimensional approximation prop ert y . Ho w ev er, in the argumen t therein, it is only used that E is compact and that I k ( E ) is dense in I k ( E ) and F k ( E ), which is true for closed and conv ex subsets of a Banac h space; see [2, Prop osition 14.7]. F or an op en or closed subset A ⊂ E , the restriction T A is a ﬂat k -curren t satisfying M p ( T A ) = ∥ T ∥ p ( A ) and T = T A + T ( E \ A ) . It follo ws from the construction that there exists a sequence T n ∈ I k ( E ) such that for all closed or op en sets A ⊂ E we hav e F p ( T n A − T A ) → 0 and M ( T n A ) → M ( T A ) as n → ∞ . Using this, one easily pro v es that ∥ T A ∥ p = ∥ T ∥ p A for all closed and op en sets A ⊂ E and ∥ φ # T ∥ p ≤ Lip( φ ) k φ # ∥ T ∥ p for all Lipsc hitz maps φ : E → F , where F is a compact and con v ex subset of a Banac h space. W e deﬁne spt p T = { x ∈ E : ∥ T ∥ p ( B ( x, r )) > 0 for all r > 0 } . By the ab o v e, we hav e spt p ( φ # T ) ⊂ φ (spt p ( T )). W e also need the reduction of an in teger rectiﬁable curren t mo dulo p . Let T ∈ I k ( X ) and let { φ i , K i , θ i } b e a parameterization of T . A r e duction mo dulo p of T is an y in teger rectiﬁable k -curren t T p with a parametrization { φ i , K i , η i } that satisﬁes η i ( x ) = θ i ( x ) mod p for almost all x ∈ K i and every i ∈ N . Notice that a reduction T p mo dulo p of T is not unique. How ev er, w e alw ays ha ve T = T p mo d p and M ( T p ) is uniquely determined. Since T p is giv en by a parametrization { φ i , K i , η i } satisfying | η i ( x ) | ≤ p for almost all x ∈ K i and every i ∈ N , we ha v e M ( T p ) ≤ pk k/ 2 H k set T . Notice that set T p ⊂ set T . If X is a compact length space, then [2, Theorem 10.5] implies that M p ( T ) = M ( T p ) for each reduction T p of T mo dulo p . Using the injective h ull E ( X ), we can extend this to compact metric spaces. Lemma 3.3. L et X b e a c omp act metric sp ac e and T ∈ I k ( X ) . Then M p ( T ) = M ( T p ) , wher e T p ∈ I k ( X ) is any of T mo dulo p . In p articular, ther e exists a ﬁnite, non-ne gative and σ -additive Bor el me asur e ∥ T ∥ p such that M p ( T B ) = ∥ T ∥ p ( B ) = ∥ T p ∥ ( B ) for every Bor el set B ⊂ X . The measure ∥ T ∥ p coincides with the p -mass measure introduced ab o v e. Pr o of. Let ι : X → E ( X ) b e an isometric embedding. Recall that E ( X ) is compact b ecause X is compact. F urthermore, E ( X ) is geo desic since it is an injective metric space. Let T p ∈ I k ( X ) b e any reduction of T mo dulo p . Notice that ι # T p is also a reduction mo dulo p of ι # T . Therefore, ∥ T p ∥ ( E ) = ∥ ι # T p ∥ ( E ( X )) = M p ( ι # T ) ≤ M p ( T ) . Since T = T p mo d p we hav e M p ( T ) = M p ( T p ) ≤ M ( T p ) and th us, M p ( T ) = M ( T p ). Now, let B ⊂ X b e Borel. Then T p B is a reduction mo dulo p of ( T B ). Hence, the ﬁrst part of the pro of implies that M ( T p B ) = ∥ T p ∥ ( B ). It follows that ∥ T ∥ p : = ∥ T p ∥ is a well-deﬁned, ﬁnite, non-negativ e and σ -additive Borel measure satisfying M p ( T B ) = ∥ T ∥ p ( B ) = ∥ T p ∥ ( B ) for every Borel set B ⊂ X . □ THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 13 As a consequence of the previous lemma, w e obtain the follo wing equiv alen t char- acterization of the ﬂat norm mo dulo p in a compact metric space X . If T ∈ F k ( X ), then F p ( T ) is equal to inf { M p ( R ) + M p ( S ) : T = R + ∂ S + pQ, R ∈ I k ( X ) , S ∈ I k +1 ( X ) , Q ∈ F k ( X ) } . W e conclude the section with the description of the fundamental class mo dulo 2 of a closed, non-orientable Riemannian n -manifold M . Similarly as in the orientable case M supp orts an in tegral curren t J M K ∈ I n ( M ) satisfying ∂ J M K = 0 mo d 2 and ∥ J M K ∥ 2 = H n ; see e.g. [2, Theorem 13.1]. W e call J M K the fundamental class mo dulo 2 of M . Let τ is an y Borel c hoice of orthonormal bases spanning the tangent spaces of M . Then, the integer rectiﬁable current deﬁned by ( f , π 1 , . . . , π n ) 7→ Z M f ⟨ τ , D π ⟩ d H n for all ( f , π 1 , . . . , π n ) ∈ D n ( M ) is equiv alen t to J M K modulo 2. The fundamen tal class mo dulo 2 is unique in the follo wing sense. Whenev er T ∈ I n ( X ) is an integer rectiﬁable current satisfying ∂ T = 0 mo d 2, then T = k · J M K mo d 2 for k equal to either 0 or 1. 3.5. Slicing. Let π : X → R b e a Lipsc hitz function. F or an in tegral curren t T ∈ I k ( X ), the slice of T at t ∈ R is deﬁned as (3.4) ⟨ T , π , t ⟩ = ∂ ( T { π < t } ) − ( ∂ T ) { π < t } . It follo ws that for almost all t ∈ R the slice ⟨ T , π , t ⟩ is an integral ( k − 1)-current satisfying spt ⟨ T , π , t ⟩ ⊂ spt T ∩ π − 1 ( t ) and Z s r ∥⟨ T , π , t ⟩∥ ( B ) dt ≤ Lip( π ) · ∥ T ∥ ( B ∩ { r < π < s } ) , for ev ery Borel B ⊂ X and all r, s ∈ R with r < s ; see [3, Theorem 5.6 and Theorem 5.7]. W e refer to this inequality as the slicing ine quality . W e ha v e ∂ ⟨ T , π , t ⟩ = −⟨ ∂ T , π , t ⟩ and in particular, if T has zero boundary , then ∂ ⟨ T , π , t ⟩ = 0. W e will need the slicing inequality for the p -mass measure. Lemma 3.4. L et E b e a c omp act and c onvex subset of l ∞ . F urthermor e, let T ∈ I k ( E ) and let π : E → R b e Lipschitz. Then, Z s r ∥⟨ T , π , t ⟩∥ p ( B ) dt ≤ Lip( π ) · ∥ T ∥ p ( B ∩ { r < π < s } ) , for every Bor el B ⊂ X and al l r, s ∈ R with r < s . The lemma is known to b e true in case l ∞ is replaced by a separable Banach space; see [2, Theorem 10.6]. Notice that E ( X ) satisﬁes the conditions in the lemma in case X is compact. Pr o of. Let T p ∈ I k ( E ) b e a reduction of T mo dulo p . It is a direct consequence of [3, Theorem 9.7] that for almost all t ∈ R , the slice ⟨ T p , π , t ⟩ is a reduction mo dulo p of ⟨ T , π , t ⟩ . Therefore, by Lemma 3.3, we hav e Z s r ∥⟨ T , π , t ⟩∥ p ( B ) dt = Z s r ∥⟨ T p , π , t ⟩∥ ( B ) dt ≤ ∥ T p ∥ ( B ∩ { r < π < s } ) = ∥ T ∥ p ( B ∩ { r < π < s } ) for every Borel B ⊂ X and all r, s ∈ R with r < s . This completes the pro of. □ 3.6. Pro duct curren ts and cone constructions. W e equip [0 , 1] × E with the Euclidean pro duct metric. Giv en a function f : [0 , 1] × X → R and t ∈ [0 , 1], w e let f t : X → R , x 7→ f ( t, x ). F or T ∈ I k ( X ), w e deﬁne the in tegral current 14 DENIS MAR TI J t K × T ∈ I k ([0 , 1] × E ) by J t K × T ( f , π 1 , . . . , π k ) := T ( f t , π 1 t , . . . , π kt ) for all ( f , π 1 , . . . , π k ) ∈ D k ([0 , 1] × E ). F urthermore, the m ulti-linear functional J 0 , 1 K × T on D k +1 ([0 , 1] × X ) assigning ( f , π 1 , . . . ,π k +1 ) 7→ k +1 X i =1 Z 1 0 ( − 1) k +1 T  f t ∂ π it ∂ t , π 1 t , . . . , π ( i − 1) t , π ( i +1) t , . . . , π ( k +1) t  dt deﬁnes an element of I k +1 ([0 , 1] × X ) and satisﬁes (3.5) ∂ ( J 0 , 1 K × T ) + J 0 , 1 K × ∂ T = J 1 K × T − J 0 K × T . See [3, Theorem 2.9] and [50, Theorem 2.9]. No w, assume that E is a closed and con v ex subset of a Banach space. Then, I k ( E ) is dense in I k ( E ) and F k ( E ) with resp ect to the mass norm and ﬂat norm, resp ectiv ely . It follo ws that F ( T ) = inf { M ( R ) + M ( S ) : T = R + ∂ S, R ∈ I k ( E ) , S ∈ I k +1 ( E ) } for all T ∈ F k ( E ). Moreo v er, by (3.5) we hav e J 0 , 1 K × ( R × ∂ S ) = J 0 , 1 K × R + J 1 K × S − J 0 K × S − ∂ ( J 0 , 1 K × S ) , for all R ∈ I k ( E ) and every S ∈ I k +1 ( E ). Therefore, using a density argument, we can deﬁne J t K × T and J 0 , 1 K × T for ﬂat currents T ∈ F k ( E ) in such a w ay that (3.5) still holds. It is not diﬃcult to show that for T , S ∈ F k ( E ) with T = S mod p we ha ve (3.6) J t K × T = J t K × S mo d p and J 0 , 1 K × T = J 0 , 1 K × S mo d p. Let H : [0 , 1] × E → X b e Lipschitz suc h that for a ﬁxed t ∈ [0 , 1] the map x 7→ H ( t, x ) is L -Lipsc hitz and for a ﬁxed x ∈ E the map t 7→ H ( t, x ) is γ -Lipschitz. A computation exactly as in the pro of of [50, Prop osition 2.10] shows that (3.7) M ( H # ( J 0 , 1 K × T )) ≤ ( k + 1) γ L k M ( T ) , for all T ∈ I k ( E ). It follo ws that if T ∈ F k ( E ) has ﬁnite p -mass M p , then (3.7) holds for T with M replaced by M p . Using this, it is not diﬃcult to show that (3.8) F p ( H # ( J 0 , 1 K × T )) ≤ ( k + 1) γ L k F p ( T ) for all T ∈ F k ( E ) with ﬁnite p -mass M p . Next, we explain the cone construction. W e refer to [50] for more details. Fix some p oint x ∈ E . W e denote by H : [0 , 1] × E → E the straight-line homotopy b et w een the iden tit y on E and the constant map equal to x . F or T ∈ I k ( E ), the cone ov er T with vertex x is deﬁned as { x } × T : = H # ( J 0 , 1 K × T ) ∈ I k +1 ( E ). By (3.5) and (3.7) it has the follo wing prop erties ∂ ( { x } × T ) = T − { x } × ∂ T and M p ( { x } × T ) ≤ 2 r M p ( T ) , where r > 0 is the radius of the smallest closed ball that contains the supp ort of T . Moreo ver, if T , R ∈ I k ( E ) and S ∈ I k +1 ( E ) are suc h that T = R + ∂ S , then { x } × T = { x } × ( R + ∂ S ) = { x } × ( R + S ) + ∂ ( { x } × S ) . Therefore, F ( { x } × T ) ≤ 2 diam( E ) F ( T ). This leads to F p ( { x } × T ) ≤ 2 diam( E ) F p ( T ) for all T ∈ F k ( E ). In particular, if a sequence T i ∈ F k ( E ) satisﬁes F p ( T i − T ) → 0 for some T ∈ F k ( E ), then also the cones conv erge F p ( { x } × T i − { x } × T ) → 0. 3.7. Compactness theorem for in tegral currents mo d p . W e essentially follo w the pro of of the compactness theorem in [1]. The deﬁnition of ﬂat chains used in THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 15 [1] diﬀers from the deﬁnition of ﬂat currents used in this article. Thus, we pro vide the pro of of the compactness theorem. Theorem 3.5. L et E b e a c omp act and c onvex subset of l ∞ . Supp ose that T i ∈ I k ( E ) is a se quenc e satisfying sup i M p ( T i ) + M p ( ∂ T i ) < ∞ . Then, ther e exists a subse quenc e  T i ( j )  j ∈ N of ( T i ) i ∈ N and T ∈ F k ( E ) such that F p ( T i ( j ) − T ) → 0 as j → ∞ . Pr o of. Deﬁne C = sup i M p ( T i ) + M p ( ∂ T i ). W e argue by induction ov er the dimen- sion k . Let k = 0. Put K = { T ∈ F k ( E ) : M p ( T ) + M p ( ∂ T ) ≤ C } . It follo ws from the low er semicon tinuit y of the p -mass that K is closed. Given ε > 0, w e cov er E by a ﬁnite family of balls { B ( x i , ε ) } N i =1 and write Q ⊂ K for the subset of ﬂat curren ts of the form N X i q i J x i K , | q i | < p and N X i | q i | ≤ C. Here, J x K denotes the 0-current deﬁned b y J x K ( f ) = f ( x ) for all f ∈ LIP b ( X ). Since E is conv ex, there exists S ∈ Q for ev ery T ∈ K satisfying F p ( T − S ) ≤ C ε . This sho ws that K is totally b ounded and hence, compact. W e pass to the induction step. Let x ∈ E and r > 0. W e claim that for almost all δ ∈ ( r, 2 r ), there exists a subsequence T i ( j ) suc h that for every j ∈ N w e ha v e T i ( j ) B ( x, δ ) ∈ I k ( E ) and M p ( T i ( j ) B ( x, δ )) + M p ( ∂ ( T i ( j ) B ( x, δ ))) ≤ 2 C δ + C . Indeed, deﬁne π : E → R , y → d ( x, y ), then Lemma 3.4 and F atou’s lemma imply Z 2 δ δ lim inf i →∞ M p ( ⟨ T i , π , t ⟩ ) dt ≤ lim inf i →∞ Z 2 δ δ M p ( ⟨ T i , π , t ⟩ ) dt ≤ sup i M p ( T i ) ≤ C. Therefore, for almost every δ ∈ ( r, 2 r ), w e hav e lim inf i M p ( ⟨ T i , π , δ ⟩ ) ≤ C /δ . In particular, we can ﬁnd a subsequence T i ( j ) satisfying M p ( ⟨ T i ( j ) , π , δ ⟩ ) ≤ 2 C /δ and ⟨ T i ( j ) , π , δ ⟩ ∈ I k − 1 ( E ) for every j ∈ N . By the deﬁnition of the slice op erator we ha ve M p ( ∂ ( T i ( j ) B ( x, δ ))) ≤ M p ( ⟨ T i ( j ) , π , δ ⟩ ) + M p (( ∂ T i ( j ) ) B ( x, δ )) and T i ( j ) B ( x, δ ) ∈ I k ( E ) for ev ery j ∈ N . Since sup i M p ( T i ) + M p ( ∂ T i ) ≤ C , the claim follows. Now, let η > 0 b e such that (1 + 8 C ) η < ε . Applying the claim rep eatedly , w e ﬁnd a ﬁnite co v er of E that consists of balls { B j } n j with radii in ( η , 2 η ) and a subsequence of ( T i ) i (still denoted by ( T i ) i ) suc h that T i B j ∈ I k ( E ) and (3.9) M p ( T i B j ) + M p ( ∂ ( T i B j )) ≤ 2 C δ + C . for all i, j ∈ N . W e decompose E \ S N j =1 ∂ B j in to a ﬁnite partition { U l } N l =1 . F or i ∈ N and l = 1 , . . . , N , w e write T l i = T i U l . Lemma 3.3 implies that the mass p -measure of each T i is additive and hence, M p ( T i ) = N X l M p ( T l i ) , for every i ∈ N . By (3.9) we ha v e sup i M p ( ∂ T l i ) < ∞ for each l = 1 , . . . , N . Th us, we conclude from the induction h yp othesis that we ma y pass to another subsequence (w e k eep writing ( T i ) i ) suc h that ∂ T l i is Cauc h y with resp ect to F p for every l = 1 , . . . , N . W e ﬁx a p oin t x l ∈ U l for each U l . It follows that ev ery 16 DENIS MAR TI sequence { x l } × ∂ T l i is Cauch y as well. Therefore, there exists I ∈ N such that F p ( { x l } × ∂ T l i − { x l } × ∂ T l i ′ ) < η N for every l = 1 , . . . , N and all i, i ′ ≥ I . Fix t wo integers a, b greater than I for the momen t. Then, for eac h l = 1 , . . . , N , there exist R l ∈ I k ( E ), S l ∈ I k +1 ( E ) and Q l ∈ F k ( E ) satisfying { x l } × ∂ T l a − { x l } × ∂ T l b = R l + ∂ S l + pQ l and M p ( S l ) + M p ( R l ) ≤ η N . Therefore, using that ∂ ( { x } × T ) = T − { x } × ∂ T for each T ∈ I k ( E ), w e conclude ( T l a − T l b ) = ∂ ( { x l } × T l a − { x l } × T l b ) − R l + ∂ S l + pQ l . Since each U l has radius at most 2 η we hav e M p ( { x l } × T l a − { x l } × T l b ) ≤ 4 η ( M p ( T l a ) + M p ( T l b )) , for every l = 1 , . . . , N . It follows F p ( T l a − T l b ) ≤ M p ( { x l } × T l a − { x l } × T l b ) + M p ( S l ) + M p ( R l ) ≤ 4 η ( M p ( T l a ) + M p ( T l b )) + η N . Finally , F p ( T a − T b ) ≤ N X l F p ( T l a − T l b ) ≤ N X l 4 η ( M p ( T l a ) + M p ( T l b )) + η N ≤ 4 η N X l ( M p ( T l a ) + M p ( T l b )) + η ≤ 4 η ( M ( T a ) + M ( T b )) + η ≤ (1 + 8 C ) η < ε. Since ε > 0 w as arbitrary this completes the pro of. □ 3.8. Rectiﬁabilit y of ﬂat currents. Let E be a compact con v ex subspace of a Banac h space. It was shown in [4] that the p -mass ∥ T ∥ p of a ﬂat k -current T in E with ﬁnite p -mass is concentrated on a k -rectiﬁable set if the Banach space satisﬁes a strong ﬁnite-dimensional appro ximation prop ert y . W e are able to relax the con- dition on the ambien t Banach space but we need to assume stronger conditions for ∥ T ∥ p . W e note that for ﬂat chains with ﬁnite mass there is an ev en more general rectiﬁabilit y result av ailable [15, Corollary 8.2.3]. Ho w ev er, in order to use this result, we need to connect the theories of ﬂat currents and ﬂat chains. In addition, the pro of of the rectiﬁability result presented here uses a diﬀeren t approach which in the opinion of the author is of indep endent in terest. W e need the following lemma. Lemma 3.6. L et T ∈ F k ( l ∞ ) . Supp ose that T has ﬁnite p -mass and spt p T is c omp act. Then, for every ε > 0 ther e exist a ﬁnite dimensional line ar subsp ac e V ⊂ l ∞ and a 1 -Lipschitz map π : l ∞ → V such that F p ( T − π # T ) ≤ 2( k + 1) ε F p ( T ) . Pr o of. Let ε > 0. Set K = spt p T . Since l ∞ has the metric appro ximation prop ert y , there exist a ﬁnite dimensional linear subspace V ⊂ l ∞ and a 1-Lipschitz map π : l ∞ → V suc h that ∥ π ( x ) − x ∥ l ∞ < ε for each x ∈ K ; see e.g. [14, Prop osition A.6]. Let H : [0 , 1] × K → V b e the straight line homotop y betw een the inclusion THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 17 ι K of K into l ∞ and π . By (3.5) we hav e ∂ ( H # ( J 0 , 1 K × T )) − H # ( J 0 , 1 K × ∂ T ) = T − π # T . Notice that for a ﬁxed t the map x 7→ H ( t, x ) is 1-Lipschitz and for a ﬁxed x ∈ K the map t 7→ H ( t, x ) is ε -Lipschitz. Therefore, (3.8) implies F p ( T − π # T ) ≤ F p ( H # ( J 0 , 1 K × T )) + F p ( H # ( J 0 , 1 K × ∂ T )) ≤ 2( k + 1) ε F p ( T ) . This completes the pro of. □ W e can now pro v e the follo wing rectiﬁability result for ﬂat currents with ﬁnite p -mass. Prop osition 3.7. L et E b e a c omp act and c onvex subset of l ∞ and let T ∈ F k ( E ) . Supp ose that T has ﬁnite p -mass and ∥ T ∥ p is c onc entr ate d on a set with ﬁnite Hausdorﬀ k -me asur e. Then, ∥ T ∥ p is c onc entr ate d on a k -r e ctiﬁable set A ⊂ E . Mor e over, ther e exists S ∈ I k ( E ) with S = T mo d p . Pr o of. Let C ⊂ E b e the set with ﬁnite Hausdorﬀ k -measure ∥ T ∥ p is concentrated on, and let P ⊂ C b e purely k -unrectiﬁable. W e need to prov e that ∥ T ∥ p ( P ) = 0. Since ∥ T ∥ p is a Borel measure, it follo ws from inner regularit y that it suﬃces to sho w that ∥ T ∥ p ( P ′ ) = 0 for all closed subsets P ′ ⊂ P in order to show that ∥ T ∥ p ( P ) = 0. Therefore, we may supp ose that P is closed. W e claim that φ # T P = 0 mo d p for all Lipschitz maps φ : E → V whenever V is a ﬁnite dimensional Banach space. Clearly , the statement is inv ariant under bi-Lipschitz changes of the metric on V . Hence, it suﬃces to prov e the claim with the additional assumption that V is equal to some R N equipp ed with the standard Euclidean distance. Let φ : E → R N b e L - Lipsc hitz and let ε > 0. It follows from Theorem 2.3 that there exists a L -Lipschitz map ψ : E → R N satisfying d ( φ, ψ ) < ε and H k ( ψ ( P )) = 0. Notice that spt p ( ψ # ( T P )) ⊂ ψ (spt p ( T P )) ⊂ ψ ( P ) . F urthermore, ψ # ( T P ) is a ﬂat k -curren t in R N with ﬁnite p -mass M p . By [4, Corollary 1.3], we ha v e that ∥ ψ # ( T P ) ∥ p is absolutely contin uous with resp ect to the Hausdorﬀ k -measure H k and hence, ψ # ( T P ) = 0 mo d p . Let H : [0 , 1] × E → R N b e the straight line homotopy b et w een φ and ψ . By (3.5) we hav e ∂ ( H # ( J 0 , 1 K × T P )) + H # ( J 0 , 1 K × ∂ ( T P )) = ψ # T P − φ # T P . F or a ﬁxed t , the map x 7→ H ( t, x ) is L -Lipsc hitz and for a ﬁxed x ∈ K , the map t 7→ H ( t, x ) is ε -Lipschitz. It follows from (3.8) that F p ( H # ( J 0 , 1 K × T P )) ≤ ( k + 1) εL k F p ( T P ) and F p ( H # ( J 0 , 1 K × ∂ ( T P ))) ≤ ( k + 1) εL k − 1 F p ( ∂ ( T P )) . Therefore, F p ( φ # ( T P )) = F p ( φ # ( T P ) − ψ # ( T P )) ≤ ( k + 1) ε ( L k − 1 + L k ) F p ( T P ) . Since ε > 0 was arbitrary , we conclude that φ # T P = 0 mo d p . This prov es the claim. Lemma 3.6 implies that there exists a sequence of ﬁnite dimensional vector subspaces V i ⊂ l ∞ and Lipschitz maps π i : E → V i suc h that F p ( T P − π i # ( T P )) ≤ ( k + 1) ε F p ( T P ). It follo ws from the claim that π i # ( T P ) = 0 mo d p for all i ∈ N . By passing to the limit w e get T P = 0 mo d p and in particular, ∥ T ∥ p ( P ) = 0. W e conclude that ∥ T ∥ p is concentrated on a k -rectiﬁable set. The second part of the statement can now be prov en exactly as [4, Corollary 1.3]. □ 4. Linear local contractibility Throughout this section, X denotes a metric space. Let R > 0 and λ ≥ 1. A subset B ⊂ X is called ( λ, R )-linearly locally con tractible if for every x ∈ B and 18 DENIS MAR TI ev ery r ∈ (0 , R ), the ball B ( x, r ) is con tractible within B ( x, λr ). W e say X is ( λ, R )-linearly lo cally contractible around a p oin t x ∈ X if the ball B ( x, R ) is ( λ, R )-linearly lo cally con tractible. Finally , supp ose that X has a ﬁnite Hausdorﬀ n -measure. Then, X is said to be almost everywhere linearly lo cally con tractible if X is linearly lo cally contractible around almost all p oin ts x ∈ X . Notice that in the previous deﬁnition, w e do allo w the linear lo cal contractibilit y constants to depend on the p oin t. The follo wing easy example sho ws that this deﬁnition is indeed w eaker than the standard deﬁnition. Let C b e a h yperb olic cone that b ecomes thinner to w ards the tip y of the cone. Then C is (1 , R )-linearly lo cally con tractible around eac h x  = y , pro vided that R is c hosen suﬃciently small such that B ( x, 2 R ) do es not wrap around the cone. Ho wev er, C is not ( λ, R )-linearly lo cally contractible around the tip y . Linear lo cal contractibilit y is a useful to ol for constructing and extending con tinuous functions in a con trolled manner. Here, w e extend these classical techniques using the weak er v ersion of linear local con tractibility that was in tro duced ab o v e. W e need the follo wing deﬁnition. A map f : X → Y b et w een t w o metric spaces is called ε -contin uous if there exists δ > 0 such that d ( f ( x ) , f ( y )) < ε for all x, y ∈ X with d ( x, y ) < δ . Prop osition 4.1. L et X , Y b e c omp act metric sp ac es such that X has top olo gic al dimension at most n and B ⊂ Y is a close d subset that is ( λ, R ) -line arly lo c al ly c ontr actible. L et A b e a (p ossibly empty) subset of X . Ther e exists Q ≥ 1 such that if f : X → Y is ε -c ontinuous on X with ε < R/Q and f is c ontinuous at every p oint of A and f ( X \ A ) ⊂ B , then ther e is a c ontinuous map g : X → Y which c oincides with f on A and satisﬁes d ( f , g ) < Q · ε . Her e, Q dep ends only on λ and n . W e record the following consequence of the proposition: if tw o maps f 0 , f 1 : X → B are contin uous and satisfy d ( f 0 , f 1 ) < ε , then they are homotopic through a homotop y H : [0 , 1] × X → Y satisfying d ( H ( t, x ) , f 0 ( x )) < (1 + Q ) · ε for every x ∈ X and each t ∈ [0 , 1]. The prop osition can b e pro ven analogously to the corresponding result in [45]; see also [46, Prop osition 5.4]. W e therefore only provide a sketc h of the pro of. Pr o of. Let Q ≥ 1 b e suﬃcien tly large to b e determined later and let δ > 0 b e such that d ( f ( x ) , f ( y )) < ε < R /Q whenever d ( x, y ) < δ . Since X is compact and has top ological dimension at most n , there exists an op en cov ering { U i } i ∈ N of X \ A with multiplicit y at most n + 1 and (4.1) diam U i < 1 2 min { δ, d ( U i , A ) } for all i ∈ N . F urthermore, there exist a simplicial complex Σ of dimension ≤ n and a con tin uous map φ : X \ A → Σ with the following property . Whenever x ∈ U i and y ∈ U j , then φ ( x ) and φ ( y ) are contained in a common simplex if and only if U i ∩ U j  = ∅ . It follo ws that for each i ∈ N , there is a vertex v i ∈ Σ suc h that φ ( U i ) is contained in a union of simplices with v i as a vertex. The simplicial complex Σ is usually referred to as the nerve of the cov er { U i } i ∈ N ; see e.g. [45, Section 3]. Let Σ 0 = { v 1 , v 2 , . . . } be the 0-sk eleton of Σ, where eac h v i corresp onds to U i as explained b efore. F or each i ∈ N , ﬁx a p oin t x i ∈ U i and deﬁne ψ 0 : Σ 0 → B by ψ 0 ( v i ) = f ( x i ) for each vertex v i . Let v i , v j b e t w o vertices of the same simplex. By (4.1), we hav e d ( x i , x j ) < δ and hence, d ( ψ 0 ( v i ) , ψ 0 ( v j )) < ε . Therefore, if Q > 0 is suﬃciently large with resp ect to λ , n and R w e can use the linear lo cal con tractibility to extend ψ 0 inductiv ely o ver the skeleton of Σ to a contin uous map ψ : Σ → Y suc h that the following holds. F or each i ∈ N and ev ery y ∈ Σ that is THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 19 con tained in a simplex with v i as a vertex, we ha v e (4.2) d ( g ( y ) , g ( x i )) ≤ C diam U i , where C dep ends only on n and λ . Finally , w e deﬁne g : X → Y as g = f on A and as ψ ◦ φ on X \ A . Using (4.1) and (4.2), it is not diﬃcult to show that g is con tinuous and satisﬁes d ( f , g ) < Qε . □ 4.1. Linear lo cal contractibilit y and manifolds. F or the remainder of the sec- tion, supp ose that X has ﬁnite Hausdorﬀ n -measure and is homeomorphic to a compact smo oth n -manifold M . W e denote the homeomorphism by ϱ : M → X . First, w e describ e the construction of the orientable double cov er of a manifold with- out b oundary . Recall that for each x ∈ X , the relative homology group H n ( X, X \ x ) is inﬁnite cyclic. The orientable double cov er of X is deﬁned as ˜ X =  ( x, o x ) : x ∈ X , o x is a generator of H n ( X, X \ x )  . F or each x ∈ X , there exist tw o p oints ( x, o + x ) , ( x, o − x ) ∈ ˜ X that corresp ond to the p ositiv e and negativ e lo cal orientations at x . F urthermore, there exists a top ology on ˜ X such that the map π : ˜ X → X, ( x, o x ) 7→ x is a (topological) double co v er. Therefore, for each x ∈ X there exists an open connected neighborho od U ⊂ X such that π − 1 ( U ) consists of t w o components U + , U − ⊂ ˜ X and π restricted to either of these comp onen ts is a homeomorphism. The orien table double cov er is a closed, orien table manifold and is connected if and only if X is non-orientable. Finally , we describ e how to construct a metric on ˜ X . Let { U i } N i =1 b e an op en cov er of X such that for each i = 1 , . . . , N , the preimage π − 1 ( U i ) consists of t w o comp onents that are homeomorphic to U i . W e endow each U + i and U − i with a metric suc h that π restricts to an isometry on each member of { U + i , U + i } N i =1 . Notice that if t w o sets V , W ∈ { U + i , U + i } N i ha ve non-empty in tersection, then for all x, y ∈ V ∩ W , the distance betw een x and y in V and in W is equal. It follo ws that there exists a metric on ˜ X that agrees with the top ology , such that the map π : ˜ X → X is a lo cal isometry . The next lemma is an immediate consequence of the deﬁnitions and the fact that π is a lo cal isometry . Lemma 4.2. If X is line arly c ontr actible ar ound x , then ˜ X is line arly c ontr actible ar ound ( x, o + x ) and ( x, o + x ) . In p articular, if X is (almost everywher e) line arly lo c al ly c ontr actible, then ˜ X is (almost everywher e) line arly lo c al ly c ontr actible. W e can now easily prov e the v alidity of the relative isop erimetric inequality in non-orien table metric manifolds that are Ahlfors n -regular and linearly lo cally con- tractible. Pr o of of The or em 1.8. Let X be a metric space that is homeomorphic to a closed, non-orien table smo oth n -manifold. F urthermore, supp ose that X is Ahlfors n - regular and linearly lo cally contractible. Let ˜ X b e the orientable double cov er of X and π : ˜ X → X b e the co vering map. By Lemma 4.2 the orien table double co v er ˜ X is linearly lo cally contractible as well. Moreov er, the cov ering map π is a lo cal isometry and thus, ˜ X is also Ahlfors n -regular. It follo ws from [5, Theorem 1.5] that ˜ X satisﬁes a relativ e isop erimetric inequality . Therefore, using again that π is a lo cal isometry , w e conclude that X satisﬁes a lo cal relative isop erimetric inequality . That is, min  H n  E ∩ B  , H n  B \ E  ≤ C · M −  E | λB  n n − 1 holds for every Borel subset E ⊂ X and every open ball B ⊂ X with suﬃcien tly small radius. Since X is compact the lo cal relative isop erimetric inequality upgrades to a global inequalit y . Notice the Ahlfors regularity and linear contractibilit y con- stan ts of X and ˜ X are comparable. Hence, the constants C , λ ≥ 1 dep end only on the data of X . This completes the pro of. □ 20 DENIS MAR TI Next, we explain how to glue tw o copies of X along their b oundaries. Deﬁnition 4.3. Supp ose that X has b oundary. L et ˆ X b e the sp ac e obtaine d as the quotient ˆ X = X 1 ⊔ X 2 / ∼ , wher e X 1 and X 2 ar e two c opies of X and x ∼ y if and only if x ∈ ∂ X 1 , y ∈ ∂ X 2 and x = y . We e quip ˆ X with the quotient metric ˆ d , which in this c ase has the fol lowing form ˆ d ( x, y ) = ( d ( x, y ) if x, y ∈ X 1 or x, y ∈ X 2 , inf z ∈ ∂ X d ( x, z ) + d ( z , y ) if x ∈ X 1 , y ∈ X 2 or y ∈ X 1 , x ∈ X 2 . It is not diﬃcult to sho w that ˆ X is a closed manifold and orien table in case X is orientable. W e write ι 1 , ι 2 : X → ˆ X for the isometric embeddings of X into ˆ X that identify X with X 1 and X 2 , resp ectiv ely . Given a map φ : X → M satisfying φ ( ∂ X ) ⊂ ∂ M , we let ˆ φ : ˆ X → ˆ M b e the map such that ˆ φ ( ι X i ( x )) = ι M i ( φ ( x )) for ev ery x ∈ X and i = 0 , 1. M 1 ˆ M M 2 X 1 ˆ X X 2 ι M 1 ι M 2 ι X 1 φ ˆ φ ι X 2 φ It is not diﬃcult to sho w that if φ is contin uous or Lipschitz, then ˆ φ is con tin uous or Lipsc hitz as well. In particular, ˆ X is homeomorphic to ˆ M via the homeomorphism ˆ ϱ : ˆ M → ˆ X . F urthermore, if X is orientable, then the degree deg( φ ) of φ is w ell- deﬁned and we hav e deg ( φ ) = deg ( ˆ φ ). Finally , we record the following lemma for later use. Lemma 4.4. If X is ( λ, R ) -line arly c ontr actible ar ound x ∈ X \ ∂ X , then ˆ X is ( λ, ˆ R ) -line arly c ontr actible ar ound ι 1 ( x ) and ι 2 ( x ) , wher e ˆ R = min { R, d ( x, ∂ X ) / 3 λ } . In p articular, if H n ( ∂ X ) = 0 and X is almost everywher e line arly lo c al ly c on- tr actible, then ˆ X is almost everywher e line arly lo c al ly c ontr actible. Pr o of. Put ˆ x = ι 1 ( x ). Let ˆ y ∈ B ( ˆ x, ˆ R ) and r ∈ (0 , ˆ R ). By assumption, the ball B ( y , r ) is contractible within B ( y , λr ), where y = ι − 1 1 ( ˆ y ). Notice that B ( y , λr ) ⊂ B ( x, 2 λ ˆ R ) and B ( x, 2 λ ˆ R ) ∩ ∂ X = ∅ . Therefore, ι 1 restricted to B ( x, 2 λ ˆ R ) is an isometry and in particular, B ( ˆ y , r ) is contractible within B ( ˆ y , λr ). The proof for ι 2 ( x ) is analogous. Finally , since H n ( ι 1 ( ∂ X )) = 0 the second part of the statement follo ws directly from the ﬁrst part. □ W e conclude the discussion with the following lemma, which will b e useful to pro ve the main theorems. Lemma 4.5. Supp ose that ˆ X has a metric fundamental class ˆ T ∈ I n ( ˆ X ) (mo dulo 2 ). Then, T = ι − 1 1# ( ˆ T ι 1 ( M 1 )) ∈ I n ( X ) deﬁnes a metric fundamental class (mo dulo 2 ) of X . Mor e over, supp ose that ˆ T has the fol lowing uniqueness pr op erty: whenever S ∈ I n ( ˆ X ) is an n -cycle (mo dulo 2 ), then ther e exists k ∈ Z (or k = 0 , 1 ) such that S = k · ˆ T (or S = k · ˆ T mo d 2 ). Then, T satisﬁes the same uniqueness pr op erty for inte gr al curr ents S ∈ I n ( X ) with spt ∂ S ⊂ ∂ X or ( spt 2 ∂ S ⊂ ∂ X ). Pr o of. Let ˆ T b e a metric fundamental class of ˆ X . Then there exists C > 0 such that ∥ ˆ T ∥ ≤ C · H n ˆ X . It follows that T satisﬁes the same inequality b ecause ι X 1 is an isometric embedding. No w, let φ : X → M b e Lipsc hitz such that φ ( ∂ X ) ⊂ ∂ M . Then there exists a Lipschitz map ˆ φ : ˆ X → ˆ M satisfying deg( φ ) = deg ( ˆ φ ) and THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 21 ˆ φ ( ι X i ( x )) = ι M i ( φ ( x )) for every x ∈ X and i = 0 , 1. Therefore, φ # T = (( ι M 1 ) − 1 ◦ ˆ φ ◦ ι X 1 ) # T = (( ι M 1 ) − 1 ◦ ˆ φ ) # ( ˆ T ι X 1 ( X )) = ( ι M 1# ) − 1 (deg( ˆ φ ) · ( J ˆ M K ι M 1 ( M ))) = deg( φ ) · J M K . Finally , let S ∈ I n ( X ) b e such that spt( ∂ S ) ⊂ ∂ X . Then ˆ S = ι X 1# S − ι X 2# S deﬁnes an integral n -cycle in ˆ X . Hence, there exists k ∈ Z such that k · ˆ T = ˆ S and in particular, k · T = k · ι − 1 1# ( ˆ T ι X 1 ( X )) = ι − 1 1# ( ˆ S ι X 1 ( X )) = S. In case ˆ T is a metric fundamental class mo dulo 2, the pro of is analogous. □ 4.2. Linear lo cal contractibilit y and degree. Let X b e a closed metric n - manifold with ﬁnite Hausdorﬀ n -measure. The next lemma provides a contin uous extension of bi-Lipschitz maps from Euclidean space into X that are compatible with the top ology of X . Lemma 4.6. L et K ⊂ R n b e c omp act and φ : K → X b e bi-Lipschitz. Supp ose that the image of φ is c ontaine d in a line arly lo c al ly c ontr actible subset B ⊂ X . Then, ther e exists a c ontinuous extension φ : U → X of φ to an op en neighb orho o d U of K and C > 0 such that (4.3) d ( φ ( x ) , φ ( y )) ≤ C ∥ x − y ∥ for e ach x ∈ K and every y ∈ U . Her e, C > 0 dep ends only on n , the bi-Lipschitz c onstant of φ and the line ar lo c al c ontr actibility c onstants of B . The lemma is a lo calized version of [5, Lemma 5.3] and is based on the same argumen t. Pr o of. Let { Q i } i b e a Whitney cub e decomposition of R n \ K , see [49, Theorem VI.1.1]. That is, the cub es { Q i } i co ver R n \ K , ha ve disjoin t interiors and (4.4) n diam Q i ≤ d ( Q i , K ) ≤ 4 n diam Q i for each i ∈ N . Let δ > 0 b e suﬃcien tly small to be determined later. W e deﬁne U as the union of all cub es Q i that intersect the δ -neighborho od. Let U 0 = { v 1 , v 2 , . . . } b e the set of all vertices of U . F or each l ∈ N , let x l ∈ K b e suc h that d ( v l , x l ) ≤ 2 d ( Q i , K ), where Q i is a cub e with v l as a vertex. W e deﬁne φ : U 0 → X b y φ ( v l ) = φ ( x l ). Let v l , v k ∈ Q i b e t w o v ertices and let L be the bi-Lipschitz constan t of φ . Then d ( φ ( v l ) , φ ( v k )) ≤ L ∥ x l − x k ∥ ≤ L (diam Q i + 4 d ( Q i , K )) ≤ 5 Ld ( Q i , K ) < 5 Lδ. Therefore, if δ is suﬃciently small, we can use that φ ( U 0 ) is con tained in a linearly lo cally contractible subset B ⊂ X to extend φ to a con tinuous map on U such that diam φ ( Q i ) ≤ cd ( Q i , K ) holds for each cub e Q i ⊂ U . Here, c dep ends only on n , L and the linear lo cal con tractibilit y constan ts of B . Finally , let x ∈ K and y ∈ U . Cho ose a cub e Q i ⊂ U containing y , and let v l b e a vertex of Q i . Then d ( φ ( x ) , φ ( y )) ≤ L ∥ x − x l ∥ + c diam Q i ≤ (4 L + c ) d ( Q i , K ) ≤ (4 L + c ) ∥ x − y ∥ . This completes the pro of. □ W e record the following consequence of the previous lemma. Lemma 4.7. L et φ : U → X b e a c ontinuous extension of a bi-Lipschitz map φ : K → X as in L emma 4.6. F urthermor e, let π : X → R n b e Lipschitz and let ψ : R n → R n b e a Lipschitz extension of π ◦ φ . Then for almost al l y ∈ R n and for every z ∈ K ∩ ψ − 1 ( y ) the lo c al de gr e es deg( φ, z ) , deg ( π , φ ( z )) ar e wel l-deﬁne d and satisfy (4.5) deg( φ, z ) · deg( π , φ ( z )) = sgn(det D z ψ ) . 22 DENIS MAR TI Notice that (4.5) implies that, for almost all y ∈ R n and for ev ery x ∈ π − 1 ( y ) ∩ φ ( K ), the lo cal degree deg( π , x ) is equal to 1 or − 1. Moreo v er, taking for π a Lipschitz extension of φ − 1 and ψ = id R n , we conclude that the lo cal degree deg( φ, z ) is equal to 1 or − 1 for almost all z ∈ K . Pr o of. Let C b e the constan t from (4.3) and let L ≥ 1 b e the bi-Lipschitz constan t of φ . F or the momen t, ﬁx a density p oin t z ∈ K suc h that ψ is diﬀerentiable at z . Then there exists r > 0 such that whenever u ∈ B ( z , r ), then there exists v ∈ K satisfying ∥ u − v ∥ < (2 C L ) − 1 ∥ u − z ∥ . F or such u and v we ha ve d ( φ ( z ) , φ ( u )) ≥ L − 1 ∥ z − v ∥ − C ∥ u − v ∥ ≥ (2 L ) − 1 ∥ u − z ∥ . Therefore, φ ( u )  = φ ( z ) for all u ∈ B ( z , r ) \ z and in particular, the lo cal degree deg( φ, z ) is w ell-deﬁned. F urthermore, for ε > 0 there exists r > 0 such that for ev ery u ∈ B ( z , r ) the following holds: (1) ∥ ψ ( z ) − ψ ( u ) − D z ψ ( u − z ) ∥ ≤ 2 − 1 ε ∥ u − z ∥ ; (2) there exists v ∈ K satisfying ∥ u − v ∥ < (2(Lip( ψ ) + C ) − 1 ε ∥ u − z ∥ . Since ψ and f = π ◦ φ agree on K w e hav e ∥ f ( z ) − f ( u ) − D z ψ ( u − z ) ∥ ≤ ∥ ψ ( z ) − ψ ( u ) − D z ψ ( u − z ) ∥ + ∥ ψ ( u ) − f ( u ) ∥ ≤ (2 ε ) − 1 ∥ u − z ∥ + ∥ ψ ( u ) − ψ ( v ) ∥ + ∥ f ( v ) − f ( u ) ∥ ≤ ε ∥ u − z ∥ for all u ∈ B ( z , r ). It follows that π ◦ φ is diﬀerentiable at z with D z ( π ◦ φ ) = D z ψ . The area and coarea formula imply that for almost all y ∈ R n the preimage ψ − 1 ( y ) ∩ K is ﬁnite and ψ is diﬀeren tiable at each z ∈ ψ − 1 ( y ) ∩ K with non-degenerate deriv ative. Notice that for such z the lo cal degrees deg( ψ , z ) and deg ( π, φ ( z )) are w ell-deﬁned. Since almost every z ∈ K is a density p oin t, the degree implies sgn(det D z ψ ) = sgn(det D z ( π ◦ φ )) = deg( π ◦ φ, z ) = deg( φ, z ) · deg( π , φ ( z )) for almost all y ∈ R n and for every z ∈ K ∩ ψ − 1 ( y ). This completes the pro of. □ The local degree in the previous lemma is deﬁned through singular homology groups with in teger co eﬃcien ts. How ev er, the statement remains true if the degree is deﬁned using the singular homology groups with coeﬃcients in Z / 2 Z , denoted by deg( φ, x, Z / 2 Z ), as explained in Section 2.2. More precisely , Lemma 4.7 states that for almost all y ∈ R n and for ev ery x ∈ π − 1 ( y ) ∩ φ ( K ) w e hav e deg( π , x, Z / 2 Z ) = [1] ∈ Z / 2 Z . W e obtain the following corollary , which is a well-kno wn result if X is smo oth. Corollary 4.8. L et X b e a close d metric n -manifold with ﬁnite Hausdorﬀ n - me asur e and let π : X → R n b e Lipschitz. If X is almost everywher e line arly lo c al ly c ontr actible and n -r e ctiﬁable, then # π − 1 ( y ) is even for almost al l y ∈ R n . Observ e that, once we prov ed Corollary 1.5, the assumption that X is n -rectiﬁable is redundant. Pr o of. Since X is n -rectiﬁable, there exist coun tably many compact sets K i ⊂ R n and bi-Lipschitz maps φ i : K i → X suc h that the φ i ( K i ) are pairwise disjoint and H n X \ [ i φ i ( K i ) ! = 0 . Applying Lemma 4.7 to each φ i implies that for almost all y ∈ R n and for ev ery x ∈ π − 1 ( y ), the preimage π − 1 ( y ) is ﬁnite and the lo cal degree deg( π , x, Z / 2 Z ) is non-zero. Therefore, b y the additivity form ula of the degree deg( π , Z / 2 Z ) = X x ∈ π − 1 ( y ) deg( π , x, Z / 2 Z ) =  # π − 1 ( y )  ∈ Z / 2 Z THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 23 for almost all y ∈ R n . Since X is closed w e ha ve deg ( π , Z / 2 Z ) = [0] ∈ Z / 2 Z ; see e.g. [17, P age 269]. It follows that # π − 1 ( y ) is even for almost all y ∈ R n . □ An analogous argumen t sho ws that whenever π : X → M is Lipschitz and M X is a closed Riemannian manifold, then # π − 1 ( y ) is even in case deg( π , Z / 2 Z ) is zero and o dd otherwise. 5. Orient able manifolds The goal of this section is to prov e Theorem 1.2. The pro of is split into tw o parts. W e ﬁrst sho w that there exists a metric fundamen tal cycle satisfying the upper b ound in (1.1). Lemma 5.1. L et X b e a close d, oriente d metric n -manifold with ﬁnite Hausdorﬀ n -me asur e. Supp ose that X is almost everywher e line arly lo c al ly c ontr actible. Then X has a metric fundamental class T ∈ I n ( X ) with a p ar ametrization { φ i , K i , θ i } satisfying | θ i | = 1 for almost every x ∈ K i and e ach i ∈ N . Pr o of. Let X = E ∪ P be a partition where E is n -rectiﬁable and P is purely n -unrectiﬁable. There exists a coun table collection of compact set K i ⊂ R n and bi-Lipsc hitz maps φ i : K i → Y suc h that the images φ i ( K i ) are pairwise disjoint and H n  E \ [ i ∈ N φ i ( K i )  = 0 . By partitioning each K i if necessary , we ma y assume that the image of each φ i is fully contained in a linearly lo cally contractible subset of X . F or each i ∈ N , let φ i : U i → C b e a contin uous extension of φ i giv en b y Lemma 4.6. It follows from Lemma 4.7 that the lo cal degree deg( φ i , z ) exists and is equal to 1 or − 1 for almost all z ∈ K i and each i ∈ N . W e deﬁne T = X i ∈ N φ i # J θ i K where θ i = deg( φ i , z ). Now, let π : X → R n b e a Lipsc hitz map and supp ose that H n ( π ( P )) = 0. Fix i ∈ N for the momen t. Let ψ : R n → R n b e a Lipsc hitz extension of π ◦ φ i . Lemma 4.7 implies that for almost all y ∈ R n and for ev ery z ∈ K ∩ ψ − 1 ( y ), the lo cal degrees deg ( φ i , z ) , deg( π , φ ( z )) are well-deﬁned and satisfy deg( φ, z ) · deg( π , φ ( z )) = sgn(det D z ψ ) . Therefore, φ i # J θ i K (1 , π ) = Z K i θ i ( z ) det( D z ψ ) d L n ( z ) = Z K i deg( φ i , z ) sgn(det D z ψ ) | det( D z ψ ) | d L n ( z ) = Z K i deg( π , φ ( z )) | det( D z ψ ) | d L n ( z ) . By the area formula and b ecause φ i is bi-Lipschitz we ha ve φ i # J θ i K (1 , π ) = Z R n   X x ∈ π − 1 ( y ) 1 φ i ( K i ) ( x ) deg ( π , x )   d L n ( y ) . 24 DENIS MAR TI Recall that H n ( π ( P )) = 0 and hence, by the additivity property of the degree T (1 , π ) = X i ∈ N φ i # J θ i K (1 , π ) = Z R n   X x ∈ π − 1 ( y ) deg( π , x )   d L n ( y ) = Z R n deg( π ) d L n ( y ) . Since X is closed w e ha v e deg( π ) = 0 and in particular, T (1 , π ) = 0 for ev ery Lipsc hitz map π : X → R n satisfying H n ( π ( P )) = 0. If π : X → R n is any Lipschitz map, then Theorem 2.3 implies that there exists a sequence of Lipsc hitz maps π i : X → R n con verging uniformly to π and satisfying Lip π i = Lip π as w ell as H n ( π i ( P )) = 0 for eac h i ∈ N . It follows from the contin uit y prop erty of metric curren ts that T (1 , π ) = lim i →∞ T (1 , π i ) = 0 and thus ∂ T = 0. It remains to show that ψ # T = deg( ψ ) · J M K for every Lipschitz map ψ : X → M . Again, w e ﬁrst supp ose that H n ( ψ ( P )) = 0. Then, a computation as ab o v e using the area formula and Lemma 4.7 shows ψ # φ i # J θ i K ( g , τ ) = Z R n   X x ∈ ψ − 1 ( τ − 1 ( y )) 1 φ i ( K i ) ( x ) g ( ψ ( x )) deg( τ ◦ ψ , x )   d L n ( y ) for ev ery i ∈ N and all ( g , τ ) ∈ D n ( M ). F urthermore, it follo ws from the area form ula and the coarea formula that for almost ev ery y ∈ R n the preimage τ − 1 ( y ) is ﬁnite, do es not intersect ψ ( P ), and for every z ∈ τ − 1 ( y ) w e hav e that ψ − 1 ( z ) is ﬁnite and τ is diﬀeren tiable at z with non-degenerate deriv ative. Therefore, by the m ultiplicity property of the lo cal degree we hav e deg( τ ◦ ψ , x ) = deg( τ , z ) · deg ( ψ , x ) = sgn(det D z τ ) · deg( ψ , x ) , for every suc h y and each z ∈ τ − 1 ( y ) and x ∈ ψ − 1 ( z ). Since H n ( ψ ( P )) = 0 and by the additivity prop ert y of the degree we get ψ # T ( g , τ ) = Z R n   X z ∈ τ − 1 ( y )   X x ∈ ψ − 1 ( z ) deg( ψ , x )   · g ( z ) sgn(det D z τ )   d L n ( y ) = deg( ψ ) Z R n   X z ∈ τ − 1 ( y ) g ( z ) sgn(det D z τ )   d L n ( y ) . The c hange of v ariables form ula implies that ψ # T = deg( ψ ) · J M K . Finally , let ψ : X → M be any Lipsc hitz map. Since M is a closed Riemannian n -manifold, there exists a bi-Lipsc hitz embedding ι of M in to Euclidean space and a Lipschitz retraction η of an op en neigh borho o d of ι ( M ). W e can apply Theorem 2.3 to the comp osition ι ◦ ψ to obtain a Lip( ι ◦ ψ )-Lipsc hitz map ψ ′ : X → M arbitrarily close to ψ and satisfying H n ( ψ ′ ( P )) = 0. Using the Lipsc hitz retraction η , we conclude that there exists a sequence ψ i : X → M con verging uniformly to ψ such that Lip( ψ i ) ≤ Lip( η ◦ ι ◦ ψ ) and H n ( ψ i ( P )) = 0 for each i ∈ N . It follows from Lemma 2.2 that for i ∈ N large enough we hav e deg( ψ ) = deg( ψ i ). Therefore, by the contin uit y prop ert y of metric currents ψ # T ( g , τ ) = lim i →∞ ψ i # T ( g , τ ) = lim i →∞ deg( ψ i ) · J M K ( g , τ ) = deg( ψ ) · J M K ( g , τ ) . for all ( g , τ ) ∈ D n ( M ). This completes the pro of. □ The low er b ound in (1.1) will b e a direct consequence of the following lemma. THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 25 Lemma 5.2. L et X b e a close d, oriente d metric n -manifold with ﬁnite Hausdorﬀ n -me asur e. Supp ose that X has a metric fundamental class T ∈ I n ( X ) . If X is ( λ, R ) -line arly lo c al ly c ontr actible ar ound x , then (5.1) ∥ T ∥ ( B ( x, r )) ≥ c · r n , for every r ∈ (0 , R ) . Her e, c dep ends only on λ and n . The argument essentially follo ws the pro of of [5, Prop osition 5.7]. Pr o of. By the slicing inequality Z r 0 M ( ∂ ( T B ( x, s ))) ds ≤ ∥ T ∥ ( B ( x, r )) , for all r > 0. Therefore, it suﬃces to pro v e that there exists a constan t c , dep ending only on λ and n , such that (5.2) M ( ∂ ( T B ( y , r ))) ≥ cr n − 1 , for almost every r ∈  0 , R / 2  . Let r ∈  0 , R / 2  b e such that T B ( x, r ) ∈ I n ( X ) and ∂ T B ( x, r ) is supp orted in { y ∈ X : d ( x, y ) = r } . Supp ose that (5.2) is not true for c = n (4(5 Q + 3) D 1+ 1 n ) − ( n − 1) . Here, D ≥ 1 is the constant from Theorem 3.2 for k = n − 1 and Q ≥ 1 is the constant from Proposition 4.1 for λ and n . Embed X into E ( X ) and let V ∈ I n ( E ( X )) b e a minimal ﬁlling of U = ∂ ( T B ( x, r )) giv en b y Theorem 3.2. Therefore, M ( V ) ≤ D M ( U ) n n − 1 < Dcr n and ∥ V ∥ ( B ( y , r )) ≥ D − n r n for each y ∈ spt V and all r ∈ (0 , d ( y , spt U )). It follows that (5.3) d H (spt V , spt U ) ≤ c ′ · r, where d H denotes the Hausdorﬀ distance and c ′ = D 1+ 1 n · ( cn ) 1 n − 1 . F urthermore, w e hav e set V = spt V . W e conclude that the Hausdorﬀ dimension of spt V is b ounded b y n and th us, its top ological dimension is b ounded by n as w ell. Let Y = X ∪ spt V and let π 0 : Y → X be a map suc h that π 0 ( x ) = x for all x ∈ X and d ( π 0 ( y ) , y ) ≤ 2 d ( X , y ) for all y ∈ spt V . Using (5.3), it is not diﬃcult to show that π 0 is (5 c ′ r )-con tin uous. Recall that B ( x, R ) is ( λ, R )-linearly lo cally con tractible. Since 5 c ′ r < Q − 1 R , Proposition 4.1 implies that there exists a con tin uous map π : Y → X satisfying π ( x ) = x for all x ∈ X and d ( π ( y ) , π 0 ( y )) ≤ 5 Qc ′ r for all y ∈ spt V . Therefore, d ( π ( y ) , y ) ≤ d ( π ( y ) , π 0 ( y )) + d ( π 0 ( y ) , y )) ≤ (5 Q + 3) c ′ r for each y ∈ spt V . Th us, by our choice of c , we hav e (5.4) d H ( π (spt V ) , spt U ) ≤ r 4 . Let ϱ : X → M be a homeomorphism of degree one, where M is a closed, oriented, smo oth n -manifold. Using that ϱ is a homeomorphism and X is compact, we conclude that there exists ε ∈ (0 , r / 2) such that (5.5) N ε ( ϱ ( A )) ⊂ ϱ  N r 4 ( A )  for all A ⊂ X . F urthermore, by [5, Lemma 5.8], and by decreasing ε if necessary , ev ery Lipschitz map φ : Y → M with d ( φ, ϱ ◦ π ) < ε satisﬁes (5.6) φ ( X \ B ( x, r )) ⊂ ϱ ( X \ B ( x, r / 2)) . Lemma 2.1 and Lemma 2.2 imply that there exists φ : Y → M Lipsc hitz with d ( φ, ϱ ◦ π ) < ε and deg ( φ ) = 1. Set W = φ # ( T B ( x, r ) − V ) ∈ I n ( M ) and notice 26 DENIS MAR TI that ∂ W = 0. It follows from (5.4) together with (5.5) that spt( φ # V ) ⊂ N ε ( ϱ ( π (spt V ))) ⊂ ϱ  N r 4 ( π (spt V ))  ⊂ ϱ  y ∈ X : r 2 < d ( x, y ) < 3 r 2  . Therefore, spt W ⊂ ϱ ( B ( x, r ))  = M and the constancy theorem in M ,[19, Corol- lary 3.13], implies that W = 0. F urthermore, φ # ( T B ( x, r )) = J M K − φ # ( T ( X \ B ( x, r ))) and, hence by (5.6) spt  φ # ( T ( X \ B ( x, r ))  ⊂ φ ( X \ B ( x, r )) ⊂ ϱ ( X \ B ( x, r / 2)) . W e conclude that ϱ ( B ( x, r / 2)) ⊂ spt  φ # ( T B ( x, r ))  and thus, W  = 0. This yields a contradiction and completes the pro of. □ W e can now pro ve the existence of a metric fundamen tal class for compact, orien ted metric manifolds. Pr o of of The or em 1.2. Let X b e a metric space with Hausdorﬀ n -measure that is almost everywhere linearly locally contractible and homeomorphic to a compact, orien ted smo oth n -manifold. W e ﬁrst pro v e the theorem under the additional as- sumption that X has no b oundary . Let T ∈ I n ( X ) b e the in tegral n -cycle given by Lemma 5.1. Since X is almost everywhere linearly lo cally contractible, Lemma 5.2 implies that lim inf r → 0 ∥ T ∥ ( B ( x, r )) r n > 0 for almost all x ∈ X . Therefore, the characteristic set of T is equal to X . F urther- more, T is giv en by a parametrization { φ i , K i , θ i } with | θ i ( x ) | = 1 for almost all x ∈ K i and ev ery i ∈ N . It follo ws that ∥ T ∥ = λ H n , where λ ∈ L 1 ( X ) satisﬁes n − n/ 2 ≤ λ ≤ n n/ 2 . Finally , [5, Proposition 5.5] implies that T is a generator of H IC n ( X ). This completes the pro of in case X has no b oundary . No w, supp ose that X has boundary and H n ( ∂ X ) = 0. Let ˆ X b e the closed manifold obtained by gluing t wo copies of X along their b oundaries and equip ˆ X with the quotien t metric; see Deﬁnition 4.3. It follows from Lemma 4.4 that ˆ X is almost ev erywhere linearly lo cally contractible. Therefore, by the ﬁrst part of the pro of ˆ X has a metric fundamental class ˆ T ∈ I n ( ˆ X ) generating H IC n ( ˆ X ) and satisfying n − n/ 2 H n ˆ X ≤ ∥ ˆ T ∥ ≤ n n/ 2 H n ˆ X . Let ι : X → ˆ X an isometric em bedding that identiﬁes X with one of its copies in ˆ X . By Lemma 4.5 the n -current T = ι − 1 # ( ˆ T ι ( X )) ∈ I n ( X ) deﬁnes a metric fundamen tal class of X satisfying n − n/ 2 H n X ≤ ∥ T ∥ ≤ n n/ 2 H n X and whenev er S ∈ I n ( X ) is such that spt S ⊂ ∂ X , then there exists k ∈ Z with k · T = S . This completes the pro of □ W e conclude the section with the pro ofs of the rectiﬁability result for metric mani- folds. Pr o of of Cor ol lary 1.5. As mentioned be fore, w e only assume that X is almost ev- erywhere linearly contractible. Let ˆ X b e the manifold obtained b y gluing t w o copies of X along their b oundary . It follows from Lemma 4.4 that ˆ X is almost everywhere linearly contractible as w ell. Since X em b eds isometrically in to ˆ X , it suﬃces to pro ve the theorem for closed manifolds. If X is orientable, it follows from Theorem 1.2 that there exists T ∈ I n ( X ) with set T = X and th us, X is n -rectiﬁable. Finally , THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 27 if X is non-orientable, then Lemma 4.2 implies that the orientable double co ver ˜ X is almost everywhere linearly lo cally contractible. By the abov e ˜ X is n -rectiﬁable. W e conclude that X is n -rectiﬁable as well b ecause ˜ X locally isometric to X . □ 6. Non-orient able manifolds In this section w e pro vide the proofs for the diﬀeren t existence results for the metric fundamen tal class mo dulo 2. 6.1. Linear lo cally contractibilit y. W e b egin with the following lemma. Lemma 6.1. L et X b e a c omp act metric sp ac e and let T ∈ I n ( X ) b e an inte gr al n -curr ent satisfying φ # T = 0 mod 2 for al l Lipschitz maps φ : X → R n . Then, ∂ T = 0 mo d 2 . Pr o of. W e ﬁrst pro ve the result under the additional assumption that X is a subset of a ﬁnite dimensional v ector space V . Clearly , the statemen t is unaﬀected by bi- Lipsc hitz changes of the metric on V . Therefore, w e may assume that V is equal to some R N equipp ed with the standard Euclidean distance. Let E ⊂ X b e the ( n − 1)-rectiﬁable set that ∥ ∂ T ∥ is concentrated on. By Theorem 2.4 there exists a 1-Lipsc hitz map φ : R N → R n suc h that H n − 1 ( φ ( E )) = H n − 1 ( E ). It follows that φ is mass-preserving, that is, H n − 1 ( φ ( B )) = H n − 1 ( B ) for every Borel set B ⊂ E ; see e.g. [39, Lemma 3.1]. Combining this with [7, Lemma 7.2], we conclude that there exist countably many pairwise disjoint subsets E i ⊂ E cov ering E up to a set of H n − 1 -measure zero and such that φ : E i → R n is bi-Lipsc hitz for each i ∈ N and the φ ( E i ) are pairwise disjoint as well. Hence, ∂ T = X i ( ∂ T ) E i = X i φ − 1 # ( φ # ( ∂ T ) E i ) . Since φ # ∂ T = 0 mo d 2 and the φ ( E i ) are pairwise disjoin t, we hav e φ # ( ∂ T ) E i = 0 mod 2 for each i ∈ N and in particular, ∂ T = 0 mo d 2. W e now prov e the general statemen t. Embed X into l ∞ . It follo ws from Lemma 3.6 that there exist coun tably man y ﬁnite dimensional linear subspaces V i ⊂ l ∞ and 1-Lipschitz maps π i : X → V i suc h that F 2 ( ∂ T − π i # ∂ T ) → 0 as i → ∞ . Fix i ∈ N for the momen t. If φ : V i → R n is Lipschitz, then there exists a Lipschitz extension ˜ φ : l ∞ → R n of φ and we hav e φ # ( π i # T ) = ( ˜ φ ◦ π i ) # T = 0 mo d 2. Therefore, φ # ( π i # T ) = 0 mo d 2 for every Lipsc hitz map φ : V i → R n . By the ﬁrst part of the pro of this implies ∂ ( π i # T ) = 0 mo d 2. W e conclude that ∂ T = 0 mod 2. □ Next, we prov e the existence of a metric fundamen tal class mo dulo 2 in a closed metric manifold. Prop osition 6.2. L et X b e a close d, non-orientable metric n -manifold that is almost everywher e line arly lo c al ly c ontr actible. Then, X has a metric fundamental class T ∈ I n ( X ) mo dulo 2 satisfying C − 1 H n ≤ ∥ T ∥ 2 = ∥ T ∥ ≤ C H n . Her e, C > 0 dep ends only on n . Pr o of. Let ˜ X be the orien table double cov er of X and let π : ˜ X → X b e the cov er- ing map. It follows from Lemma 4.2 that ˜ X is almost ev erywhere linearly lo cally con tractible. Hence, Theorem 1.2 implies that ˜ X has a metric fundamen tal class ˜ T ∈ I n ( ˜ X ) satisfying (6.1) C − 1 · H n ˜ X ≤ ∥ ˜ T ∥ ≤ C · H n ˜ X , where C > 0 dep ends only on n . There exists R > 0 suc h that for eac h x ∈ X and all 0 < r < R , the preimage π − 1 ( B ( x, r )) consists of tw o comp onen ts and π restricted either comp onen t is an isometry . Let x 1 , . . . , x N ∈ X and r 1 , . . . , r N ∈ (0 , R ) b e a 28 DENIS MAR TI collection of points and radii with the follo wing properties. The balls B i = B ( x i , r i ) co ver X and for eac h i = 1 , . . . , N , the slice ⟨ ˜ T , π i , r i ⟩ is an in tegral ( n − 1)-cycle, where π i : ˜ X → R , x 7→ d ( x i , π ( x )). Notice that for each i ∈ N , ∂ ( ˜ T π − 1 ( B i )) = ∂ ( ˜ T B ( x + i , r i )) + ∂ ( ˜ T B ( x − i , r i )) = ⟨ ˜ T , π i , r i ⟩ , where π − 1 ( x i ) = { x + i , x − i } and hence, ˜ T π − 1 ( B i ) ∈ I n ( ˜ X ). Let U 1 = B 1 and for i = 2 , . . . , N set U i = B i \ S i − 1 j =1 U j . It follows that { U i } N i =1 is a partition of X and for eac h i = 1 , . . . , N the preimage π − 1 ( U i ) consists of t w o components U + i , U − i ⊂ ˜ X suc h that ˜ T U + i , ˜ T U − i ∈ I n ( ˜ X ). F or eac h i = 1 , . . . , N choose one comp onent ˜ U i of π − 1 ( U i ) and deﬁne T = N X i π # ( ˜ T ˜ U i ) ∈ I n ( X ) . Since the ˜ U i are pairwise disjoin t and π is an isometry on each ˜ U i it follo ws from (6.1) that C − 1 H n X ≤ ∥ T ∥ ≤ C H n X . Moreo ver, by Theorem 1.2 there exists a parametrization { φ j , K j , θ j } of ˜ T such that | θ j ( x ) | = 1 for almost all x ∈ K j and ev ery j ∈ N . W e conclude that the integral n -curren t T is already a reduction modulo 2 and hence, Lemma 3.3 implies that ∥ T ∥ 2 = ∥ T ∥ . Next, we sho w that ∂ T = 0 mo d 2. Let f : X → R n b e Lipsc hitz. By Corollary 4.8 and Corollary 1.5 w e hav e that # f − 1 ( z ) is even for almost all z ∈ R n . Therefore, there exists a parametrization { ( φ k,l , φ N ) , ( K k,l , N ) , ( θ k,l , θ N ) } of T such that | θ k,l ( y ) | = 1 for almost all y ∈ K k,l and every k , l ∈ N and H n ( f ( N )) = 0 as w ell as [ l φ k,l ( K k,l ) = f − 1 ( { z ∈ R n : # f − 1 ( z ) = 2 k } for all k ∈ N . Notice that changing the sign of some θ k,l on a subset of K k,l induces the same current mo dulo 2. Th us, we ma y assume that for eac h k ∈ N we ha v e f # X l φ k,l # J θ k,l K ! = 2 k · J A k K , where A k = { z ∈ R n : # f − 1 ( z ) = 2 k } and therefore f # T = 2 S = 2 ∞ X k =1 k · J A k K ! . Since f is Lipschitz, the area formula implies that S ∈ I n ( R n ) has ﬁnite mass and hence, f # T = 0 mod 2. Lemma 6.1 implies that ∂ T = 0 mo d 2. Finally , let M b e a smo oth n -manifold homeomorphic to X and let ψ : X → M be Lipschitz. It can b e shown analogously as in Corollary 4.8, that for almost all z ∈ M the n um b er of preimages # ψ − 1 ( z ) is even if deg ( ψ, Z / 2 Z ) = [0] and o dd if deg( ψ , Z / 2 Z ) = [1]. The same argument as ab o v e, using a parametrization of T adapted to ψ , shows that there exists S ∈ I n ( M ) with ψ # T = deg( ψ , Z / 2 Z ) · J M K + 2 S . In particular, ψ # T = deg( ψ , Z / 2 Z ) · J M K mo d 2. This completes the pro of. □ Finally , we prov e the uniqueness of the metric fundamental class modulo 2 in a non-orien table metric manifold. The pro of is an adaptation of [5, Prop osition 5.5]. Lemma 6.3. L et X b e a close d, non-orientable metric n -manifold with ﬁnite Haus- dorﬀ n -me asur e. Supp ose that X has a metric fundamental class T ∈ I n ( X ) mo dulo 2 . Then, every S ∈ I n ( X ) with ∂ S = 0 mo d 2 satisﬁes k T = S mo d 2 for k either e qual to 0 or 1 . Pr o of. Let ϱ : X → M b e a homeomorphism, where M is a closed non-orientable Riemannian manifold. By Lemma 2.2 there exists δ > 0 such that whenev er THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 29 φ, ψ : X → M are Lipschitz maps satisfying d ( φ, ψ ) < 2 δ then there exists a Lips- c hitz homotopy H : [0 , 1] × X → M betw een φ and ψ . It follows from (3.5) ∂ H # ( J 0 , 1 K × S ) + H # ( J 0 , 1 K × ∂ S ) = φ # S − ψ # S. Since I n +1 ( M ) = 0, we hav e ∂ H # ( J 0 , 1 K × S ) = 0. F urthermore, ∂ S = 0 mo d 2 and thus, φ # S = ψ # S mo d 2. By the uniqueness of J M K , we conclude that there exists k equal to either 0 or 1 such that φ # S = k · J M K mod 2 for every Lipschitz map φ : X → M with d ( φ, ϱ ) < δ . Embed X into l ∞ . W e claim that for every ε > 0 there exists U ∈ I n +1 ( l ∞ ) with spt U ⊂ N ε ( X ) and U is a ﬁlling mo dulo 2 of S − k T , that is, ∂ U = S − k T mo d 2. Let ε > 0. W e ﬁrst apply Lemma 2.1 to ϱ − 1 and obtain a Lipsc hitz map g : M → l ∞ with d ( ϱ − 1 , g ) < ε/ 2. Then we use Lemma 2.1 again to ﬁnd f : X → M Lipsc hitz with d ( ϱ, f ) < ( ε/ 2 Lip( g )). It follows that d ( g ◦ f , ϱ − 1 ◦ ϱ ) ≤ d ( g ◦ f , g ◦ ϱ ) + d ( g ◦ ϱ, ϱ − 1 ◦ ϱ ) < ε. Set Q = S − kT . W e ma y supp ose that deg ( f , Z / 2 Z ) = 1. Therefore, since T is a metric fundamen tal class mo dulo 2, w e hav e f # Q = 0 mo d 2 and in particular, ( g ◦ f ) # Q = 0 mo d 2. Let H : [0 , 1] × X → l ∞ b e the straight line homotopy b etw een ( g ◦ f ) and the inclusion ι X of X into l ∞ . Put U = H # ( J 0 , 1 K × Q ) ∈ I n +1 ( l ∞ ). W e ha ve H ( t, x ) ∈ N ε ( X ) for all x ∈ X and every t ∈ [0 , 1] and hence, spt U ⊂ N ε ( X ). Moreo ver, using (3.5) we get ∂ U = Q − ( g ◦ f ) # Q − H # ( J 0 , 1 K × ∂ Q ) . Since ∂ Q = 0 mo d 2 and ( g ◦ f ) # Q = 0 mo d 2 we conclude that U is a ﬁlling mo dulo 2 of Q = kT − S . This prov es the claim. Finally , w e show that k T = S mo d 2. Lemma 3.6 implies that there exist coun tably man y ﬁnite dimensional linear subspaces V i ⊂ l ∞ and 1-Lipsc hitz maps π i : l ∞ → V i suc h that F 2 ( Q − π i # Q ) → 0 as i → 0. Fix i ∈ N for the momen t. Let N = dim V i , Y = π i ( X ) and Q ′ = π i # Q ∈ I n ( V i ). It follows from the claim that for each ε > 0 there exists a ﬁlling U mo dulo 2 of Q ′ satisfying spt U ⊂ N ε ( Y ) ⊂ V i . W e apply a reﬁned v ersion of F ederer and Fleming’s deformation theorem [18, Theorem 4.2.9 ν ] for in teger rectiﬁable currents mo d 2. More precisely , for η > 0 we choose a cubical subdivision of V i in to Euclidean cub es of side length η . Then we use radial pro jections to push Q ′ ﬁrst into the ( N − 1)-dimensional sk eleton, then into the ( N − 2)-dimensional skeleton and so on till w e arriv e at the sk eleton of dimension n . Since H n +1 ( Y ) = 0 we can ac hieve this with pro jections that hav e their pro jection cen ters outside of Y ; c.f. [28, Chapter 3] and [40, Chapter 5]. It follo ws that the resulting current P is the pushforw ard of Q ′ b y a Lipschitz map p deﬁned on Y and F 2 ( Q ′ − P ) ≤ C η M ( Q ′ ), where C dep ends only on N . Because the n -sk eleton of the cubical sub division is an absolute Lipschitz neigh b orhoo d retract, we can extend p to an op en neighborho o d of Y . Therefore, there exists ﬁlling U modulo 2 of Q ′ suc h that the pushforward p # U is well-deﬁned. This implies that there exists an in teger rectiﬁable curren t U ′ ∈ I n +1 ( V i ) in the n -sk eleton with ∂ U ′ = P mo d 2. Ho w ev er, every such U ′ has to b e zero and th us, P = 0 mod 2. Since η > 0 w as arbitrary , we conclude that π i # Q = Q ′ = 0 mod 2. Using that F 2 ( Q − π i # Q ) → 0 as i → 0, w e get Q = 0 mo d 2 and in particular, k T = S mo d 2. This completes the pro of. □ The pro of of Theorem 1.4 is a direct consequence of the preceding results. Pr o of of The or em 1.4. Let X b e a metric space with ﬁnite Hausdorﬀ n -measure that is almost ev erywhere linearly con tractible and homeomorphic to a compact, non-orien table smooth n -manifold. First, we assume that X is closed. It follo ws from Prop osition 6.2 that there exists T ∈ I n ( X ) with ∂ T = 0 mo d 2 satisfying (6.2) C − 1 H n ≤ ∥ T ∥ = ∥ T ∥ 2 ≤ C H n , 30 DENIS MAR TI where C > 0 dep ends only on n . By Lemma 6.3 the curren t T is unique in the follo wing sense. Whenever S ∈ I n ( X ) satisﬁes ∂ S = 0 mo d 2, then k T = S mo d 2 for k equal to either 0 or 1. Now, supp ose that X has b oundary and satisﬁes H n ( ∂ X ) = 0. Let ˆ X be the manifold obtained b y gluing tw o copies of X along the b oundary . Lemma 4.4 implies that ˆ X is almost everywhere linearly contractible as w ell. Let ˆ T ∈ I n ( ˆ X ) b e the metric fundamental class mo dulo 2 of ˆ X obtained in the ﬁrst part of the pro of. It follo ws from Lemma 4.5 that ( ι − 1 ) # ˆ T ι ( X ) deﬁnes a unique metric fundamen tal class modulo 2 in X . Here, ι : X → ˆ X is the embedding that identiﬁes X with one of its copies in ˆ X . Since ι is an isometric embedding, T satisﬁes (6.2) as well. This concludes the pro of. □ 6.2. Nagata dimension. Throughout this section, let X denote a metric space with ﬁnite Hausdorﬀ n -measure and Nagata dimension less than N . F urthermore, let ϱ : M → X b e a homeomorphism, where M is a compact, non-orientable Rie- mannian n -manifold. Em b ed X in to its injective hull E ( X ). Lemma 6.4. Supp ose that M is close d. Then, for every ε > 0 ther e exists a Lipschitz map η : M → E ( X ) such that d ( ϱ, η ) ≤ ε and M 2 ( η # J M K ) ≤ C H n ( X ) . Her e, C dep ends only on the data of X . Pr o of. Let ε > 0. By Theorem 2.6 there exists a ﬁnite, n -dimensional simplicial complex Σ equipp ed with the l 2 distance | · | l 2 and each simplex in Σ is a standard Euclidean simplex of side length ε . F urthermore, there exist C > 0, dep ending only on the data of X , and Lipschitz maps φ : X → Σ, ψ : Σ → E ( X ) with the following prop erties: Hull( φ ( X )) = Σ, ψ is C -Lipschitz on eac h simplex, d ( ψ ( φ ( x )) , x ) ≤ C ε for ev ery x ∈ X and H n ( φ ( X )) ≤ C H n . Recall that Hull( φ ( X )) is the smallest sub complex of Σ con taining φ ( X ) and th us, H n (Σ) ≤ C H n ( X ). Since Σ is an absolute Lipschitz neigh b orhoo d retract, Lemma 2.1 implies that there exists a Lipsc hitz map ξ : M → Σ satisfying d ( ξ , φ ◦ ϱ ) < ε and ξ ( z ) , ( φ ◦ ϱ )( z ) are contained in neighboring simplices for all z ∈ M . Put η = ψ ◦ ξ and let z ∈ M . It follows from [5, Lemma 2.3] that there exists y ∈ Σ con tained in a common simplex with ξ ( z ) and ( φ ◦ ϱ )( z ) and such that | ξ ( z ) − y | l 2 + | y − ( φ ◦ ϱ )( z ) | l 2 ≤ 4 √ n | ξ ( z ) − ( φ ◦ ϱ )( z ) | l 2 . Therefore, using that ψ is C -Lipschitz on each simplex, we ha v e d ( η ( z ) ,ϱ ( z )) ≤ d ( ψ ( ξ ( z )) , ψ ( φ ( ϱ ( z )))) + d ( ψ ( φ ( ϱ ( z ))) , ϱ ( z )) ≤ C  | ξ ( z ) − y | + | y − φ ( ϱ ( z )) |  + C ε ≤ (4 √ n + 1) C ε. By replacing ξ # J M K with a reduction mo dulo 2 if necessary , w e may assume that M 2 ( η # J M K ) ≤ c n H n ( η ( M )), where c n dep ends only on n . Therefore, since ψ is C -Lipsc hitz on each simplex of Σ M 2 ( η # J M K ) ≤ c n H n ( η ( M )) ≤ c n H n ( ψ (Σ)) ≤ c n C n H n (Σ) ≤ c n C n +1 H n ( X ) . This completes the pro of. □ W e can now prov e Theorem 1.6. Pr o of of The or em 1.6. Let ˆ X be the manifold obtained b y gluing tw o copies of X along their b oundaries. It follo ws that the Nagata dimension of ˆ X is b ounded b y N as w ell. Thus, by Lemma 4.5 it suﬃces to prov e the theorem with the additional assumption that X is closed in order to pro v e the general statement. F or k ∈ N , let η k : M → E ( X ) b e the map given by Lemma 6.4 for ε k = 1 /k and put T k = η k # J M K ∈ I n ( E ( X )). W e hav e sup i M p ( T i ) + M p ( ∂ T i ) < C H n ( X ) < ∞ , THE METRIC FUNDAMENT AL CLASS OF NON-ORIENT ABLE MANIFOLDS 31 where C dep ends only on the data of X . By Theorem 3.5 there exist a subsequence of ( T k ) k ∈ N (still denoted by ( T k ) k ∈ N ) and S ∈ F n ( E ( X )) such that F 2 ( T k − S ) → 0 as k → ∞ . Clearly , S is a cycle modulo 2. Since d ( ϱ, η k ) < ε k for eac h k ∈ N , the supp ort of eac h T k is contained in the ε k neigh b orhoo d of X . W e conclude that ∥ S ∥ p is supp orted on X . Therefore, Prop osition 3.7 implies that there exists T ∈ I n ( X ) such that S = T mo d 2. By Lemma 3.3 and by passing to a reduction of T mo dulo 2 if necessary , we may assume that ∥ T ∥ = ∥ T ∥ 2 ≤ C H n , where C > 0 dep ends only on n . Now, let φ : X → M b e a Lipschitz map. Since M is an absolute Lipschitz neighborho od retract, w e can extend φ to some neigh- b orhoo d of X . Hence, for k ∈ N suﬃciently large the comp osition φ ◦ η k : M → M is well-deﬁned and for such k we ha v e φ # T k = ( φ ◦ η k ) # J M K = deg( φ ◦ η k , Z / 2 Z ) · J M K . Since d ( η k , ϱ ) < ε k for eac h k ∈ N , the composition φ ◦ η k con verges uniformly to φ ◦ ϱ . It follo ws from Lemma 2.2 that for k suﬃciently large, the maps φ ◦ η k and φ ◦ ϱ are homotopic. Thus, using the homotop y inv ariance of the degree and that ϱ is a homeomorphism, we get φ # T k = deg( φ ◦ η k , Z / 2 Z ) · J M K = deg( φ ◦ ϱ, Z / 2 Z ) · J M K = deg( φ, Z / 2 Z ) · J M K , for all k suﬃciently large. Passing to the limit w e obtain φ # T = deg( φ, Z / 2 Z ) · J M K . This completes the pro of. □ 6.3. Surfaces. Let X b e a metric space with ﬁnite Hausdorﬀ 2-measure that is homeomorphic to a closed smooth surface M . It follo ws from [43, Theorem 1.3] that there exists a Riemannian metric g on M of constant curv ature and a contin uous, surjectiv e, monotone map ψ ∈ W 1 , 2 ( M , X ) that is w eakly conformal; see also [42, 44]. A map is said to b e monotone if it is the uniform limit of homeomorphism from M to X . W e refer to [43] for more details on weakly conformal maps and to [23] for more information of the theory of Sob olev maps in metric spaces. One can pro ve exactly as in [5, Proposition 6.1] that there exist countably many Borel sets A k ⊂ M suc h that ψ restricted to each A k is 2 k -Lipsc hitz and H n ( M \ A k ) ≤ ε k /k 2 , where ε k is some sequence con verging to 0. Using this, the pro of of Theorem 1.7 is not to o diﬃcult. Pr o of of The or em 1.7. By Lemma 4.5 it suﬃces to prov e the theorem when X is homeomorphic to a closed, non-orientable smo oth surface M . Let ψ ∈ W 1 , 2 ( M , X ) and A k ⊂ M , k ∈ N , b e as explained ab o v e. Em bed X into E ( X ). Since ψ restricted to each A k is 2 k -Lipsc hitz, there exists 2 k -Lipsc hitz extensions ψ k : M → E ( X ) of ψ | A k . F or k ∈ N , we deﬁne T k = ψ k # J M K ∈ I 2 ( E ( X )). W e hav e M 2 ( T k ) ≤ M 2 ( ψ k # ( J M K A k )) + M 2 ( ψ k # ( J M K M \ A k )) ≤ v ol ∗ ( ψ | A k ) + 4 ε k . for all k ∈ N . Here, vol ∗ ( ψ | A k ) denotes the (parametrized) Gromo v mass ∗ v olume of ψ ; see [37, 38]. Since ψ b elongs to W 1 , 2 ( M , X ) we ha v e that vol ∗ ( ψ ) is uniformly b ounded. Notice that ∂ T k = 0 mod 2 for all k ∈ N . Therefore, Theorem 3.5 implies that there exists S ∈ F 2 ( E ( X )) suc h that F 2 ( T k − S ) → 0 as k → ∞ . It is not diﬃcult to sho w that the ψ k con verge uniformly to ψ : M → X . Hence, spt T k ⊂ N δ k ( X ) for some δ k → 0 and in particular, spt 2 T ⊂ X . It follo ws from Prop osition 3.7 that there exists T ∈ I 2 ( X ) suc h that S = T mo d 2. By Lemma 3.3 and replacing T b y a reduction modulo 2 if necessary , w e may supp ose that ∥ T ∥ 2 = ∥ T ∥ ≤ 2 H 2 . Now, let φ : X → M be Lipschitz. Recall that ψ is the uniform limit of homeomorphism from M → X and hence, deg( ψ , Z / 2 Z ) = [1] ∈ Z / 2 Z . W e can extend φ as a Lipsc hitz map to an op en neighborho o d of X in E ( X ). This is p ossible b ecause M is an absolute Lipsc hitz neigh b orhoo d retract. It follows that for k ∈ N suﬃciently large the comp osition φ ◦ ψ k : M → M is w ell-deﬁned and 32 DENIS MAR TI con verges uniformly to φ ◦ ψ . Hence, for k ∈ N suﬃciently large, Lemma 2.2 and the multiplicit y prop ert y of the degree imply φ # T k = ( φ ◦ ψ k ) # J M K = deg( φ ◦ ψ k , Z / 2 Z ) · J M K = deg( φ, Z / 2 Z ) · J M K . W e conclude that φ # T = deg ( φ, Z / 2 Z ) · J M K . Finally , the uniqueness (modulo 2) of T follows from Lemma 6.3. This completes the pro of. □ References [1] T arn Adams. Flat chains in Banac h spaces. J. Ge om. 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