One-dimensional and codimension one homology of metric manifolds

ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIF OLDS DENIS MAR TI Abstract. W e compare singular homology and homology via integral currents in metric spaces that are homeomorphic to smooth manifolds. F or suc h spaces, we pro vide sufficient conditions that guarantee the existence of a surjective homomorphism from the codimension one homology group via integral curren ts to the codimension one singular homology group. Moreov er, w e show that a one-dimensional isop erimetric inequality for integral curren ts implies that the one-dimensional homology groups coincide. 1. Introduction In their seminal paper [7], F ederer and Fleming introduced integral currents in Eu- clidean space and the associated homology groups. Ambrosio and Kirchheim later extended this theory to metric spaces [2], making it possible to consider homology groups via in tegral currents in a more general con text. These homology groups hav e b ecome an imp ortant topic in metric geometry; see e.g. [6, 13, 28, 29]. They are a useful to ol for studying the analytic and geometric structure of metric spaces. F or instance, in a closed Lipschitz manifold, the homology groups via in tegral cur- ren ts and the singular homology groups coincide. This provides a measure-theoretic and analytic representation of singular homology classes. The aim of this article is to explore the relationship b etw een singular homology and homology via in tegral curren ts in more general spaces. More sp ecifically , w e consider metric spaces that are homeomorphic to a smo oth manifold. Such spaces, known as metric manifolds, ha ve a strong connection b et w een their topological and analytic structures, mak- ing them suitable for our purposes. Indee d, in join t w ork with Basso and W enger [4], the author pro ved that, under mild assumptions, the top-dimensional homology group via integral curren ts in a metric manifold is infinite cyclic. The generator, called the metric fundamental class, w as further studied in [19, 20]. 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, c onne cte d R iemannian n -manifold M . A metric fundamental class of X is an inte gr al curr ent T ∈ I n ( X ) with ∂ T = 0 such that (a) ther e exists C > 0 such that ∥ T ∥ ≤ C · H n ; (b) ϕ # T = deg ( ϕ ) · J M K for every Lipschitz map ϕ : X → M . Here, J M K denotes the fundamen tal class of M represented b y an integral curren t, deg( ϕ ) is the (topological) degree of ϕ , and ∥ T ∥ is the mass measure of T . It follo ws from [4, Prop osition 5.5] that if a metric manifold X has a metric fundamen tal class T , then T generates the top-dimensional homology group via integral curren ts H I C n ( X ). In particular, H I C n ( X ) is isomorphic to the top-dimensional singular homology group. W e sho w that the existence of a metric fundamental class also has implications for the co dimension one homology group via integral curren ts. 2020 Mathematics Subje ct Classific ation. Primary 53C23; Secondary 49Q15, 28A75. Key words and phr ases. Metric geometry , Metric manifolds, Homology , In tegral currents. D.M. was supp orted by Swiss National Science F oundation gran t 212867. 1 2 DENIS MAR TI Theorem 1.2. 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, c onne cte d smo oth n -manifold. Supp ose that X has a metric fundamental class. Then, ther e exists a surje ctive homomorphism fr om H IC n − 1 ( X ) to H n − 1 ( X ) . Here, H I C n − 1 ( X ) and H n − 1 ( X ) denote the ( n − 1)th homology group via integral curren ts and the ( n − 1)th singular homology group of X , resp ectiv ely . W e refer to Section 2.3 for the precise definitions. The following type of metric manifolds satisfies the conditions of Theorem 1.2. By [4, Theorem 1.3], ev ery metric surface ( n = 2) has a metric fundamen tal class. In addition, an y metric manifold that is the Gromo v–Hausdorff limit of Lipschitz manifolds with b ounded volume has a metric fundamen tal class [20, Theorem 1.2]. Finally , [4, Theorem 1.1] implies that linearly lo cally contractible metric manifolds supp ort a metric fundamen tal class. A metric space is said to b e linearly lo cally con tractible if there exists λ > 0 suc h that ev ery ball of radius 0 < r < diam( X ) /λ is con tractible within the ball with the same center and radius λr . W e note that, in general, the homomorphism in Theorem 1.2 is not an isomorphism. Indeed, in Section 5, w e construct t w o examples of geodesic metric spaces with finite Hausdorff 2-measure that are homeomorphic to the tw o-sphere, but their 1-dimensional homology groups via in tegral currents are not trivial. The first example is Ahlfors 2-regular but not linearly locally con tractible. On the other hand, the second example is linearly locally contractible but not Ahlfors 2-regular. W e call a metric space X Ahlfors 2-regular if there exists a constan t c > 0 suc h that for every ball B ( x, r ) in X , we ha ve c − 1 r 2 ≤ H 2 ( B ( x, r )) ≤ cr 2 . Both examples in Section 5 hav e a metric fundamental class. Therefore, for a large group of metric surfaces, the one-dimensional homology group via integral currents is strictly larger than the one-dimensional singular homology group. Ho w ev er, in certain situations, they coincide. Theorem 1.3. L et X b e a quasic onvex metric sp ac e that is home omorphic to a close d, c onne cte d smo oth manifold. If X satisfies a 1 -dimensional smal l mass isop erimetric ine quality for inte gr al curr ents, then H IC 1 ( X ) and H 1 ( X ) ar e isomor- phic. A metric space X is said to be quasiconv ex if there exists a constant c > 0 such that for all x, y ∈ X , there exists a curve γ from x to y with length( γ ) ≤ cd ( x, y ). The definition of the isop erimetric inequalit y for in tegral currents is presented in Section 2.4. There, we also in tro duce the lo cal quadratic isop erimetric inequality for Lipsc hitz curv es. This t yp e of isoperimetric inequalit y p lays an imp ortan t role in analysis in metric spaces; see e.g. [17, 18, 22]. It is w ell-kno wn that a metric surface that is Ahlfors 2-regular and linearly locally con tractible satisfies a lo cal quadratic isop erimetric inequalit y for Lipschitz curves. Since a lo cal quadratic isoperimetric inequalit y for Lipsc hitz curves implies a 1-dimensional small mass isop erimetric inequalit y for in tegral currents (Prop osition 2.5), w e obtain the following corollary . Corollary 1.4. L et X b e a metric sp ac e home omorphic to a close d, oriente d, c on- ne cte d smo oth surfac e. Supp ose that X is Ahlfors 2 -r e gular and line arly lo c al ly c ontr actible. Then, H IC k ( X ) and H k ( X ) ar e isomorphic for every k ≥ 0 . The examples in Section 5 show that all the assumptions in the corollary are nec- essary to obtain an isomorphism b etw een H IC 1 ( X ) and H 1 ( X ). The article is structured as follows. In Section 2, we fix the notation and present the theory of metric currents, homology theory , and isop erimetric inequalities. W e also define the slice op erator T 7→ ⟨ T , g, p ⟩ for Lipschitz maps g : X → S 1 in this section. In the next section, we prov e Theorem 1.2. The pro of is based on the ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 3 follo wing classical fact. Let M be a closed, oriented, connected Riemannian n - manifold. Given a primitive homology class [ η ] ∈ H n − 1 ( M ), there exists a smo oth map f : M → S 1 suc h that for eac h regular v alue p ∈ S 1 , the preimage N p = f − 1 ( p ) is a smooth submanifold that represen ts [ η ]. Using that H IC n − 1 ( M ) and H n − 1 ( M ) are isomorphic, we can pro v e the analogous statement for elements in H IC n − 1 ( M ). No w, let X be a metric space homeomorphic to M and suppose that X has a metric fundamen tal class T ∈ I n ( X ). W e approximate the homeomorphism  : X → M b y a Lipschitz map ϕ : X → M and consider the slices ⟨ T , f ◦ ϕ, p ⟩ and ⟨ J M K , f , p ⟩ for p ∈ S 1 . Recall that J M K denotes the fundamental class of M represen ted by an integral current. It follows that for almost all p ∈ S 1 , the slice ⟨ J M K , f , p ⟩ is equal to the integral current J N p K induced b y the smo oth submanifold N p . Since the pushforward and the slice op erator commute and T is a metric fundamen tal class of X , we ha ve ϕ # ⟨ T , f ◦ ϕ, p ⟩ = ⟨ ϕ # T , f , p ⟩ = ⟨ J M K , f , p ⟩ = J N p K for almost all p ∈ S 1 . Therefore, for almost all p ∈ S 1 , the slice ⟨ T , f ◦ ϕ, p ⟩ induces a homology class [ ξ ] ∈ H IC n − 1 ( X ) that is sen t to [ η ] b y the homomorphism H IC n − 1 ( X ) → H IC n − 1 ( M ) induced b y ϕ . W e rep eat this for each generator of H IC n − 1 ( M ) and obtain a surjectiv e homomorphism H IC n − 1 ( X ) → H IC n − 1 ( M ). Finally , w e note that H IC n − 1 ( M ) and H n − 1 ( M ) as w ell as H n − 1 ( M ) and H n − 1 ( X ) are isomorphic, resp ectively . This concludes the pro of of Theorem 1.2. The proof of Theorem 1.3 is contained in Section 4. First, we show that an y homology class [ η ] ∈ H IC 1 ( X ) can b e represented b y an in tegral 1-current induced b y finitely many Lipsc hitz loops. This follows from a decomp osition result for in tegral 1-curren ts and the 1-dimensional isop erimetric inequalit y . A finite collection of Lipschitz lo ops induces a class in H 1 ( X ). W e define a homomorphism Θ : H IC 1 ( X ) → H 1 ( X ) by sending a class [ η ] ∈ H IC 1 ( X ) to the singular homology class induced b y a representation of [ η ] b y finitely many Lipschitz lo ops. It is not difficult to prov e that t wo different representations of [ η ] induce the same homology class in H 1 ( X ). Since X is geo desic, a straigh tforw ard argumen t sho ws that Θ is surjective. Injectivit y is more complicated. Let [ η ] ∈ H IC 1 ( X ) b e suc h that Θ([ η ]) = 0. Then, there exists a represen tative T ∈ I 1 ( X ) of [ η ] giv en b y finitely many Lipsc hitz loops and a singular 2-c hain c that b ounds T . W e use c and the isop erimetric inequalit y to construct an in tegral 2-current V ∈ I 2 ( X ) that satisfies ∂ V = T . Thus, [ η ] = 0, whic h shows that Θ is injectiv e. Finally , in Section 5, w e construct the tw o examples men tioned in the in tro duction. 2. Preliminaries 2.1. Metric notions. Let ( X, d ) b e a metric space. W e write B ( x, r ) for the op en ball with center x ∈ X and radius r > 0. F or A ⊂ X and r > 0, w e denote the open r -neigh borho o d of A by N X r ( A ) = { x ∈ X : d ( A, x ) < r } . If the ambien t space X is clear from the context, w e simply write N r ( A ). A map f : X → Y betw een metric spaces is said to be L -Lipschitz if d ( f ( x ) , f ( y )) ≤ Ld ( x, y ) for all x, y ∈ X . The smallest L ≥ 0 satisfying the previous inequalit y is called the Lipschitz constan t of f and is denoted by Lip( f ). If f is injectiv e and the in v erse f − 1 is L -Lipsc hitz as well, we say f is L -bi-Lipschitz . W e let Lip( X ) b e the set of all Lipschitz functions from X to R . The Hausdorff k -measure on X is denoted by H k . Given tw o maps f , g : X → Y , we define the uniform distance b et w een f and g by d ( f , g ) = sup x ∈ X d ( f ( x ) , g ( x )) . W e need the follo wing Lipsc hitz appro ximation result. 4 DENIS MAR TI Lemma 2.1. L et f : X → M b e a c ontinuous map fr om a c omp act metric sp ac e X to a close d R iemannian manifold M . Then, for every ε > 0 , ther e exists a Lipschitz map g : X → M with d ( f , g ) < ε . The lemma is a particular instance of [ 4, Lemma 2.1] since every closed Riemannian manifold is an absolute Lipschitz neigh b orho od retract [12, Theorem 3.1]. W e will use the previous lemma frequen tly in the following context. Let  : X → M b e a homeomorphism, where X is a metric space and M is a closed Riemannian manifold. By Lemma 2.1, there exist Lipschitz maps X → M that are arbitrarily close to  . If w e choose the Lipsc hitz map ϕ : X → M sufficien tly close to  , then ϕ is homotopic to  , and the comp osition  − 1 ◦ ϕ is homotopic to the iden tit y on X . Indeed, embed M in l ∞ , and let U ⊂ l ∞ b e an op en neigh b orhoo d of M suc h that there exists a retraction π : U → M . Whenev er d ( ϕ,  ) is sufficiently small, the straight-line homotop y H b et w een ϕ and  sta ys within U . Hence, π ◦ H is w ell-defined and is a homotop y b etw een ϕ and  as maps from X to M . Since X is a top ological manifold, it is an absolute neighborho o d retract. Th us, an analogous argument shows that  − 1 ◦ ϕ is homotopic to the iden tity on X , in case ϕ : X → M is sufficien tly close to  . 2.2. Metric curren ts. W e pro vide a brief in tro duction to the theory of currents in metric spaces and record some facts w e will need later. F or more information, w e refer to [2] and [16]. Let X b e a complete metric space and k ≥ 0 b e an integer. Let D k ( X ) = Lip b ( X ) × Lip( X ) k , where Lip b ( X ) denotes the space of b ounded Lipsc hitz functions X → R . A m ultilinear functional T on D k ( X ) is said to b e a metric k -curren t if it satisfies a certain con tin uit y , locality , and finite mass prop ert y . F or eac h k -current T , there exists an associated finite Borel measure ∥ T ∥ , called the mass measure of T . This measure can be thought of as the volume measure on the generalized surface T . W e write M k ( X ) for the space of metric k -currents on X . Let T ∈ M k ( X ). W e denote by M ( T ) = ∥ T ∥ ( X ) the mass of T . There exists a unique extension of T to L 1 ( X, ∥ T ∥ ) × Lip( X ) k . F or a Borel set B ⊂ X , we define the restriction T B ∈ M k ( X ) b y T B ( f , π 1 , . . . , π k ) = T ( 1 B f , π 1 , . . . , π k ) for all ( f , π 1 , . . . , π k ) ∈ D k ( X ). The b oundary of T is given b y ∂ T ( f , π 1 , . . . , π k − 1 ) = T (1 , f , π 1 , . . . , π k − 1 ) for all ( f , π 1 , . . . , π k − 1 ) ∈ D k − 1 ( X ). If the boundary ∂ T is a ( k − 1)-current, then w e call T a normal k -current. The space of normal k -currents in X is denoted b y N k ( X ). W e define the normal mass by N ( T ) = M ( T ) + M ( ∂ T ). The space N k ( X ) equipp ed with the normal mass N is a Banach space. Given a Lipschitz map ϕ : X → Y , the pushforw ard of T under ϕ is the metric k -current in Y defined as 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 ). In Euclidean space, every function θ ∈ L 1 ( R k ) induces a metric k -current as follo ws 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 ). A normal k -curren t T ∈ N k ( X ) is called an in tegral k -curren t if there exist coun tably man y compact sets K i ⊂ R k and θ i ∈ L 1 ( R k , Z ) with spt θ i ⊂ K i and bi-Lipsc hitz 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 denote b y I k ( X ) the space of in tegral k -currents in X . An in tegral k -current T is called an integral k -cycle if it has zero b oundary ∂ T = 0. Imp ortant examples of ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 5 in tegral cycles are currents induced by Riemannian manifolds. Let M b e a closed, orien ted Riemannian k -manifold. Then J M K ( f , π ) = Z M f det( Dπ ) d H k for all ( f , π ) ∈ D k ( M ), defines an in tegral k -current on M . Stokes’ theorem implies that J M K is a cycle. In case M is connected, the integral cycle J M K generates H IC k ( M ) and is called the fundamen tal class of M . Next, w e explain the slicing of a normal current. This is an imp ortant technique that allows us to represen t a current with lo w er dimensional pieces of the curren t. Let T ∈ N k ( X ) and let f : X → R b e Lipschitz. F or almost all t ∈ R , the slice ⟨ T , f , t ⟩ of T by f at t is a normal ( k − 1)-curren t c haracterized by the following prop ert y: (2.1) Z R ⟨ T , f , t ⟩ ψ ( t ) dt = T ( ψ ◦ f ) d f for all ψ ∈ C c ( R ); see [2, Theorem 5.6]. Here, T ( ψ ◦ f ) d f is the ( k − 1)-current defined b y T ( ψ ◦ f ) d f ( g, π ) = T (( ψ ◦ f ) g , f , π ) for all ( g , π ) ∈ D k − 1 ( X ). Using the characterizing prop ert y of the slice, w e show that slicing commutes with the pushforw ard b y a Lipsc hitz map. Lemma 2.2. L et ϕ : X → Y b e a Lipschitz map b etwe en two c omplete metric sp ac es. F urthermor e, let T ∈ N k ( X ) and f : Y → R b e Lipschitz. Then ϕ # ⟨ T , f ◦ ϕ, t ⟩ = ⟨ ϕ # T , f , t ⟩ for almost al l t ∈ R . Pr o of. Set g = f ◦ ϕ . Fix ( h, π ) ∈ D k − 1 ( Y ) and ψ ∈ C c ( R ) for the moment. Then ϕ # ( T ( ψ ◦ g ) dg )( h, π ) = T ( ψ ◦ g ) dg ( h ◦ ϕ, π ◦ ϕ ) . Th us, by the c haracterizing property of the slice (2.1), w e hav e ϕ # ( T ( ψ ◦ g ) dg )( h, π ) = Z R ⟨ T , g , t ⟩ ( h ◦ ϕ, π ◦ ϕ ) ψ ( t ) dt = Z R ϕ # ⟨ T , g , t ⟩ ( h, π ) ψ ( t ) dt. On the other hand, ϕ # ( T ( ψ ◦ g ) dg )( h, π ) = T ((( ψ ◦ f ) h ) ◦ ϕ, f ◦ ϕ, π ◦ ϕ ) = ( ϕ # T ) ( ψ ◦ f ) d f ( h, π ) . Therefore, by applying the c haracterizing prop erty of the slice to ϕ # T this time, w e get ( ϕ # ( T ( ψ ◦ g ) dg ))( h, π ) = Z R ⟨ ϕ # T , f , t ⟩ ( h, π ) ψ ( t ) dt. W e conclude that Z R ϕ # ⟨ T , g , t ⟩ ( h, π ) ψ ( t ) dt = Z R ⟨ ϕ # T , f , t ⟩ ( h, π ) ψ ( t ) dt for all ( h, π ) ∈ D k − 1 ( Y ) and ev ery ψ ∈ C c ( R ). The functions t 7→ ϕ # ⟨ T , g , t ⟩ ( h, π ) and t 7→ ⟨ ϕ # T , f , t ⟩ ( h, π ) are L 1 -in tegrable for all ( h, π ) ∈ D k − 1 ( Y ). Hence, the fundamen tal lemma of calculus of v ariations implies that for almost all t ∈ R ϕ # ⟨ T , g , t ⟩ ( h, π ) = ⟨ ϕ # T , f , t ⟩ ( h, π ) . This completes the pro of. □ If T is an in tegral curren t and f : X → R is Lipschitz, then for almost all t ∈ R , the slice ⟨ T , f , t ⟩ is an integral curren t as well [2, Theorem 5.7]. In this case, we hav e 6 DENIS MAR TI an explicit representation of the slice. W e first introduce some terminology and refer to [2, Section 9] and [3] for more information on the concepts discussed b elow. Supp ose that X is separable and em b ed X in to l ∞ . If E ⊂ l ∞ is k -rectifiable, then E has an approximate tangen t space T an( E , x ) at almost all x ∈ E . The appro ximate tangent space is a k -dimensional linear subspace of l ∞ . An orientation τ of E is a choice of unit simple k -vectors τ = τ 1 ∧ · · · ∧ τ k suc h that the τ i : E → l ∞ are Borel functions that span the approximate tangen t space T an( E , x ) for almost ev ery x ∈ E . If π : l ∞ → R is Lipsc hitz, then π is tangen tially differen tiable almost ev erywhere on E . The tangen tial differential of π at x is a linear map d E x π : T an( E , x ) → R that conv erges to the difference quotient of π at x in a suitable sense. F or π = ( π 1 , . . . , π k ) : l ∞ → R k Lipsc hitz, we let V k d E x π b e the simple k -co v ector induced b y the tangen tial differentials of the comp onen ts of π . Let T ∈ I k ( X ). It follo ws from [2, Theorem 9.1] that there exist a k -rectifiable set E ⊂ X ⊂ l ∞ with finite Hausdorff k -measure, a Borel function θ : E → N , and an orien tation τ of E such that for all ( g , π ) ∈ D k ( l ∞ ) T ( g , π ) = Z E g ( x ) θ ( x ) D ^ k d E x π , τ ( x ) E d H k ( x ) , where  V k d E x π , τ ( x )  denotes the standard duality pairing b etw een k -vectors and k -co v ectors. W e write T = [ E , θ , τ ]. This represen tation can b e seen as a measure theoretic v ersion of a curren t J M K induced b y a closed, orien ted Riemannian mani- fold M . Indeed, w e hav e J M K = [ M , θ , τ ] with θ = 1 and τ is the usual orien tation of M . With this representation of an in tegral current T ∈ I k ( X ), we can no w give an explicit description of the slices of T . Let f : E → R b e Lipschitz. Then, for almost all t ∈ R , the preimage E t = E ∩ f − 1 ( t ) is ( k − 1)-rectifiable and T an( E t , x ) = k er( d E x f ) for H k − 1 -almost all x ∈ E t . By [2, Theorem 9.1], for almost all t ∈ R and H k − 1 -almost ev ery x ∈ E t , w e can write τ ( x ) = ξ ( x ) ∧ τ t ( x ) suc h that τ t is an orien tation of E t and (2.2) ⟨ T , f , t ⟩ = [ E t , θ , τ t ] The orien tation τ t is c haracterized by the follo wing equalit y (2.3) D ^ k − 1 d E t x π , τ t ( x ) E J k ( d E x ( π , f )) J k − 1 ( d E t x π ) = D ^ k d E x ( π , f ) , τ ( x ) E for H k − 1 -almost all x ∈ E t and every Lipschitz map π : E → R k − 1 . Here, the Jacobian is equal to J k ( d E x ( π , f )) =    D ^ k d E x ( π , f ) , τ ( x ) E    . A similar equality is true for J k − 1 ( d E t x π ), but we will not need it here. The next goal is to define the slice for a cycle T ∈ I k ( X ) with resp ect to a Lipsc hitz map g : X → S 1 . Let p ∈ S 1 and let ϕ : X → R , x → d S 1 ( g ( x ) , p ). W e hav e ⟨ T , ϕ, t ⟩ = ∂ ( T { ϕ < t } ) = ∂ ( T g − 1 ( B ( p, t ))) and spt ( ⟨ T , ϕ, t ⟩ ) ⊂ g − 1 ( { ϕ = t } ) for almost all t ∈ R ; see [2, Theorem 5.6]. W e conclude that, for a given differen tiable c hart ψ of S 1 , w e can find an op en set U ⊂ S 1 in the domain of ψ with the follo wing prop erties: ∂ U has zero H 1 -measure, T V ∈ I k ( X ) for V = g − 1 ( U ) and spt( ∂ ( T V )) ⊂ g − 1 ( ∂ U ). W e fix an orientation of S 1 and for p ∈ S 1 , w e define (2.4) ⟨ T , g , p ⟩ = ⟨ T g − 1 ( U ) , ψ ◦ g , ψ ( p ) ⟩ ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 7 whenev er the right hand side mak es sense and ψ : U ⊂ S 1 → R is any p ositively orien ted chart as abov e. Using [2, Theorem 5.6], we get spt( ∂ ⟨ T , g , p ⟩ ) = spt( ⟨ ∂ ( T V ) , ψ ◦ g , ψ ( p ) ⟩ ) ⊂ g − 1 ( ∂ U ) ∩ g − 1 ( p ) for almost all p ∈ U and in this case, ⟨ T , g , p ⟩ is a cycle. W e claim that the definition do es not dep end on the choice of the chart. Letting T = [ E , θ , τ ], then (2.3) gives that for almost all p ∈ U ⟨ T , g , p ⟩ = ⟨ T V , ψ ◦ g , ψ ( p ) ⟩ = [ E ∩ V ∩ g − 1 ( p ) , θ , τ p ] = [ E p , θ , τ p ] , where E p = E ∩ g − 1 ( p ) and V = g − 1 ( U ). In particular, E p do es not dep end on ψ . It remains to sho w that the orien tation τ p is also independent of ψ . Let ϕ : U → R b e another p ositiv ely orien ted chart. A direct computation using the chain rule (whic h also holds for the tangen tial differential) sho ws that for every Lipsc hitz map π : E → R k − 1 , almost all p ∈ S 1 and H k − 1 -almost all x ∈ E p D ^ k d E x ( π , ψ ◦ g ) , τ E = d φ ( g ( x )) ( ψ ◦ ϕ − 1 ) D ^ k d E x ( π , ϕ ◦ g ) , τ E as w ell as J k ( d E x ( π , ψ ◦ g )) = | d φ ( g ( x )) ( ψ ◦ ϕ − 1 ) | J k ( d E x ( π , ϕ ◦ g )) . Since ψ and ϕ are positively oriented, we ha v e d φ ( g ( x )) ( ψ ◦ ϕ − 1 ) > 0 for ev ery x ∈ V . Therefore, (2.3) implies that the orientation τ p do es not dep end on the c hart ψ . Hence, for almost all p ∈ S 1 , the slice ⟨ T , g , p ⟩ is a well-defined integral ( k − 1)-cycle and is indep endent of the c hart used in (2.4). Notice that the ambien t structure of l ∞ and the represen tation T = [ E , θ, τ ] are only used to show that the slice ⟨ T , g , p ⟩ is w ell-defined for almost all p ∈ S 1 . Moreov er, by [2, Theorem 9.6], the definition of the slice is intrinsic in the sense that it do es not depend on the isometric em b edding of X in to l ∞ . Th us, w e use the slice for Lipschitz maps g : X → S 1 in separable metric spaces without embedding them into l ∞ . W e conclude the section with the following lemma. Lemma 2.3. L et ϕ : X → Y b e a Lipschitz map b etwe en two sep ar able metric sp ac es. F urthermor e, let T ∈ I k ( X ) b e a cycle and g : Y → S 1 b e Lipschitz. Then ϕ # ⟨ T , g ◦ ϕ, p ⟩ = ⟨ ϕ # T , g , p ⟩ for almost al l p ∈ S 1 . Pr o of. Let ψ : U ⊂ S 1 → R b e a p ositiv ely orien ted c hart suc h that ∂ U has zero H 1 -measure, T W ∈ I k ( X ) and ( ϕ # T ) V ∈ I k ( Y ) for W = ( g ◦ ϕ ) − 1 ( U ) and V = g − 1 ( U ), resp ectively . Recall that by [2, Theorem 5.6] w e can find, for each c hart ψ , an op en set in its domain that has these prop erties. It follo ws from the definition (2.4) and Lemma 2.2 that ϕ # ⟨ T , g ◦ ϕ, p ⟩ = ϕ # ⟨ T W , ψ ◦ g ◦ ϕ, ψ ( p ) ⟩ = ⟨ ϕ # ( T W ) , ψ ◦ g , ψ ( p ) ⟩ for almost all p ∈ S 1 . On the other hand, ⟨ ϕ # T , g , p ⟩ = ⟨ ( ϕ # T ) V , ψ ◦ g , ψ ( p ) ⟩ for almost all p ∈ S 1 . F or ( f , π ) ∈ D k ( Y ) we ha ve ( ϕ # T ) V ( f , π ) = T (( 1 V ◦ ϕ )( f ◦ ϕ ) , π ◦ ϕ ) = T ( 1 φ − 1 ( V ) f ◦ ϕ, π ◦ ϕ ) and ϕ # ( T W )( f , π ) = T ( 1 W f ◦ ϕ, π ◦ ϕ ) . Since W = ϕ − 1 ( V ) we ha ve ( ϕ # T ) V = ϕ # ( T W ) and in particular, ϕ # ⟨ T , g ◦ ϕ, p ⟩ = ⟨ ϕ # T , g , p ⟩ for almost all p ∈ S 1 . This completes the pro of. □ 2.3. Homology. Next, w e discuss singular homology and homology groups via in tegral currents. W e refer to [9] and [25] for more details. Let X be a complete 8 DENIS MAR TI metric space and let k ≥ 0 b e an integer. The standard k -dimensional Euclidean simplex is denoted by ∆ k . W e call a contin uous map ϕ : ∆ k → X a singular k - simplex. A singular k -chain c is a formal sum c = N X i =1 n i · ϕ i , where each ϕ i is a singular k -simplex and n i ∈ Z . The boundary of c is the ( k − 1)- c hain bc defined as bc = N X i =1 n i · bϕ i , where bϕ i = P j ( − 1) j ϕ i,j and the ϕ i,j are ϕ i restricted to the differen t b oundary comp onen ts of ∆ k . At this p oint we fixed an enumeration of the b oundary compo- nen ts for eac h ∆ k . W e sa y c is a cycle if bc = 0. The space of singular k -chains in X is denoted by C k ( X ). It follows that these groups, together with the boundary op erator, form a chain complex. W e write H k ( X ) for the k th homology group of this c hain complex and call it the k th singular homology group of X . Two singular k -cycles c and c ′ that represent the same class in H k ( X ) are said to be homologous, that is, there exists v ∈ C k +1 ( X ) such that bv = c − c ′ . Let f : X → Y b e contin uous and let c = P N i =1 n i · ϕ i b e a singular k -chain. The pushforward f # c of c by f is the singular k -c hain in Y given b y f # c = N X i =1 n i · ( f ◦ ϕ i ) . It is not difficult to see that this induces a homomorphism at the level of homology f ∗ : H k ( X ) → H k ( Y ). In particular, f ∗ [ c ] = [ f # c ] for all c ∈ C k ( X ), where [ c ] and [ f # c ] are the homology classes of c and f # c , resp ectiv ely . If tw o contin uous maps f , g : X → Y are homotopic, then they induce the s ame homomorphism H k ( X ) → H k ( Y ). Giv en T ∈ I k ( X ), the b oundary-rectifiabilit y theorem of Am brosio and Kirc hheim [2, Theorem 8.6] implies that ∂ T ∈ I k − 1 ( X ). Therefore, the b oundary operator ∂ for in tegral currents maps I k ( X ) in to I k − 1 ( X ) and in particular, · · · ∂ k +1 − → I k ( X ) ∂ k − → I k − 1 ( X ) ∂ k − 1 − → · · · ∂ 1 − → I 0 ( X ) is a c hain complex. W e denote b y H IC k ( X ) the k th homology group of this chain complex and call H IC k ( X ) the k th homology group via in tegral curren ts. Two cycles S, T ∈ I k ( X ) hav e the same homology class [ T ] = [ S ] if and only if there exists V ∈ I k +1 ( X ) with ∂ V = S − T . Let f : X → Y b e Lipschitz. The b oundary op erator and pushforw ard comm ute, that is, ∂ ( f # T ) = f # ( ∂ T ) for every T ∈ I k ( X ). Hence, f induces a homomorphism f ∗ : H IC k ( X ) → H IC k ( Y ). As in the case of singular homology groups, t w o Lipsc hitz maps f , g : X → Y induce the same homomorphism H IC k ( X ) → H IC k ( Y ) if there exists a Lipschitz homotop y betw een f and g . A singular k -c hain c is called a singular Lipsc hitz k -c hain if eac h singular simplex of c is a Lipschitz map. The space of singular Lipschitz k -c hains in X is denoted b y C Lip k ( X ). The boundary of a singular Lipsc hitz c hain is again a Lipschitz chain. Th us, we can define the singular Lipschitz homology groups H Lip k ( X ) via these Lipsc hitz chains. Notice that ev ery singular Lipschitz k -chain induces an in tegral k - curren t as well as a singular k -chain. It follows that the inclusions C Lip k ( X )  → I k ( X ) and C Lip k ( X )  → C k ( X ), resp ectiv ely , are c hain maps. It is well known that if the underlying space is lo cally Lipsc hitz contractible, then these inclusion maps induce isomorphisms on the level of homology; see [25, 26]. ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 9 Theorem 2.4. L et X b e a c omp act metric sp ac e. Supp ose that ther e exist R, λ > 0 such that every b al l B in X of r adius 0 < r < R is c ontr actible via a Lipschitz map ϕ : [0 , 1] × B → X satisfying d ( ϕ ( s, x ) , ϕ ( t, y )) ≤ λr | s − t | + λd ( x, y ) for al l s, t ∈ [0 , 1] and every x, y ∈ B . Then, H IC k ( X ) ∼ = H Lip k ( X ) ∼ = H k ( X ) for al l k ∈ N . The isomorphisms ar e induc e d by the inclusions C Lip k ( X )  → I k ( X ) and C Lip k ( X )  → C k ( X ) , r esp e ctively. Clearly , every closed Riemannian manifold satisfies the conditions of the theorem, as it is lo cally bi-Lipsc hitz to Euclidean space. 2.4. Isop erimetric inequalities. Let X b e a complete metric space and let T ∈ I 1 ( X ) b e an integral 1-cycle. An integral 2-curren t V ∈ I 2 ( X ) is called a filling of T if ∂ V = T . W e say X satisfies a 1-dimensional (Euclidean) small mass isoperimetric inequalit y with constants C, Λ > 0 if the following holds. F or every cycle T ∈ I 1 ( X ) with M ( T ) < Λ, there exists a filling V ∈ I 2 ( X ) of T satisfying M ( V ) ≤ C M ( T ) 2 . Next, w e in troduce the local isop erimetric inequalit y for Lipsc hitz curves and com- pare it to the isop erimetric inequality for integral curren ts. The space of Sobolev maps from the 2-dimensional unit disk D into X is denoted b y W 1 , 2 ( D , X ). F or a detailed exp osition of the theory on Sob olev maps in metric spaces, see [11]. Let ϕ ∈ W 1 , 2 ( D , X ). Then, for almost all x ∈ D , the map ϕ has an approximate metric deriv ative apmd x ϕ at x , whic h is a seminorm on R 2 . W e need t w o different Jaco- bians for a seminorm s on R 2 . If s defines a norm, then w e put J ( s ) = H 2 ( R 2 ,s ) ([0 , 1] 2 ) and J ∗ ( s ) = 4 / L 2 ( P ), where L 2 ( P ) is the Leb esgue measure of the parallelogram of smallest area con taining the unit ball with respect to s . In case s is degenerate, w e set J ( s ) = J ∗ ( s ) = 0. W e note that J ∗ ( s ) ≤ 2 J ( s ) for every norm s on R 2 [2, Lemma 9.2]. The (parameterized) Hausdorff area of ϕ is defined as Area( ϕ ) : = Z D J (apmd x ϕ ) d L 2 ( x ) . Analogously , the (parameterized) Gromov mass ∗ area, denoted b y Area ∗ ( ϕ ), is defined by replacing J with J ∗ in the previous definition. F or a subset A ⊂ D , the area Area( ϕ | A ) of ϕ restricted to A is defined by integrating only o v er A . W e refer to [1, 17] for more information on Jacobians and area. F or almost all v ∈ S 1 , the curv e t 7→ ϕ ( tv ) is absolutely con tinuous on [1 / 2 , 1). The trace of ϕ is giv en b y tr( ϕ )( v ) = lim t ↗ 1 ϕ ( v t ) for almost every v ∈ S 1 . W e say X satisfies a lo cal quadratic isop erimetric inequal- it y for Lipschitz curves with constants D , Γ > 0 if the follo wing holds. F or ev ery Lipsc hitz curve γ : S 1 → X with length( γ ) < Γ, there exists ϕ ∈ W 1 , 2 ( D , X ) satis- fying tr( ϕ ) = γ and Area( ϕ ) ≤ D length( γ ) 2 . If this holds for all Lipsc hitz curv es of arbitrary length, then we sa y X satisfies a quadratic isoperimetric inequality for Lipsc hitz curves with constan t D . It is well-kno wn to exp erts that a lo cal qua- dratic isop erimetric inequality for Lipsc hitz curv es implies a 1-dimensional small mass isoperimetric inequality for integral curren ts. Ho wev er, we could not find a reference and thus, w e provide a pro of. Prop osition 2.5. L et X b e a c omplete and sep ar able metric sp ac e. Supp ose that X satisfies a lo c al quadr atic isop erimetric ine quality for Lipschitz curves with c onstants D , Γ > 0 . Then, X satisfies a 1 -dimensional smal l mass isop erimetric ine quality for inte gr al curr ents with c onstants 2 D , Γ . 10 DENIS MAR TI The main idea of the pro of is to appro ximate a giv en Sob olev map ϕ ∈ W 1 , 2 ( D , X ) b y Lipsc hitz maps with uniformly bounded area, similar as in [18, Proposition 3.1]. W e can construct a curren t in X using these approximations and a pushforw ard argumen t. This approach has also been used in [4, 14, 19]. Pr o of. Let γ : S 1 → X be Lipschitz. Supp ose that there exists a Sob olev map ϕ : D → X with tr( ϕ ) = γ . W e claim that there exists U ∈ I 2 ( X ) satisfying ∂ U = γ # J S 1 K and M ( U ) ≤ 2 Area( ϕ ). Let B = B (0 , 2) b e the open ball of radius 2 in R 2 . W e extend ϕ to the closed ball B as follo ws ˜ ϕ ( v ) = ( ϕ ( v ) , v ∈ D ϕ  v ∥ v ∥  , v / ∈ D . Here, ∥·∥ denotes the standard Euclidean norm. Notice that ˜ ϕ b elongs to W 1 , 2 ( B , X ) and Area( ˜ ϕ ) = Area( ϕ ). It follows that there exists g ∈ L 2 ( B ) such that (2.5) d ( ˜ ϕ ( x ) , ˜ ϕ ( y )) ≤ ∥ x − y ∥ ( g ( x ) + g ( y )) for almost all x, y ∈ B ; see [11, Theorem 8.1.7]. F or k ∈ N , we define A k = { x ∈ B : g ( x ) ≤ k } and ε k = Z B \ A k g ( x ) 2 d L 2 ( x ) . Since g ∈ L 2 ( B ), Chebyshev’s inequality and the absolute contin uity of in tegrals imply that L 2 ( B \ A k ) ≤ ε k k 2 and ε k → 0 as k → ∞ . In particular, eac h A k is √ ε k k -dense in B . It follows from (2.5) that the map ˜ ϕ is 2 k -Lipschitz on eac h A k . Moreo ver, by construction, the map ˜ ϕ is Lipschitz on B \ B (0 , 1). Therefore, for k ∈ N sufficien tly large, ˜ ϕ is 4 k -Lipschitz on B k = A k ∪  B \ ( B (0 , 3 2 )  . T o simplify the notation, w e assume that ˜ ϕ is 4 k -Lipschitz on B k for all k ∈ N . Em bed X into l ∞ . F or each k ∈ N , let ϕ k : B → l ∞ b e a 4 k -Lipschitz extension of ˜ ϕ | B k . W e define a sequence U k = ϕ k # J B K ∈ I 2 ( l ∞ ). W e ha ve M ( ϕ k # J B \ B k K ) ≤ Lip( ϕ k ) 2 · L 2 ( B \ B k ) ≤ 16 ε k , and M ( ϕ k # J B k K ) ≤ Z B k J ∗ (apmd x ϕ k ) d L 2 ( x ) = Area ∗ ( ϕ | B k ) for all k ∈ N . Therefore, M ( U k ) ≤ M ( ϕ k # J B k K ) + M ( ϕ k # J B \ B k K ) ≤ Area ∗ ( ϕ ) + 16 ε k for eac h k ∈ N . F urthermore, ϕ k (2 v ) = γ ( v ) for all k ∈ N and almost every v ∈ S 1 , and hence ∂ U k = ϕ k # ( ∂ J B K ) = γ # J S 1 K for each k ∈ N . In particular, N ( U k ) is uniformly bounded. It follo ws from the compactness and closure theorem for in tegral curren ts [2, Theorem 5.2 and Theorem 8.5] that there exists U ∈ I 2 ( l ∞ ) suc h that the U k con verge weakly to U . Clearly , ∂ U = γ # J S 1 K . Since the B k are √ ε k k -dense in B and eac h ϕ k is 4 k -Lipschitz, w e hav e spt U k ⊂ ϕ k ( B ) ⊂ N l ∞ 4 √ ε k ( X ) for all k ∈ N . Thus, U b elongs to I 2 ( X ). Finally , the low er semicontin uity of mass implies M ( U ) ≤ lim inf k →∞ M ( U k ) ≤ Area ∗ ( ϕ ) ≤ 2 Area( ϕ ) . This pro ves the claim. No w, let T ∈ I 1 ( X ) b e such that M ( T ) < Γ. By [5, Theorem 5.3], there exist coun tably many injectiv e Lipschitz lo ops γ k : S 1 → X suc h that (2.6) T = ∞ X k =1 γ k # J S 1 K and M ( T ) = ∞ X k =1 M  γ k # J S 1 K  . W e conclude that M  γ k # J S 1 K  = length( γ k ) < Γ for each k ∈ N . Because X satisfies a lo cal quadratic isop erimetric inequality for Lipschitz curves with constants ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 11 D , Γ > 0, there exist coun tably many ϕ k ∈ W 1 , 2 ( D , X ) satisfying tr( ϕ k ) = γ k and Area( ϕ k ) ≤ D length( γ k ) 2 = D M  γ k # J S 1 K  2 for each k ∈ N . It follows from the claim that there exist countably man y U k ∈ I 2 ( X ) with ∂ U k = γ k # J S 1 K and M ( U k ) ≤ 2 D M ( γ k # J S 1 K ) 2 for ev ery k ∈ N . F or m ∈ N , w e define V m = P m k =1 U k ∈ I 2 ( X ). The second part of (2.6) implies that the sequence ( V m ) m is Cauch y in I 2 ( X ). W e denote the limit by V ∈ I 2 ( X ). W e hav e ∂ V = T . Finally , b y the con vexit y of x 7→ x 2 , M ( V ) ≤ lim m →∞ m X k =1 M ( U k ) ≤ 2 D lim m →∞ m X k =1 M ( γ k # J S 1 K ) 2 ≤ 2 D lim m →∞ m X k =1 M ( γ k # J S 1 K ) ! 2 = 2 D M ( T ) 2 . This completes the pro of. □ F or the remainder of the section, let X b e a metric space that is homeomorphic to a closed, oriented, connected smo oth surface. F urthermore, w e suppose that X is Ahlfors 2-regular and linearly lo cally contractible. It was shown in [27] that X supp orts a weak 1-Poincar ´ e inequality; see also [4, Corollary 1.6]. It is well-kno wn that this implies that X is quasiconv ex; see e.g. [11, Theorem 8.3.2]. Poincar ´ e in- equalities are strongly related to relativ e isoperimetric inequalities. More precisely , b y [15, Theorem 1.1] there exist D , λ ≥ 1 such that (2.7) min {H 2 ( B ∩ E ) , H 2 ( B \ E ) } ≤ D · H 1 ( λB ∩ ∂ E ) 2 for every Borel set E ⊂ X and every ball B ⊂ X . Here, λB denotes the ball with the same cen ter as B and radius equal to λ times the radius of B . Notice that in [15] the inequality (2.7) is form ulated for the Hausdorff measure of co dimension one H defined as H ( E ) = lim R → 0 inf ( ∞ X i =0 H 2 ( B ( x i , r i )) r i : E ⊂ ∞ [ i =1 B ( x i , r i ) , r i ≤ R ) for all Borel sets E ⊂ X . Since X is Ahlfors 2-regular, w e hav e H ≤ c · H 1 , where c > 0 denotes the Ahlfors 2-regularit y constant of X . Thus, the relative isop erimetric inequalit y as in (2.7) follows. Theorem 2.6. L et X b e a metric sp ac e that is home omorphic to a close d, ori- ente d, c onne cte d smo oth surfac e. Supp ose that X is A hlfors 2 -r e gular and line arly lo c al ly c ontr actible. Then, X satisfies a lo c al quadr atic isop erimetric ine quality for Lipschitz curves. Pr o of. Let Y ⊂ X b e a subset that is home omorphic to the closed disk D with diam Y < diam( X ) / 3 and let Ω ⊂ Y be a Jordan domain. A Jordan domain Ω is an open set homeomorphic to D such that ∂ Ω is homeomorphic to S 1 . The relativ e isop erimetric inequalit y (2.7) implies that H 2 (Ω) ≤ D · H 1 ( ∂ Ω) 2 = D · length( ∂ Ω) 2 . It follows from [17, Theorem 1.4] that Y satisfies a quadratic isoperimetric inequality for Lipsc hitz curv es with constant D . Co v ering X b y finitely man y subsets Y as ab o v e, we conclude that X satisfies a lo cal quadratic isop erimetric inequality for Lipsc hitz curves. □ 3. Codimension one homology Throughout this section, let X be a metric space with finite Hausdorff n -measure that is homeomorphic to a closed, orien ted, connected Riemannian n -manifold M . W e denote by  : X → M a homeomorphism of degree one. It is a classical fact that 12 DENIS MAR TI ev ery primitiv e co dimension one singular homology class in M can b e represented b y an embedded submanifold. A homology class is called primitive if it is the zero class or if it is not a nontrivial multiple of another class. W e need a v ersion of this statemen t for the homology via integral curren ts. Lemma 3.1. L et [ η ] ∈ H IC n − 1 ( M ) b e primitive and non-zer o. Then, ther e exists a surje ctive smo oth map f : M → S 1 with the fol lowing pr op erty. F or almost al l p ∈ S 1 , the pr eimage f − 1 ( p ) = N p is a close d, oriente d smo oth ( n − 1) − manifold and  J N p K  = [ η ] = [ ⟨ J M K , f , p ⟩ ] . Here, ⟨ J M K , f , p ⟩ is the slice of the fundamental class J M K of M b y f at p ∈ S 1 as defined in Section 2.2, and J N p K ∈ I n − 1 ( M ) denotes the in tegral ( n − 1)-cycle induced b y N p . Pr o of. The homology groups H IC n − 1 ( M ) and H n − 1 ( M ) are isomorphic; see Theorem 2.4. By abuse of notation, we also denote by [ η ] the singular homology class giv en by the isomorphism H n − 1 ( M ) → H IC n − 1 ( M ). It follo ws from the Poincar ´ e dualit y that [ η ] can b e represented by a surjective smo oth map f : M → S 1 . The standard duality theorems imply that for each regular v alue p ∈ S 1 , the preimage f − 1 ( p ) = N p is a closed, oriented submanifold representing [ η ]; see [21]. Using a triangulation of N p and Theorem 2.4, one easily chec ks that the singular homology class induced b y N p and the homology class in H I C n − 1 ( M ) induced b y J N p K agree under the isomorphism H n − 1 ( M ) → H IC n − 1 ( M ). No w, let p ∈ S 1 b e a regular v alue such that the slice ⟨ J M K , f , p ⟩ exists. Then, ⟨ J M K , f , p ⟩ has a represen tation [ M ∩ f − 1 ( p ) , θ , τ p ] with θ = 1 and τ p is an orientation of M ∩ f − 1 ( p ) = N p induced by f and the orientation of M ; see (2.2). It follows from the pro of of [2, Theorem 9.7] that τ p is the same orien tation as the pullback orientation of N p giv en b y f ; see [8, p. 100]. Therefore, ⟨ J M K , f , p ⟩ = [ N p , θ , τ p ] = J N p K . In particular, [ J N p K ] = [ η ] = [ ⟨ J M K , f , p ⟩ ] for almost all p ∈ S 1 . This completes the pro of. □ Theorem 1.2 will b e a direct consequence of the following proposition. Prop osition 3.2. Supp ose that X has a metric fundamental class. Then, ther e exists ε > 0 such that every Lipschitz ϕ : X → M satisfying d ( , ϕ ) < ε induc es a surje ctive homomorphism ϕ ∗ : H IC n − 1 ( X ) → H IC n − 1 ( M ) . Recall that a metric fundamental class of X is a cycle T ∈ I n ( X ) satisfying ϕ # T = deg ( ϕ ) · J M K for every Lipschitz map ϕ : X → M , where deg( ϕ ) denotes the (top ological) degree of ϕ . In fact, T also satisfies a second prop ert y that w e do not need here; see Definition 1.1. Pr o of of Pr op osition 3.2. Since M is a closed Riemannian manifold, there exists a bi-Lipsc hitz embedding ι : M  − → R N and a Lipschitz retraction π : N 2 ε ( ι ( M )) → ι ( M ) for some N ≥ n and ε > 0. See e.g. [10, Theorem 3.10] and [12, Theorem 3.1]. Let ϕ, ψ : X → M b e tw o Lipschitz maps satisfying d ( ϕ,  ) , d ( ψ ,  ) < ε . The straigh t-line homotop y H betw een ( ι ◦ ϕ ) and ( ι ◦ ψ ) stays inside N 2 ε ( ι ( M )). Therefore, the comp osition ( ι − 1 ◦ π ◦ H ) : [0 , 1] × X → M is w ell-defined, Lipsc hitz, and a homotopy betw een ϕ and ψ . W e conclude that ϕ ∗ and ψ ∗ are equal as homomorphism H IC n − 1 ( X ) → H IC n − 1 ( M ). The homology group H n − 1 ( M ) is finitely generated b ecause M is an absolute neigh- b orhoo d retract; see e.g. [9, Corollary A.8]. Th us, H IC n − 1 ( M ) is finitely generated as well. W e fix a generating set of H IC n − 1 ( M ) that consists of primitiv e elements of H IC n − 1 ( M ). Clearly , it suffices to show that for each generator [ η ] ∈ H IC n − 1 ( M ), there exists [ ξ ] ∈ H IC n − 1 ( X ) with ϕ ∗ [ ξ ] = [ η ] for all Lipsc hitz maps ϕ : X → M satisfying d ( , ϕ ) < ε . Let [ η ] ∈ H IC n − 1 ( M ) b e a generator. It follo ws from Lemma 3.1 that there exists a smooth map f : M → S 1 suc h that [ ⟨ J M K , f , p ⟩ ] = [ η ] for almost all ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 13 p ∈ S 1 . By Lemma 2.1, w e can approximate the homeomorphism  : X → M by a Lipsc hitz map ϕ : X → M that is homotopic to  , satisfies d ( ϕ,  ) < ε and such that the composition f ◦ ϕ is surjectiv e. The degree is a homotop y inv ariant and th us, deg ( ϕ ) = deg(  ) = 1. As explained in Section 2.2, the slice ⟨ T , g , p ⟩ exists and is an integral ( n − 1)-cycle for almost all p ∈ S 1 . Lemma 2.3 implies ϕ ∗ [ ⟨ T , f ◦ ϕ, p ⟩ ] = [ ϕ # ⟨ T , f ◦ ϕ, p ⟩ ] = [ ⟨ ϕ # T , f , p ⟩ ] = [ ⟨ J M K , f , p ⟩ ] = [ η ] , for almost every p ∈ S 1 . Here, w e used that T is the metric fundamen tal class of X and hence, ϕ # T = deg ( ϕ ) · J M K = J M K . This completes the pro of. □ W e conclude the section with the short pro of of Theorem 1.2 Pr o of of The or em 1.2. Let ε > 0 b e given by Proposition 3.2. Applying Lemma 2.1 to the homeomorphism  : X → M w e obtain a Lipsc hitz map ϕ : X → M with d ( ϕ,  ) < ε . Therefore, the induced homomorphism ϕ ∗ : H IC n − 1 ( X ) → H IC n − 1 ( M ) is surjectiv e. W e denote by Θ : H IC n − 1 ( M ) → H n − 1 ( M ) the isomorphism of Theorem 2.4. Since  is a homeomorphism, the induced map  − 1 ∗ : H n − 1 ( M ) → H n − 1 ( X ) is an isomorphism. W e conclude that (  − 1 ∗ ◦ Θ ◦ ϕ ∗ ) : H IC n − 1 ( X ) → H n − 1 ( X ) is a surjection. □ 4. One-dimensional homology The goal of this section is to pro v e Theorem 1.3. Let X b e a quasicon v ex metric space with finite Hausdorff n -measure that satisfies a 1-dimensional small mass isop erimetric inequalit y for in tegral curren ts with constants C, Λ > 0. F urthermore, supp ose that there exists a homeomorphism  : X → M , where M is a closed, Riemannian manifold. Notice that X still satisfies a 1-dimensional small mass isop erimetric inequality for integral currents after a bi-Lipschitz change of the metric and the conclusion of Theorem 1.3 is in v ariant under bi-Lipschitz changes of the metric. Thus, w e ma y assume that X is geo desic. First, we sho w that every homology class in H IC 1 ( X ) has a representativ e given b y a finite sum of injective Lipsc hitz loops. Lemma 4.1. L et T ∈ I 1 ( X ) b e a cycle. Then, ther e exist finitely many inje ctive Lipschitz lo ops γ i : S 1 → X and V ∈ I 2 ( X ) such that ∂ V = T − N X i =1 γ i # J S 1 K . Pr o of. It follo ws from [5, Theorem 5.3] that there exist countably many injective Lipsc hitz lo ops γ i : S 1 → X suc h that T = ∞ X i T i and M ( T ) = ∞ X i M ( T i ) , where T i = γ i # J S 1 K . Let N b e sufficiently large suc h that for all i > N we hav e M ( T i ) < min(1 , Λ). Then, for eac h i > N , there exists a filling V i ∈ I 2 ( X ) of T i satisfying M ( V i ) ≤ C M ( T i ) 2 ≤ C M ( T i ) . Therefore, the sum V = P i>N V i con verges and defines an integral 2-curren t that is a filling of P i>N T i . In particular, ∂ V = T − N X i =1 T i = T − N X i =1 γ i # J S 1 K . This completes the pro of. □ 14 DENIS MAR TI Clearly , a finite sum of injective Lipschitz lo ops induces an element of C 1 ( X ). By abuse of notation, w e denote the induced singular 1-c hain also b y T , whenev er T is an in tegral 1-cycle given b y a finite sum of injective Lipschitz loops. W e w ant to define a homomorphism H IC 1 ( X ) → H 1 ( X ) by sending each class [ η ] ∈ H IC 1 ( X ) to a class in H 1 ( X ) induced b y a represen tativ e of [ η ] giv en by finitely man y Lipsc hitz lo ops. But to do so, we first need to sho w that the induced class in H 1 ( X ) does not dep end on the represen tation of [ η ]. F ortunately , this can easily b e done by factoring the different represen tations through the Riemannian manifold M . Lemma 4.2. L et β i , γ j : S 1 → X b e finitely many Lipschitz lo ops such that the inte gr al 1 -cycles T = N X i β i # J S 1 K and S = M X j γ j # J S 1 K define the same homolo gy class in H IC 1 ( X ) . Then, the induc e d singular homolo gy classes [ T ] , [ S ] ∈ H 1 ( X ) ar e e qual. In the Riemannian manifold M , the inclusions C Lip 1 ( M )  → I 1 ( M ) and C Lip 1 ( M )  → C 1 ( M ), resp ectively , induce isomorphisms on the level of homology; see Theorem 2.4. Pr o of. It follo ws from Lemma 2.1 that there exists a Lipschitz map ϕ : X → M suc h that the comp osition ψ =  − 1 ◦ ϕ is homotopic to the identit y on X . Therefore, [ ψ # T ] = [ T ] and [ ψ # S ] = [ S ] as singular homology classes. Since S and T are homologous as in tegral currents, the in tegral 1-cycles ψ # S and ψ # T represent the same homology class in H IC 1 ( M ). Notice that ϕ # T and ψ # S are Lipschitz chains. Hence, the singular homology classes of ϕ # T and ψ # S , obtained by the isomor- phism H IC 1 ( M ) → H 1 ( M ) and the classes induced by in terpreting ϕ # T and ψ # S as singular chains, coincide. Therefore, as singular homology classes, [ T ] = [ ψ # T ] =  − 1 ∗ [ ϕ # T ] =  − 1 ∗ [ ϕ # S ] = [ ψ # S ] = [ S ] This completes the pro of. □ W e define a homomorphism (4.1) Θ : H IC 1 ( X ) → H 1 ( X ) , [ η ] 7→ " N X i =1 γ i # J S 1 K # , where P N i =1 γ i # J S 1 K is an y represen tation of [ η ] b y finitely man y Lipsc hitz lo ops. By Lemma 4.1, there exists such a representation for each class [ η ] ∈ H IC 1 ( X ). F urthermore, Lemma 4.2 implies that Θ is w ell-defined. In order to pro v e Theorem 1.3, w e hav e to sho w that Θ is bijectiv e. So far, w e ha v e used representations of homology classes in H IC 1 ( X ) b y Lipschitz lo ops. How ev er, one can simply pass from suc h a representation to a represen tation b y Lipschitz curves γ : [0 , 1] → X and vice versa. In the remainder of this section, w e will use whichev er represen tation is more suitable for the situation. Lemma 4.3. The homomorphism Θ is surje ctive. Pr o of. Let ε > 0 b e such that every ball in X with radius less than ε is con tractible in X . Suc h an ε exists since X is homeomorphic to a c losed, smo oth manifold. Let [ η ] ∈ H 1 ( X ) b e a singular homology class, and let c b e a singular 1-chain that represen ts [ η ]. Then c is of the form c = P N i n i · γ i , where n i ∈ Z and γ i : [0 , 1] → X are contin uous curv es. Fix i = 1 , . . . , N for the momen t. Let 0 = t 1 < · · · < t k = 1 b e a partition of [0 , 1] suc h that eac h γ i ([ t j , t j +1 ]) is contained within an open ball of radius ε/ 2. W e define a Lipsc hitz curve β i : [0 , 1] → X that is a geo desic from γ i ( t j ) to γ i ( t j +1 ) for ev ery j = 1 , . . . , k − 1. Eac h concatenation − β i | [ t j ,t j +1 ] · γ i | [ t j ,t j +1 ] is ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 15 a closed curv e and is con tained within a ball of radius ε , and hence is con tractible. Here, − β i | [ t j ,t j +1 ] · γ i | [ t j ,t j +1 ] denotes the curv e that first follo ws γ i | [ t j ,t j +1 ] and then β i | [ t j ,t j +1 ] in the rev erse direction. W e conclude that there exists a singular 2- c hain C with bC = γ i − β i . W e rep eat this for eac h i = 1 , . . . , N and define T = P N i =1 n i · β i # J 0 , 1 K ∈ I 1 ( X ). Clearly , ∂ T = 0 and c and T are homologous as singular chains. Therefore, Θ  [ T ]  = [ c ] = [ η ]. W e conclude that Θ is surjective. □ Lemma 4.4. The homomorphism Θ is inje ctive. Pr o of. Let [ η ] ∈ H IC 1 ( X ) b e suc h that Θ  [ η ]  = 0 ∈ H 1 ( X ). Therefore, there exists a representation of [ η ] by finitely many Lipsc hitz curves γ i : [0 , 1] → X and a singular 2-c hain c that b ounds T = P N i =1 γ i # J 0 , 1 K , when T is in terpreted as a singular 1-chain. W e ma y supp ose that length( γ i ) < Λ / 3 for eac h i = 1 , . . . , N . W rite c = P m j =1 θ j ψ j , where ψ j : ∆ 2 → X are contin uous maps and θ j is equal to 1 or − 1. By passing to barycentric sub division and sub dividing the γ i accordingly , if necessary , we ma y assume that for all x, y ∈ ∆ 2 and ev ery j = 1 , . . . , m , w e ha v e d ( ψ j ( x ) , ψ j ( y )) < Λ / 3. F or j = 1 , . . . , m and k = 1 , 2 , 3, let ψ k j : [0 , 1] → X b e the restriction of ψ j to the k th b oundary comp onen t of ∆ 2 . Then (4.2) ∂ c = m X j =1 3 X k =1 ( − 1) k θ j ψ k j = N X i =1 γ i . F or eac h j = 1 , . . . , m and ev ery k = 1 , 2 , 3, w e define a Lipschitz curv e β k j : [0 , 1] → X as follows. If ψ k j is equal to one of the curv es γ i , we set β k j = γ i . Otherwise, w e let β k j : [0 , 1] → X be a geo desic from ψ k j (0) to ψ k j (1). W e can do this in a consisten t wa y such that β k j = β k ′ j ′ whenev er ψ k j = ψ k ′ j ′ . F or every j = 1 , . . . , m , let S j = P 3 k =1 ( − 1) k β k j # J 0 , 1 K ∈ I 1 ( X ). It follows M ( S j ) ≤ 3 X k =1 length( β k j ) < Λ , for all j = 1 , . . . , m . Moreov er, b y construction, each S j is a cycle. Therefore, the small mass isop erimetric inequalit y implies that for ev ery j = 1 , . . . , m there exists a filling V j ∈ I 2 ( X ) of S j . W e put V = P m j =1 θ j V j . By (4.2), w e ha v e ∂ V = m X j =1 θ j ∂ V j = m X j =1 θ j S j = m X j =1 3 X k =1 ( − 1) k θ j β j k # J 0 , 1 K = N X i =1 γ i # J 0 , 1 K = T . Hence, V is a filling of T and in particular, [ T ] = [ η ] is the trivial class in H IC 1 ( X ). This completes the pro of. □ By com bining the tw o previous lemmas, w e conclude that the homomorphism Θ : H IC 1 ( X ) → H 1 ( X ) defined in (4.1) is an isomorphism. This prov es Theorem 1.3. W e conclude the section with the pro of of the corollary . Pr o of of Cor ol lary 1.4. Let X b e a metric space homeomorphic to a closed, ori- en ted, connected smooth surface. Suppose that X is Ahlfors 2-regular and linearly lo cally contractible. It follows from Prop osition 2.5 and Theorem 2.6 that X satisfies a 1-dimensional small mass isop erimetric inequalit y for in tegral curren ts. Therefore, b y Theorem 1.3 the homology group H IC 1 ( X ) is isomorphic to H 1 ( X ). F urthermore, b y [4, Theorem 1.3] there exists T ∈ I 2 ( X ) that generates H IC 2 ( X ) and in particu- lar, H IC 2 ( X ) and H 2 ( X ) are isomorphic. Recall that X is quasicon v ex; see Section 2.4. Using this is it not difficult to sho w that H IC 0 ( X ) and H 0 ( X ) are isomorphic as w ell. Finally , X is Ahlfors 2-regular and thus I k ( X ) = 0 for all k > 2. W e conclude that H IC k ( X ) = 0 = H k ( X ) for all k > 2. This completes the pro of. □ 16 DENIS MAR TI 5. Examples W e construct t w o geo desic metric spaces X and Y that are homeomorphic to the 2-sphere S 2 and ha v e finite Hausdorff 2-measure. While X is Ahlfors 2-regular but not linearly locally contractible, Y is doubling and linearly lo cally contractible but not Ahlfors 2-regular. Ho w ev er, the 1-dimensional homology groups via integral curren ts of X and Y are not trivial. Recall from the introduction that a metric space E is linearly lo cally contractible if there exists λ > 0 such that every ball B ( x, r ) in E with r ∈ (0 , diam( E ) /λ ) is contractible within B ( x, λr ). F urthermore, E is said to b e Ahlfors 2-regular if there exists c > 0 such that for ev ery ball B ( x, r ) in E with r ∈ (0 , diam( E )) w e hav e c − 1 r 2 ≤ H 2 ( B ( x, r )) ≤ cr 2 . By [4, Theo rem 1.3], b oth spaces X and Y ha v e a metric fundamental class. There- fore, these examples show that the homology groups via in tegral curren ts can b e larger than the singular homology groups. 5.1. The first example. Let ( S k ) k ≥ 1 b e a family of disjoin t circles in S 2 of radius 1 / 2 3 k , arranged in a ”line” such that d ( S k , S k +1 ) = 1 / 2 k for all k ≥ 1. W e denote b y D k the closed disk bounded by S k . F or k ≥ 1, we define the ”mushroom” M k with the stem equal to the cylinder S k × [0 , 1 / 2 3 k ] and the cap equal to the hemisphere of radius 2 − k . Figure 1. Two subsequen t m ushrooms M k and M k +1 W e obtain X b y replacing each disk D k with the mushroom M k and we equip X with the intrinsic metric d int . Then H 2 ( X ) ≤ H 2 ( S 2 ) + ∞ X k =1 H 2 ( M k ) = 4 π + 2 π ∞ X k =1 1 2 4 k + 1 2 2 k < 5 π . Therefore, X is a geo desic metric space with finite Hausdorff 2-measure that is homeomorphic to S 2 . W e claim that X is Ahlfors 2-regular. The lo w er bound is easily verified. F or the momen t, fix x ∈ X and r ∈ (0 , 1). Let j, N ≥ 1 be such that M j is the closest mushroom to x and P ∞ k = N − 1 1 / 2 k ≤ r ≤ P ∞ k = N 1 / 2 k . In case N ≥ j + 2, then B ( x, r ) do es not intersect an y other mushroom than M j and the upp er bound H 2 ( B ( x, r )) ⪯ r 2 is immediate. Otherwise, if N < j + 2, then B ( x, r ) in tersects at most the M k for k ≥ N − 1, and hence, H 2 ( B ( x, r )) ≤ cr 2 + ∞ X k = N − 1 H 2 ( M k ) ≤ cr 2 + ∞ X k = N − 1  1 2 k  2 ≤ cr 2 + 4 π ∞ X k = N − 1 1 2 k ! 2 ≤ ( c + 4 π ) r 2 , ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 17 where c > 0 is the Ahlfors 2-regularity constan t of the standard 2-sphere. A ball cen tered at some S k that wraps around the stem of M k is only contractible within a ball that contains the whole m ushroom M k . The ratio betw een the circumference of the stem and the heigh t of M k is proportional to 1 + 2 2 k . W e conclude that X is not linearly lo cally contractible. Finally , for k ≥ 1, we define T = P ∞ k =1 2 2 k J S i K ∈ I 1 ( X ). Then, T is a cycle and M ( T ) ≤ ∞ X k =1 2 2 k M ( J S k K ) = 2 π ∞ X k =1 1 2 k < ∞ . Notice that eac h J S k K has only tw o fillings, J M k K and J M c k K . Out of these tw o p ossibilities, J M k K has smaller mass. How ev er, ∞ X k =1 2 2 k M ( J M k K ) = 2 π ∞ X k =1 2 2 k  1 2 4 k + 1 2 2 k  = ∞ . It follo ws that T has no filling with finite mass. In particular, [ T ]  = 0 and H IC 1 ( X ) is not trivial. 5.2. The second example. W e construct the next example Y out of the surface Q = ∂ [0 , 1] 3 of the unit cub e in R 3 . As before, we replace small pieces in Q with more complicated sets to obtain Y . W e b egin with the definition of those sets. Divide the unit square [0 , 1] 2 in to 9 squares of side-length 1 / 3 and replace the middle square with the surface of the cub e [0 , 1 / 3] 3 , from whic h w e ha v e remo v ed the b ottom square. The resulting surface G is called the generator and consists of 13 squares of side-length 1 / 3. Set S 0 = G . W e inductively define the surfaces S k b y replacing each of the 13 k squares in S k − 1 with a cop y of the generator scaled b y a factor of 1 / 3 k . Figure 2. The generator G and the first iteration S 1 . Notice that the S k ha ve self-intersections. T o av oid this, w e carefully round off the sides of eac h new cub e added in S k . W e equip each S k with the intrinsic length metric d int . It is not difficult to sho w that the sequence ( S k ) k ≥ 0 con verges in the Hausdorff distance. The limit S is one side of the so-called sno wsphere. The snowsphere is a 2-dimensional analogue of the snowflak e curve and has b een extensiv ely studied in [23, 24]. W e need the following facts. (1) The S k are uniformly doubling; (2) H 2 ( S k ) = (13 / 9) k for every k ∈ N and in particular, H 2 ( S k ) → ∞ as k → ∞ ; (3) d int ( x, y ) ≤ 4 for all x, y ∈ S k and k ∈ N ; (4) the S k are uniformly linearly lo cally con tractible. The first three prop erties are verified easily and w e refer to [23] for details. W e explain why the last prop ert y holds. By [23, Theorem 5.1], the limit S of the S k is homeomorphic to the unit square via a quasisymmetry f : S → [0 , 1] 2 . That is, there exists a homeomorphism η : [0 , ∞ ) → [0 , ∞ ) such that (5.1) d S ( x, y ) ≤ td S ( x, z ) = ⇒ ∥ f ( x ) − f ( y ) ∥ ≤ η ( t ) ∥ f ( x ) − f ( z ) ∥ 18 DENIS MAR TI for all x, y , z ∈ S and t ≥ 0. It follows from the construction of f that the S k are uniformly quasisymmetric to the unit square. Here, uniformly means that there exists one homeomorphism µ : [0 , ∞ ) → [0 , ∞ ) such that eac h quasisymmetry f k : S k → [0 , 1] 2 satisfies (5.1) with resp ect to that homeomorphism µ . F urthermore, it is a direct consequence of the definitions that a compact metric space X which is quasisymmetric to a linearly locally con tractible metric space is itself linearly lo cally contractible (quantitativ ely). This yields prop erty (4). Let Q be the surface of the unit cube [0 , 1] 3 and let ( Q k ) k ∈ N b e a family of disjoin t squares of side-length (3 / 4) k . W e obtain Y b y replacing each square Q k with S k (scaled appropriately). Then, Y is a top ological 2-sphere and has finite Hausdorff 2-measure H 2 ( Y ) ≤ H 2 ( Q ) + ∞ X k =1  3 4  2 k H 2 ( S k ) = 6 + ∞ X k =1  13 16  k < ∞ . Com bining prop erties (2) and (3) implies that Y is not Ahlfors 2-regular. Since the doubling prop ert y and linear lo cal con tractibilit y are inv arian t under scaling, w e conclude that Y is doubling and linearly locally con tractible. Finally , let T = P ∞ k =1  13 10  k ∂ J S ′ k K , where the S ′ k denote the scaled copies of the S k w e attached to Y . Then, T is an integral 1-cycle with M ( T ) ≤ ∞ X k =1  13 10  k M ( ∂ J S ′ k K ) = 4 ∞ X k =1  39 40  k < ∞ . F or eac h k ≥ 1, the in tegral 1-cycle ∂ J S ′ k K has exactly t wo fillings, J S ′ k K and J ( S ′ k ) c K . W e hav e M ( J S ′ k K ) ≤ M ( J ( S ′ k ) c K ) for each k ≥ 1. How ever, ∞ X k =1  13 10  k M ( J S ′ k K ) = ∞ X k =1  169 160  k = ∞ . Therefore, T has no filling with finite mass. W e conclude that [ T ]  = 0 and H IC 1 ( X ) is not trivial. References [1] J. C. ´ Alv arez P aiv a and A. C. Thompson. V olumes on normed and Finsler spaces. In A sampler of Riemann-Finsler ge ometry , v olume 50 of Math. Sci. R es. Inst. Publ. , pages 1–48. Cambridge Univ. Press, Cambridge, 2004. [2] Luigi Ambrosio and Bernd Kirchheim. Curren ts in metric spaces. A cta Math. , 185(1):1–80, 2000. [3] Luigi Ambrosio and Bernd Kirc hheim. Rectifiable sets in metric and Banach spaces. Math. Ann. , 318(3):527–555, 2000. [4] Giuliano Basso, Denis Marti, and Stefan W enger. Geometric and analytic structures on metric spaces homeomorphic to a manifold. Duke Math. J. , 174(9):1723–1773, 2025. [5] Paolo Bonicatto, Giacomo Del Nin, and Enrico Pasqualetto. Decomposition of integral metric currents. J. F unct. A nal. , 282(7):P aper No. 109378, 28, 2022. [6] Th. De Pau w, R. M. Hardt, and W. F. Pfeffer. Homology of normal chains and cohomology of charges. Mem. Amer. Math. So c. , 247(1172):v+115, 2017. [7] Herb ert F ederer and W endell H. Fleming. Normal and integral currents. A nnals of Mathe- matics , 72(3):458–520, 1960. [8] Victor Guillemin and Alan P ollack. Differential top olo gy . Pren tice-Hall, Inc., Englewoo d Cliffs, NJ, 1974. [9] Allen Hatcher. Algebr aic top olo gy . Cam bridge Universit y Press, Cambridge, 2002. [10] Juha Heinonen. Ge ometric emb e ddings of metric sp ac es , volume 90 of R eport. University of Jyv¨ askyl¨ a Dep artment of Mathematics and Statistics . Univ ersity of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, 2003. [11] Juha Heinonen, Pekk a Kosk ela, Nagesw ari Shanmugalingam, and Jerem y T. T yson. Sob olev sp ac es on metric me asur e sp ac es , volume 27 of New Mathematic al Mono gr aphs . Cambridge Universit y Press, Cambridge, 2015. [12] Aarno Hohti. On absolute Lipschitz neighbourho o d retracts, mixers, and quasiconv exit y . T op ol. Pr o c. , 18:89–106, 1993. ONE-DIMENSIONAL AND CODIMENSION ONE HOMOLOGY OF METRIC MANIFOLDS 19 [13] Jingyin Huang, Bruce Kleiner, and Stephan Stadler. Morse quasiflats I. J. Reine Angew. Math. , 784:53–129, 2022. [14] T oni Ikonen. Pushforward of currents under sobolev maps. arXiv pr eprint arXiv:2303.15003 , 2023. [15] Riikk a Korte and Pan u Lahti. Relative isop erimetric inequalities and sufficient conditions for finite p erimeter on metric spaces. A nn. Inst. H. Poinc ar´ e C A nal. Non Lin´ eaire , 31(1):129– 154, 2014. [16] Urs Lang. Lo cal currents in metric spaces. J. Ge om. Anal. , 21(3):683–742, 2011. [17] Alexander Lytchak and Stefan W enger. Canonical parameterizations of metric disks. Duke Math. J. , 169(4):761–797, 2020. [18] Alexander Lytchak, Stefan W enger, and Robert Y oung. Dehn functions and H¨ older extensions in asymptotic cones. J. Reine Angew. Math. , 763:79–109, 2020. [19] Denis Marti. The metric fundamen tal class of non-orientable manifolds and manifolds with boundary . In pr ep ar ation . [20] Denis Marti and Elefterios Soultanis. Characterization of metric spaces with a metric funda- mental class. arXiv pr eprint arXiv:2412.15794 , 2024. [21] William H. Meeks, I II and Julie P atrusky . Representing co dimension-one homology classes by embedded submanifolds. Pacific J. Math. , 68(1):175–176, 1977. [22] Damaris Meier, Noa Vikman, and Stefan W enger. Monotone sob olev extensions in metric surfaces and applications to uniformization. arXiv preprint , 2025. [23] Daniel Meyer. Quasisymmetric em b edding of self similar surfaces and origami with rational maps. Ann. A c ad. Sci. F enn. Math. , 27(2):461–484, 2002. [24] Daniel Meyer. Sno wballs are quasiballs. T r ans. Amer. Math. So c. , 362(3):1247–1300, 2010. [25] Ayato Mitsuishi. The coincidence of the homologies of in tegral currents and of integral singular chains, via cosheaves. Math. Z. , 292(3-4):1069–1103, 2019. [26] Christian Riedw eg and Daniel Sc h¨ appi. Singular (lipsc hitz) homology and homology of integral currents. arXiv pr eprint arXiv:0902.3831 , 2009. [27] Stephen Semmes. Finding curv es on general spaces through quan titativ e topology , with ap- plications to Sobolev and Poincar´ e inequaliti es. Sel. Math., New Ser. , 2(2):155–295, 1996. [28] Antoine Song. Spherical v olume and spherical plateau problem. T o app ear in S´ eminair e de th´ eorie sp e ctr ale et g ´ eom ´ etrie . [29] Stefan W enger. Flat conv ergence for in tegral currents in metric spaces. Calc. V ar. Partial Differ ential Equations , 28(2):139–160, 2007. Dep ar tment of Ma thema tics, University of Fribourg, Chemin du Mus ´ ee 23, 1700 Fri- bourg, Switzerland Email address : denis.marti@unifr.ch