Cycle decompositions: from graphs to continua

Cycle decomp ositions : from graphs to con tin ua Agelos Georgak o poulos ∗ T ec hnisc he Univ ersit¨ at G raz Steyrergasse 30, 8010 Graz, Austria Octob er 27, 2021 Abstract W e generalise a fund amen tal graph - theoretical fact, stating that every elemen t of the cycle sp ace of a graph is a sum of edge-disjoint cycles, to arbitrary continua. T o achiev e this we replace graph cycles by top ologica l circles, and replace the cycle space of a graph by a new h omol ogy group for conti nua which is a qu otien t of the first singular homology group H 1 . This homology seems to b e particularly apt for study ing spaces with infinitely generated H 1 , e.g. infinite graphs or fractals. 1 In tro duction 1.1 Ov erview In a recent series of pap ers, Diestel et. al. show ed that many well-known theo- rems a b out cycles in finite graphs r emain true for infinite gra phs pr o vided one replaces the c lassical graph-theor etical concepts by top ological analo gues. F or example, instead of g raph cycles one uses top ological cir cles. This appro ac h has bee n very fruitful, not only extending theorems fro m the finite to the infinite case (see e .g . [5, 6, 1 6]) , but a lso ha v ing further applicatio ns [1 8] and op ening new directions [4, 12, 11, 10, 19]. See [7] for a survey o n this pro ject. This pa per is motiv ated by an a ttempt to gene r alise so me of these gr aph- theoretical facts to contin uous ob jects. And indeed, our main result is a g eneral- isation of one of the most basic to ols in the a fo remen tioned pro ject of Diestel et. al., Theor em 1 .3 b elow, from gr aphs to arbitrar y contin ua. In order to achieve this g eneralisation we in tro duce a new homology tha t generalise s the cycle space of gra phs to arbitra ry metric spa ces. W e use this ho mology to conjecture a char- acterisatio n of the cont inua embeddable in the plane. 1.2 Bac kground and motiv ation The cy cle space C ( G ) of a finite gr aph G coincides with its first, simplicial or singular, homology group. As an example of the usefulness of this concept in graph theory , let me men tion the following classical theorem of MacLane, providing an a lgebraic characterisation of the gr aphs embeddable in the plane. ∗ Supported by GIF grant I-879-124.6/2005 and FWF gr an t P-19115-N18. 1 Theorem 1.1 (MacLane [22], [6]) . A finite gr aph G is planar if and only if its cycle sp ac e C ( G ) has a 2-b asis. Here, a 2-basis is a set B generating C ( G ) s uc h that no edg e of G is used by more than tw o elements of B . See [6 ] for mor e. If the gr aph is infinite though, then Theorem 1.1 does no t hold any mor e if C ( G ) is still ta k en to b e the first simplicial o r singula r ho mology gro up [5]. How ever, Diestel and K ¨ uhn [8, 9] intro duced a new homolog y for infinite graphs , called the top olo gic al cycle sp ac e C ( G ) , which a llows a verbatim g eneralisation of Theor e m 1.1: Theorem 1.2 (Br uhn & Stein [5]) . A lo c ally fi nite gr aph G is plana r if and only if its top olo gi c al cycle sp ac e C ( G ) has a 2-b asis. The top ological cycle space allows for such generalis a tions o f all the funda- men tal facts ab out the cycle spa ce of a finite gra ph. It is defined as a v ector space, o ver Z 2 , co ns isting of sets of edges of the graph. Namely , it con ta ins those e dg e-sets of G that form top ological circles in the end-compactification | G | of G , as well as the s ums o f these edge-sets, where we allow sums of infinitely many summands as long a s they are w ell defined. An imp ortan t innov ation in the approa c h of Diestel and K ¨ uhn is that even if one is interested in the graph G only , it is helpful to consider the lar g er s pace | G | that als o contains the ends of G . The interested re a der can find mor e details and results ab out the top ological cycle space in [6, Chapter 8.5] or [7]; these details are how ever not ne c essary for understanding the current pap er. The to pologic a l c ycle space C ( G ) of G is larger than the fir st simplicial homolo gy gr oup of G , since the latter do es not hav e any element comprising infinitely ma ny edges. It is far less o bvious, but true [11], tha t C ( G ) is on the o ther hand sma ller than the first s ing ular homolog y gr o up of | G | . Consider for example the graph G of Figur e 1 , whic h is a one-w ay infinite ‘ladder’. The end- compactification | G | of G is in this ca se its one- p oint c o mpactification (gra phs are considered as 1-complexes throughout the paper ). Thus there is a loo p σ in | G | , depicted in Figure 1, starting at the top-left vertex v , winding around ea c h of the infinitely many 4-gona l faces of G , r eac hing the p oin t at infinity , then r eturning to v , and finally winding aro und the whole graph once in the clo c kwise direction without using any of the p erp endicular edg es. It turns o ut [11] that σ do es no t b elong to the trivial elemen t of H 1 ( | G | ), but it do es c o rresp ond to the trivial elemen t of C ( G ): it trav er ses each edge the same num b er o f times in e ac h direction; thus, seen a s an ele men t of C ( G ), it is the empt y set of edges. A s imilar example can be obtained in the Haw a iian ea ring by contracting a spa nning tree of G to a po in t. This pathologica l behaviour of σ is due to the fact that althoug h it winds around any hole the sa me num b er o f times in each direction, it do es so in such a complicated or der that one ca nnot ‘disentangle’ it by a dding only finitely many bo undaries of 2-simplices. T o put it in a different wa y , the ho mology cla ss of σ is a pro duct of infinitely ma n y commutators. This example shows that C ( G ) is indeed smaller than the first singular ho- mology group of | G | a s cla imed. How ever, this discrepancy b et ween C ( G ) and H 1 ( | G | ) sho uld b y no means b e considered a s a shortcoming of C ( G ); for e x- ample, it is imp ortant for the truth of Theor em 1.2: the set of edge-sets o f the 4-gona l fac e s of Figure 1 form a 2-bas is, but it cannot repres en t a lo op like σ . It turns out, and is not hard to chec k, that C ( G ) is ca no nically isomorphic to the first ˇ Cech homo logy gr o up of | G | ; see [1 1] for deta ils. 2 v Figure 1: A loop σ that is not null-homologous although w e w ould like it t o b e. W e w ould like to genera lise gr aph-theoretical theorems like Theore m 1 .1 to contin uous s paces. The main aim of this paper is to achieve such a generalisa - tion for the following fact, which has b een a corners tone in the a fo remen tioned pro ject of Diestel et. al. Theorem 1.3 (Diestel and K ¨ uhn [9]) . L et G b e a lo c al ly fin ite gr aph. Then every element of C ( G ) is a disjoint union of e dge-sets of cir cles in | G | . Theorem 1 .3 has found se veral applica tions in the study of C ( G ) [5, 8, 20] and else where [16], a nd at least four pro ofs have b een published; see [17] for an exp osition. Now in order to b e able to generalis e theorems like Theorem 1.3 or Theo - rem 1.1 to co n tinuous spaces, we hav e to overcome t wo ma jor difficulties: firstly , reformulate the assertio ns to rid them of an y concepts, e.g. edg es, that o nly make sense for gr a phs, and seco ndly , choose the right ho mology theor y . T o see how the first difficulty can b e ov er c ome, supp ose that the graph G in Theorem 1.3 is finite. W e could then r eform ulate the as sertion as follows: Every element of C ( G ) has a representative of minimal length. (1) Here, a r epr esentative is a forma l sum of e dge-sets of cyc le s. Indeed, this for- m ulation is equiv alen t to that of Theor em 1 .3 if G is finite: a representative o f minimal length cannot hav e tw o summands C 1 , C 2 containing the same edge e , for then we could delete e , and a ny other common edges , fr om b o th C 1 , C 2 and combine the r emaining paths in to a ne w cycle or new set o f cycles w ho se total length is smaller , since we sav ed some length by removing e . F or m ulation (1) ha s the a dv an tage that it makes sense for ob jects other than graphs if o ne replaces C ( G ) by so me suitable homolog y gro up. The question now is, which homolog y should one use to extend this a ssertion b ey ond graphs. F or example, singula r ho mology will not do b ecause of the ex a mple of Figure 1: the lo op σ has finit e length if we metrize that space using the E uclidean metric, but there are lo ops homolo gous to σ with arbitrarily sma ll length, namely , those obtained by translating σ to the r igh t by o ne o r more squares. Sing ular homology can fail to satisfy (1) even if it is finitely generated, se e Exa mple 6.3. 1.3 A new homology In view of the ab ov e discussion it is clear tha t in order to make asser tion (1 ) true in genera l we need a homology group that excludes s o me ‘r edundan t’ elements of singular homology . In fact such an approach is often follow ed when dealing with 3 ‘wild’ spac e s, e.g. spaces with an uncount a bly ge nerated fundamental g roup: in these cases many elements of the homotopy or homolo gy gro ups do not ca pture some ‘hole’ of the s pa ce but rather represent a complicated wa y to wind ar ound infinitely many ho les, and o ne w a n ts to omit these e le men ts in or der to obtain a smaller group that still reflects the structure of the space; see [10, 1 3, 14 ] fo r some examples . In many cas es the b e tter-kno w n shap e groups [2 3] a lso pr o vide such simplifications of the corres ponding homotopy or homolo gy gro ups. Earlier constructions of homolo gy gro ups are not well-suited for our purp oses as they either o bviously fail to s atisfy (1) or it is not cle ar how to a ssign lengths to their representativ es. In this pap er, we will introduce a homology gro up H d that comes with a na tural no tion of length, has the top ological cycle spa ce a s a sp ecial cas e (Sectio n 1 1) and, mor e imp ortantly , ma k es ass ertion (1 ) true for all compact metric spaces. W e define H d as a quo tient o f the fir st singular homology group H 1 . F or example, we w ould like to iden tify the clas s of σ in the example of Figure 1 with the trivia l class. In order to decide which class es should b e identifi ed, w e int r oduce a natura l distance function on H 1 , and ident ify a ny tw o elements if their distance is zero. This distance function is defined a s follows. Intuitiv ely , if t wo 1 -cycles are not homo logous, then there are some ‘holes ’ in o ur space that witness this fa ct, and we assign a dista nce to the corresp onding pair o f clas s es of H 1 reflecting the ‘size’ of these ho les. More precisely , the dista nce betw ee n t wo classe s c, d ∈ H 1 is defined to b e the minimal total area of a —po ssibly infinite— set o f metric discs and cylinders that we could glue to o ur space X to make c and d homologous. These metric discs and cylinders m ust be a r a metric such that this glueing do es not affect the metric of X . See Section 3 for the formal definitions. In Sec tio n 6 w e display some examples that justify this definition by showing that mo difying it would ma k e assertio n (1 ) fals e . An imp o rtan t feature of this distance function is that a n infinite commutator pro duct as the one of Figure 1 can have distance zero to the trivial elemen t. F or ex ample, patc hing all but finitely many of the 4-gona l faces in Figure 1 by a dding the missing trap eze w ould render σ null-homologous , and this can be acco mplishe d b y adding ar bitrarily little area if we sk ip a lo t o f the 4-g o nal faces. The afo r emen tioned distance function gives rise to a metric on H d after the ident ifications hav e taken place, whic h turns H d int o a metriz a ble top ological group. W e will also consider the completion ˆ H d of H d , which will hav e the effect of streng thening our ma in result. 1.4 Main result W e can now state our main result. Theorem 1.4. F o r every c omp act metric sp ac e X and C ∈ H d ( X ) , ther e is a σ -r epr esentative ( z i ) i ∈ N of C whose length is at most the infimum of the lengths of al l r epr esentatives of C . Here, a σ -representativ e can intuitiv ely b e thought of as a sum of infinitely many 1-cy c le s z i . F ormally , a σ -representativ e of C is defined as a sequence ( z i ) i ∈ N whose initial subsequence s g iv e rise to a se quence ( P j ≤ i z j ) i ∈ N of 1- cycles the homology classes of whic h conv er ge to C with r espect to the metric of H d ; see Section 3 for details. The length of a σ -representativ e is the sum of 4 the lengths o f the simplices in z i , the latter lengths b eing defined in the standar d wa y (see Section 2). F or example, consider the subspace X of the rea l plane depicted in Fig ure 2 . Let σ b e a closed 1-simplex σ : [0 , 1] → X that trav erse s each o f the infinitely many cir cles in this spa ce precisely o nce and has finite length. L e t β ∈ H 1 ( X ) denote the homo lo gy class of the 1-cycle 1 σ . Note that for every r epresen ta tiv e of β ther e is a further repres en tative of smaller length, obtained by avoiding to trav ers e some of the per pendicular segments. Th us no representative achiev es a minim um leng th. Still, Theorem 1.4 y ie lds a σ - representativ e ( z i ) i ∈ N of min- im um length: let for example each z i be a clo sed simplex winding aro und the i th circle o nce in a s traigh t manner. 1 2 1 2 1 2 1 4 1 4 1 4 1 8 1 8 1 8 1 16 1 16 1 16 Figure 2 : A compact subspace of th e real plane. The num b ers denote the length s of the corresp onding segments. Theorem 1.4 implies T he o rem 1 .3. This ca n b e shown by a simila r arg umen t as the one we us ed fo r the equiv alence of the la tter and (1) for finite G , except that if G is infinite w e assig n lengths to its edges to make their tota l length summable; see Section 11 for details. In fac t, we obtain a streng thened version of Theo rem 1.3. F urther more, with Theo rem 1.4 we g eneralise Theorem 1 .3 to non-lo cally-finite graphs, a chieving a goa l set by the author in [19, Se c tion 5]; see Section 11 for more. F or the pro of of Theorem 1 .4 we obtain an intermediate result w hich might b e of indep enden t interest. This r esult states that if ( H, + ) is an ab elian metriza ble top ological gr oup, and a function ℓ : H → R + is given satisfying cer tain natural prop erties that derive their intuition from the b eha viour o f leng ths in g eometry , then ev er y element h of H c an b e ‘decompos ed’ as a sum h = P h i so that ℓ ( h ) = P ℓ ( h i ) and no h i can b e decomp osed further . Se e Sec tio n 8 for details. 1.5 F urther problems and remarks In this section we discuss some r elated conjectures for which strong evidence is av ailable. With Theorem 1 .4 we extended a basic g raph-theoretical to ol to ar bitrary compact metric spaces. It r emains to try to explo it this in order to also extend results whose pro ofs ar e based o n or r elated to this to ol. A conjecture of this kind is o ffer ed in [19, Conjecture 6.1]. A further example is the fo llowing conjectur e , 5 which seeks an alg ebraic characterisa tion of the Peano contin ua embeddable in the plane, similar to that of Theorem 1.1 . Conjecture 1.5. L et X b e a c omp act, lo c al ly c onne cte d, metrizable sp ac e that is lo c al ly emb e ddable in S 2 . Then X is emb e ddable in S 2 if and only if ther e is a simple set S of cir cles in X and a m etric d inducing the top olo gy of X so that the set U := { J χ K ∈ ˆ H d ( X, d ) | χ ∈ S } sp ans ˆ H d ( X, d ) . See [19, Conjecture 6.2 ] for more on this conjecture. F o r example, X here could be the Sier pinski triangle, in which case w e could cho ose S to be the set of its triangula r face b oundaries, corr obora ting the conjecture. A further question motiv ated by our main result is whether something s imilar holds for hig her dimens io ns. It is stra igh tforward to s e e ho w to genera lise the definition of H d : instead o f top ologica l discs and cylinders one ha s to use their higher dimensiona l ana logues. Our pro of cannot prov e this, but many of our int e r mediate steps still work. Problem 1. 6. Gener alise The or em 1.4 t o higher dimensions. See Section 12 for mor e on this pro blem. Although w e can generalise o ur homolog y group H d or ˆ H d to higher di- mensions, w e do not o bta in a homolog y theor y in the sense of Eilenberg and Steenro d [15, 21], since H d ( X ) dep ends not only on the topo logy of X but a lso on its metric. F or the purp oses of the current pap er this is rather an a dv an tage of H d : since Theorem 1.4 holds for a n y choice of a compatible metric, we can affect the outcome o f the application of the theorem by v arying the metric. Still, it would b e interesting to obtain a similar homolog y theory that do es satisfy the axioms o f Eilenberg and Steenr od by eliminating the dep endence on the metric. Similarly , one could for example try to pr o ve the following: Conjecture 1.7. Every Pe ano c ontinuum X has a metric c omp atible wi t h its top olo gy such that the c orr esp onding ˆ H d c oincides with the first ˇ Ce ch homolo gy gr oup of X . Theorem 11 .1 b elow implies that this is true when X is the end-co mpa ctification of a lo cally finite g raph. The co ndition that X be a Peano contin uum is imp osed bec ause in a space that is not lo cally connected ˇ Cech ho mology may c o n tain elements not r e pr esen ted by sing ular homolo gy . 2 General definitions and basic facts In this section w e recall the standard definitions and facts tha t we will use later. Most of this is very well-known but it is included for the conv enience of the r e ader. F or other s tandard terms use d in the pap er but not found in this section we refer to the textbo oks [1] for top ology , [2 1] for algebra ic top ology and [6] for gra ph theory . F or every metric s pace M , it is p ossible to construct a complete metric space M ′ , called the c ompletion of M , which co n tains M as a dense subspace. The completion M ′ of M has the fo llo wing universal prop ert y [25]: 6 If N is a complete metric spa ce and f : M → N is a unifor mly contin uous function, then there exists a unique uniformly contin uous function f ′ : M ′ → N which extends f . The space M ′ is determined up to iso metry by this prop erty (a nd the fact that it is complete). (2.1) Next, w e recall the definition o f the length of a top ological path σ : [ a, b ] → M in a metr ic space ( M , d ). F or a finite seq uence S = s 1 , s 2 , . . . , s k of p oin ts in [ a, b ], let ℓ ( S ) := P 1 ≤ i 0 . Next, w e chec k tha t the leng ths of e lemen ts o f H 1 satisfy a triangle inequalit y : Lemma 5.4. L et X b e a metric sp ac e and let φ, χ b e two 1-chains in X . Then ℓ ([ φ + χ ]) ≤ ℓ ([ φ ]) + ℓ ([ χ ]) (and thus ℓ ([ φ − χ ]) ≥ ℓ ([ φ ]) − ℓ ([ χ ]) ). 13 Pr o of. It is a trivial fact that if φ ′ , χ ′ are 1 - c hains in X , then ℓ ( φ ′ + χ ′ ) ≤ ℓ ( φ ′ ) + ℓ ( χ ′ ). The assertion now easily follows from the definition o f ℓ ([ φ ]), s ince if b 1 , b 2 ∈ B ′ 1 then ℓ ( φ + χ + b 1 + b 2 ) ≤ ℓ ( φ + b 1 ) + ℓ ( χ + b 2 ). F ro m this we e asily o btain a tria ng le inequality for elements of ˆ H d to o: Corollary 5.5. L et X b e a metric sp ac e and let C , D ∈ ˆ H d ( X ) . Then ℓ ( C + D ) ≤ ℓ ( C ) + ℓ ( D ) (and thus ℓ ( C − D ) ≥ ℓ ( C ) − ℓ ( D ) ). 6 Examples In this section we sho w examples that explain some o f our choices in the pre - ceding definitions and sta temen ts. W e defined H d as a quotient of H 1 by iden tifying pair s of elemen ts with distance zer o . This identification e n tails the danger of identifying all of H 1 with the triv ial element, which would make H d and ˆ H d trivial a nd our main r esult void. And indee d, in certain pathological spaces X , e.g. when each element of H 1 ( X ) can be represented as an infinite pro duct of commutators, this could happ en; an exa mple of such a sp ece can b e found in [3]. How ever, the following basic example when X = S 1 shows that H d and ˆ H d are not trivia l whe n it should not b e: Theorem 6. 1 . ˆ H d ( S 1 ) ∼ = H 1 ( S 1 ) . Pr o of. Let σ b e a cir c lex in S 1 . Then H 1 ( S 1 ) is gener ated by the corr esponding homology class [ σ ]. Thus all we need to show is that [ σ ] is not identified with the trivial element 0 of H 1 ( S 1 ); in other words, that there is a low er b ound M such that every area extension of S 1 in whic h σ is null-homologous has excess area at lea s t M . So let ( S ′ , ι ) b e an area ex tension of S 1 in which σ is null-homologous. Thu s there is a 2 -c hain B in S ′ whose b oundary is σ . F rom now on w e assume for simplicity that the group of co efficients o n which H 1 ( S 1 ) is based is Z ; the in tere s ted reader will b e able to adapt our argumen ts to other gr oups of co efficien ts. W e may as s ume without loss of ge neralit y that B cons ists of a single 2- simplex ρ whose bo undary is a sub division o f σ into three subsimplices, for otherwise w e can co mbine pairs of 2- simplices o f B together to g et a shor ter 2-chain. All we need to show no w is that the area A ( P ) o f the image P o f ρ is b ounded from b elo w by some constant M indep endent of S ′ . In fact, we will show that we can ch o ose M = 1. Rec all that w e defined the area of a metric space to be its 2-dimens io nal Hausdo rff measur e, although the reader could pro bably arrive to the same conclusio ns us ing any a lternativ e co ncept o f area he is keen on. Note that this b ound M = 1 is b est pos sible, since it equals the 2-dimensio na l Hausdorff measure o f the unit disc. T o prov e the claimed bo und for A ( P ), w e subdivide S 1 , and σ , in to four equal arcs X 1 , Y 1 , X 2 , Y 2 of length π / 2 ea c h, trav ers ed by σ in that or der. Note that d ( x, X 1 ) ≥ √ 2 holds for every x ∈ X 2 , and similarly for Y 1 , Y 2 . (6.1) 14 W e will use this observ ation to prove that A ( P ) is a t least half the area of a square with side length √ 2. F or this, define the mapping f : P → [0 , √ 2] 2 by x 7→ ( ⌈ d ( x, X 1 ) ⌉ , ⌈ d ( x, Y 1 ) ⌉ ) where ⌈ d ⌉ := max { d, √ 2 } . W e ma y assume that the domain of ρ is also the square [0 , √ 2] 2 rather than the sta ndard 2-simplex. Now c onsider the function f ◦ ρ : [0 , √ 2] 2 → [0 , √ 2] 2 , which is contin uous since b oth f and ρ are. Note that the restrictio n of f ◦ ρ to the bo unda ry of the squar e [0 , √ 2] 2 is, b y (6.1 ), a homeomor phism from that b oundary o nto itself. F rom this we will infer tha t f ◦ ρ is onto. (6.2) There are p erhaps many w ays to prov e this basic fact, and the reader mig h t hav e a fa vourite o ne dep ending on their background. Here we s k etch a proo f using homo lo gy: suppose, to the contrary , that some p oin t z ∈ [0 , √ 2] 2 is not in the image I of f ◦ ρ , and let Q := [0 , √ 2] 2 \{ z } . Note tha t Q is homo top y equiv alent to S 1 , and so H 1 ( Q ) is isomorphic to H 1 ( S 1 ) [21, Co r ollary 2.11]. But f ◦ ρ is a 2-s implex of Q proving that its b oundary is n ull-homologo us, and so H 1 ( Q ) is triv ia l by the remar k preceding (6.2). This co n tradiction esta blishes (6.2). Note that f must thu s also b e onto. Now supp ose that A ( P ) < 1, which mea ns that for every δ ther e is a co un t- able cov er ( U i ) i ∈ N of P with diam ( U i ) < δ and P diam ( U i ) 2 < 1. Letting V i := f ( U i ) we obta in a cov er ( V i ) i ∈ N of [0 , √ 2] 2 since f is on to . Mo reo ver, by the definition of f and the triang le inequa lit y we have diam ( V i ) ≤ √ 2 diam ( U i ). Thu s P diam ( V i ) 2 ≤ 2 P diam ( U i ) 2 < 2 by the a bov e assumption. This means that the ar ea of [0 , √ 2] 2 is less than 2, a c on tradic tio n. This completes the pro of that any area extension of S 1 in which σ is null- homologo us ha s an exc e ss area o f at least 1, implying that ˆ H d ( S 1 ) ∼ = H 1 ( S 1 ). Using the same ar gumen ts one can g eneralise this to the following. Corollary 6.2. L et G b e a lo c al ly finite 1-c omplex. The n ˆ H d ( G ) ∼ = H 1 ( G ) . The following importa n t example sho ws that The o rem 3.4 would b ecome false if w e banned metric cy linders from the definition of an area extension. Moreov er, it s ho ws tha t Theorem 3.4 fails if we repla c e ˆ H d by the firs t sing ular homology gr oup H 1 ( X ) even if H 1 ( X ) is finitely generated. This example could also contribute to a b etter under standing of Se c tio n 10, the geo metric pa rt of the pro of of our main r e sult. Example 6. 3 . W e will define our spa ce X as a subspace of R 3 with the Eu- clidean metric. It is s imila r to a w ell- kno wn construction of [2] ca lled the har- monic ar chi p elago . The shap e of X is r eminiscen t of the sha p e of an old- fashioned folding camera: for e very even i ∈ N let D i be the cir c le { ( x, y , z ) ∈ R 3 | y = 2 − i , x 2 + z 2 = 1 } a nd for every odd i ∈ N let D i be the circle { ( x, y , z ) ∈ R 3 | y = 2 − i , x 2 + z 2 = 1 / 2 + 2 − i } . Moreov er, for every i ∈ N let X i be the closed cylinder in R 3 with bounda ry D i ∪ D i +1 that ha s minimum area among all such cy linders. Let X b e the closur e of S X i in R 3 , that is, X is the union of S X i and the cy linder { ( x, y , z ) ∈ R 3 | y = 0 , 1 / 2 ≤ x 2 + z 2 ≤ 1 } . F or every i ∈ N let σ i be a circlex that travels o nce around D i . Note that σ i is homotopic to σ j for every i, j . How ever, no σ i is homologo us to a circlex τ that trav els once ar o und the cir cle D := { ( x, y , z ) ∈ R 3 | y = 0 , x 2 + z 2 = 1 / 2 } ⊂ X , 15 bec ause no 2-s implex can meet infinitely many X i . Moreov er , the t wo homology classes corre sponding to τ and the σ i cannot b e made equiv alent by glueing discs of a rbitrarily small a rea to X without distorting its metric . Thus, if we mo dified the definition of area extension to only allow discs as comp onen ts of X ′ \ ι ( X ), then Theorem 3.4 would fail for C := J 1 σ 1 K , as C ha s repr e sen tatives with leng th arbitrar ily close to π = ℓ ( τ ), namely , the σ i , but no represe ntative of length π or less. This example also shows that we canno t r eplace ˆ H d ( X ) b y H 1 ( X ) (a nd ‘ σ -representativ e’ by ‘representativ e’) in the assertion of Theo rem 3.4 ev en if H 1 ( X ) is finitely g enerated. Indeed, H 1 ( X ) is generated by 2 elemen ts here, namely [ σ 1 ] and [ τ ], and [ σ 1 ] has no re presen tative of minimum length. If X is compa ct then in ma n y ca ses we do not g ain anything w hen we take the completion ˆ H d ( X ) of H d ( X ). F o r ex ample, if X is the space o f Fig ure 4 then H d ( X ) is already complete as the in tere s ted rea der can chec k. There are how ever compact ex a mples X wher e H d ( X ) is not complete: Example 6.4. Le t X b e a metr ic space obtained as follows. Start with a top ologist’s sine curve S , pick a coun tably infinite ‘cofinal’ sequence ( u i ) i ∈ N of po in ts of S , and attach a circle of length 2 − i at each point u i . T o s ee that H d ( X ) is not complete, let σ i be a circlex cor responding to the cir cle attached at u i , and note that ( J σ i K ) i ∈ N is a Cauch y sequence that has p o sitiv e distance from each element c of H 1 ( X ). Indeed, any such c must miss some circ le, and Theorem 6.1 yields a lo wer bo und for that distance. F or C ′ ∈ H d the element C of ˆ H d corres p onding to C ′ satisfies ℓ ( C ) ≤ inf { ℓ ( β ) | β ∈ C ′ } b y the definitions. The aim of our next example is to show that this inequa lit y can b e prop er. This means that (the first sentence of ) the assertion o f Theorem 3 .4, applied to a C ∈ H d , is in fact stro nger than that of Theorem 1.4. Example 6. 5 . C o nsider the compact space X ⊆ R 2 depicted in Fig ure 4. It is easy to construct a closed 1-simplex σ : [0 , 1] → X tha t trav e r ses each of the infinitely many circles in this spa ce precisely o nce. Let β ∈ H 1 ( X ) denote the homology cla ss of the 1-cycle 1 σ , and note that for every 1-cycle χ ∈ β there holds ℓ ( χ ) = ∞ b ecause o f the perp endicular segments. It is no t hard to see that for C ′ := J β K ∈ H d ( X ) w e hav e inf { ℓ ( β ) | β ∈ C ′ } = ∞ . Now le t τ i be a circlex that travels once ar ound the circle of length 2 − i in X , and let ψ i denote the 1-chain P j ≤ i τ j . By Lemma 4 .1 we can, for every i , ‘patc h’ a ll circles of X of leng th le s s than 2 − i to obtain an area extension X i of X o f some excess area v ( i ) < ∞ in which the 1-cyc les 1 σ and ψ i are homolog ous. Note that lim i v ( i ) = 0, thu s ( J ψ i K ) i ∈ N is a Cauc hy sequence equiv alent to the constant sequence ( C ′ ) i ∈ N , which means tha t (1 τ i ) i ∈ N is a σ -representativ e of the class C ∈ ˆ H d ( X ) containing these se q uences. Th us ℓ ( C ) ≤ P i ℓ ( τ i ) = 1. Finally , it is worth mentioning that we ca nnot re la x the asser tion o f Theo- rem 3.4 to requir e that X is just complete r ather than compact. F or example, the cylinder { ( x, y , z ) ∈ R 3 | z ≥ 1 , x 2 + y 2 = 1 + 1 / z } with the Euclidean metric is co mplete, but it is ea s y to see that no non-tr ivial element of ˆ H d has a σ -representativ e of minimu m leng th. 16 1 2 1 2 1 2 1 3 1 4 1 5 1 4 1 4 1 8 1 8 1 16 1 16 Figure 4 : A compact subspace of th e real plane. The num b ers denote the length s of the corresp onding segments. 7 Sk etc h of the main pro of The proof of our main result, Theore m 3.4, consis ts of t wo ma jor steps: the first step is algebra ic, and shows that every C ∈ ˆ H d can be ‘decomp osed’ as a sum P D i of simpler elemen ts of ˆ H d , ca lled primitive e le men ts, that are eas ier to work with. The second step is more geometric, and pr o ves the as sertion for these primitive ele ments. Our intuition b ehind a primitive element is that it is a homology class cor - resp onding to a single circle, and indeed we will prov e , in Section 10 , that every primitive element D ha s a repres e ntative consisting of a cir clex z , and in fact one of the desired length ℓ ( z ) = ℓ ( D ). W e obtain z by a g eometric co nstruction: starting from a se q uence of clo sed 1-simplices σ i representing D whose lengths conv erg e to ℓ ( D ), we exploit the co mpactness of our space to find a subsequence that conv er ges po in twise to the desired 1-s implex z , and show that J z K = D by constructing arbitrarily small metric cylinder s jo ining z to some σ i . See also Example 6.3, wher e we co uld choose τ to be the des ir ed circlex z . Now having a decomp osition C = P D i as above, we can tr y to co m bine a ll the cir clexes z i we go t as representativ es of ea c h D i to form a σ -representativ e of C . But will such a σ -representativ e hav e the desire d total length P ℓ ( z i ) = ℓ ( C )? In genera l not, if our decompo s ition is arbitrary . F or example, in the graph o f Figure 5 consider the class C = J σ + τ K . W e could write C = D 1 + D 2 where D 1 = J σ K and D 2 = J τ K are b oth primitiv e. Now σ, τ are circlex e s that do attain the length of D 1 , D 2 resp ectiv ely , but we cannot combine them into a representative of C of minimum length, b ecause ℓ ( σ ) + ℓ ( τ ) > ℓ ( C ); indeed, C has the repr esen tative ρ whose length is smaller than ℓ ( σ ) + ℓ ( τ ) bec ause it av oids the middle edge. This exa mple shows that if we wan t to follow the ab o ve plan of fir st decomp o sing C as a sum of pr imitive element s and then combine shortest representativ e s of those elements into a σ -representativ e of C of the desired total length ℓ ( C ), then our decompositio n ha s to b e ‘economical’. If our space is a gra ph then it is easy to say what ‘econo mical’ should mean: no edge s hould b e us e d in more than one summands. In a general spac e this is less obvious, but there is an elega n t wa y a r ound it des c ribed in Section 9 .2. W e will prove, in Section 9, that every C ∈ ˆ H d can b e dec omposed as a sum P D i where, not only the D i are primitive, but also the decomp osition is economical 17 in this sense. This pro of is algebra ic, and we obtain a more genera l abs tr action describ ed in the next section. Figure 5: A simple ex ample showing t hat we n eed our primitive decomp ositions to b e economical. 8 In termezzo: general ising to ab elian metriz- able top ological groups In this section we state an intermediate result, mentioned a lso in Section 1.4, that might be useful in other co n texts to o. It says that if a to pologica l gro up H and an a ssignmen t ℓ : H → R + (whic h can b e thought of as a n ass ignmen t of lengths) satisfy cer ta in axioms, then ev e r y element of H can b e written as a sum of primitive e le men ts, which we define b elo w, and this sum is in a sens e ‘economical’ (recall the dis cussion in the previous sectio n). The reade r will lose nothing b y assuming that H = ˆ H d ( X ) throughout this section. Given tw o elements C, D of H we will write C  D if ℓ ( C ) = ℓ ( D ) + ℓ ( C − D ). Note that if D  C then C − D  C , (8.1) since ℓ ( C − ( C − D )) = ℓ ( D ). W e will say that an element C o f H is primitive if C 6 = 0 and there is no D ∈ H \{ C, 0 } s uch that D  C holds. The reader may c ho ose to skip the rest of this s ection, since this is a co rollary of our main r esult rather than something that we will need later. Theorem 8. 1 . L et ( H , +) b e an ab elian metrizable top olo gic al gr oup, and su p- p ose a function ℓ : H → R + is given satisfying the fol lowing pr op erties: (i) ℓ ( C ) = 0 if and only if C = 0 ; (ii) ℓ ( C + D ) ≤ ℓ ( C ) + ℓ ( D ) for every C, D ∈ H ; (iii) if D = lim C i then ℓ ( D ) ≤ lim inf ℓ ( C i ) ; (iv) for some metric d of H t her e is a b ound U ∈ R such that d ( C , 0 ) ≤ U ℓ 2 ( C ) for every C ∈ H (i.e. an isop erimetric ine quality holds). Then every element C of H c an b e r epr esente d as a (p ossibly infinite) sum C = P D i of primitive elements D i so that ℓ ( C ) = P ℓ ( D i ) . 18 Infinite sums as in the conclusion of the theorem ar e formalised, in Sec- tion 9.1, using the concept of nets. Since it is the gro up ˆ H d we are in terested in in this paper , we will give a formal pro of of Theorem 8.1 only for the sp ecial case when H = ˆ H d (more precisely , when H is the s ubgroup of elements of ˆ H d with finite length). In this case Theorem 8.1 is ta ntamount to Co rollary 9.7 b elo w. Ho wever, the r eader int e r ested in Theorem 8.1 in its full gener alit y will easily b e able to chec k that the same pr oof applies, since no other prop erties of ˆ H d are used in the pro of o f Corollar y 9 .7 than the conditions of Theorem 8.1 . One can relax co ndition (iv) a bov e a bit b y re placing it with the following ( iv ′ ) if C ∈ H is fragmentable then C = 0 . The term fr agmentable is defined in Section 9.3 b elow. 9 Splitting homology classes in to primitiv e sub- classes The main result of this sectio n, Cor o llary 9.7 , is that every C ∈ ˆ H d can be written as a sum of pr imitive elemen ts D i  C . This is the first s tep of the pro of o f our main result as sketc hed in Section 7. Recall the definitions of primitive a nd  from Section 8. 9.1 Infinite sums in ˆ H d F or the pro of of o ur main r esult w e a re going to use some s tandard ma c hinery related to nets in o rder to be able to rigoro usly define sums of infinitely man y elements of ˆ H d . Let us first r e call the ne c essary definitions . A net in a top ological space Y is a function from some directed set A to Y . A dir e cte d set is a nonempt y set A together with a reflexive and transitive binary r elation, that is, a pr eorder, with the additional pr operty that every pa ir of elements ha s an upp er b ound in A . One can think of a net as a g e ne r alisation of the concept of a s e quence, a nd one is usually interested in the conv erg ence of such a gener alised sequence: w e say that the net ( x α ) c onver ges to the po in t y ∈ Y , if for every neighborho o d U o f y there is a β ∈ A such that x α ∈ U for every α ≥ β . See [26] for more deta ils. In o ur cas e, the to pologic a l space Y in which our nets will tak e their v a lues will alw ays b e ˆ H d , bearing the topolog y induced by the metr ic d 1 . W e will say that an infinite fa mily { C i } i ∈ I of elemen ts of ˆ H d is unc ondi- tional ly summable if for every ǫ > 0 there is a finite subset F of I so that for every t wo finite s ets A, B ⊇ F there holds d ( P i ∈ A C i , P i ∈ B C i ) < ǫ ; in other words, if the family { P i ∈ F C i } F ∈F is a Cauchy net, wher e F is the set of finite subsets of I pre ordered b y the inclusion relation. Since ˆ H d is complete, it is well known that if { C i } i ∈ I is unconditionally summable then the net { P i ∈ F C i } F ∈F conv erg es to an element C ∈ ˆ H d , see [24, Prop osition 2 .1.49]. In this case , we call C the sum of the family { C i } i ∈ I and write C = P i ∈I C i . Note that if I is countable then for ev er y enumeration a 1 , a 2 , . . . of I ther e ho lds P i ∈I C i = lim i P 1 ≤ j ≤ i C a j . (9.1) 19 Our next le mma generalise s the triangle inequa lit y for ˆ H d (Corollar y 5.5) to infinite sums using the notions we just defined. Lemma 9.1. L et { C i } i ∈ I b e an un c ondi tional ly su mmable family of elements of ˆ H d . Then ℓ ( P i ∈I C i ) ≤ P i ∈I ℓ ( C i ) . Pr o of. If P i ∈I ℓ ( C i ) = ∞ then there is nothing to show, s o supp ose that P i ∈I ℓ ( C i ) < ∞ . W e may assume w itho ut loss of g e ne r alit y tha t ℓ ( C i ) > 0 holds fo r every i ∈ I , for if ℓ ( C i ) = 0 then C i = 0 by Observ atio n 5.3 . Thus, w e may also assume that I is countable, and let a 1 , a 2 , . . . be an en umeration of I . Let C := P i ∈I C i and let D i := P 1 ≤ j ≤ i C a j for every i . W e have C = lim i D i by (9.1 ). By the definition of ℓ ( C ) we then hav e ℓ ( C ) ≤ lim ℓ ( D i ). (9.2) Applying Corollary 5.5 (s e v era l times) to D i we obtain ℓ ( D i ) ≤ P 1 ≤ j ≤ i ℓ ( C a j ) < P i ∈I ℓ ( C i ). Combining this with (9.2) yields ℓ ( C ) ≤ P i ∈I ℓ ( C i ) as desire d. 9.2 Splitting homology classes in to shorter ones W e intro duce the nota tion C = D L E to denote the asser tion that C = D + E and ℓ ( C ) = ℓ ( D ) + ℓ ( E ). Note that this definition implemen ts the intuition outlined in Sectio n 7 that D + E is an ‘economica l’ wa y to split C . It fo llows by the definitions that D  C if a nd only if C = D L ( C − D ). (9.3) More generally , the notation C = L i ∈K D i (or D 1 L . . . L D k ), where K is a p ossibly infinite set o f indices, denotes the assertion that C = P i ∈K D i and ℓ ( C ) = P i ∈K ℓ ( D i ). Our next lemma shows that, in a sense, L behaves w ell with r espect to comp osition: Lemma 9.2. L et C, D , E , F , G ∈ ˆ H d b e such t ha t C = D L E and E = F L G . Then the fol lowing assertions hold: (i) C = D L F L G ; (ii) C = ( D + F ) L G , and (iii) D + F = D L F . In p articular, F, G, ( D + F )  C . Pr o of. By the assumptions we hav e ℓ ( C ) = ℓ ( D ) + ℓ ( E ) = ℓ ( D ) + ℓ ( F ) + ℓ ( G ), which yields (i). Note that C = D + F + G . By Cor ollary 5.5 w e hav e ℓ ( C ) = ℓ (( D + F ) + G ) ≤ ℓ ( D + F ) + ℓ ( G ), and ℓ ( D + F ) ≤ ℓ ( D ) + ℓ ( F ). No w since we hav e a lr eady prov ed that ℓ ( C ) = ℓ ( D ) + ℓ ( F ) + ℓ ( G ), equality must hold in b oth ab o ve inequalities. The first of these eq ua lities yields (ii) and the s e c ond yields (iii). This nice b eha viour of L extends to infinite sums to o: 20 Lemma 9. 3. If ℓ ( C ) < ∞ and C = L i ∈K D i then for every subset M ⊆ K ther e holds P i ∈M D i = L i ∈M D i and C = P i ∈M D i L P i ∈ M D i , wher e M := K\ M . Pr o of. By Corollary 5.5 we hav e ℓ ( C ) ≤ ℓ ( P i ∈M D i ) + ℓ ( P i ∈ M D i ) and by Lemma 9.1 we hav e ℓ ( P i ∈ M D i ) ≤ P i ∈ M ℓ ( D i ). Combining the last tw o in- equalities we obtain ℓ ( X i ∈M D i ) ≥ ℓ ( C ) − ℓ ( X i ∈ M D i ) = P i ∈K ℓ ( D i ) − ℓ ( X i ∈ M D i ) ≥ P i ∈K ℓ ( D i ) − X i ∈ M ℓ ( D i ) = X i ∈M ℓ ( D i ) , where we used our as sumption that P i ∈K ℓ ( D i ) = ℓ ( C ) < ∞ . Applying Lemma 9.1 aga in we als o hav e ℓ ( X i ∈M D i ) ≤ X i ∈M ℓ ( D i ) , hence equality holds in the las t tw o inequalities, which prov es that X i ∈M D i = M i ∈M D i . Similarly , we hav e P i ∈ M D i = L i ∈ M D i . The as s ertion C = X i ∈M D i M X i ∈ M D i now easily follows fr om the definitions. 9.3 Exploiting the isoperimetric inequalit y W e will say that an element C ∈ ˆ H d is δ -fr agmentable , for some δ ∈ R + , if there is a finite family { D i } i ∈K , D i ∈ ˆ H d , such that C = L i ∈K D i and for every i there holds ℓ ( D i ) < δ . W e will call C fr agmentable if it is δ -frag men table fo r arbitrar ily small δ . It turns out that the only frag men table element of ˆ H d is 0 : Lemma 9.4. If C ∈ ˆ H d is fr agmentable then C = 0 . Pr o of. Supp ose C is fragmentable, and fix some ǫ > 0 for which we want to show that d 1 ( C, 0 ) < ǫ . Let { D i } i ∈K be a family witnessing the fact that C is δ -fragmentable for some par ameter δ tha t we w ill spe c ify la ter. F or every i ∈ N , we can, by the definition of ℓ ( D i ), find elements of H 1 arbitrar ily close (with resp ect to d 1 ) to D i whose lengths ar e ar bitrarily close to ℓ ( D i ); more forma lly , it follows by the definitions that there is a class α i ∈ H 1 with ℓ ( α i ) < ℓ ( D i ) + min( δ, ǫ/ 2 |K| ) such that d 1 ([ α i ] , D i ) < ǫ/ 2 |K| . (9.4) By the definition of ℓ ( α i ) there is an 1- c hain χ i ∈ α i such that ℓ ( χ i ) < ℓ ( α i ) + min( δ, ǫ/ 2 |K| ). Com bining this with our a ssumption that ℓ ( D i ) < δ and the choice o f α i we obtain ℓ ( χ i ) < ℓ ( α i ) + δ < ℓ ( D i ) + 2 δ < 3 δ. (9.5) 21 By Lemma 4.1 there is an extension X i of X of exc e s s area at most U ℓ 2 ( χ i ) in whic h χ i is nu ll-homolog o us. Combining these extensions X i for ev ery i we obtain an extens io n X ′ of X o f excess area V at most U P i ∈K ℓ 2 ( χ i ) in which the 1-chain P i ∈K χ i is null-homologous. Note that b y the choice of the χ i and the α i we have X ℓ ( χ i ) < X ( ℓ ( α i )+ ǫ/ 2 |K | ) < X ( ℓ ( D i )+ ǫ/ 2 |K | + ǫ/ 2 |K| ) = ( X ℓ ( D i ))+ ǫ = ℓ ( C )+ ǫ, where we used o ur assumption that ℓ ( C ) = P ℓ ( D i ). This means that P ℓ ( χ i ) is b ounded from ab o ve; th us by (4.2) a nd (9.5) ch o osing δ small enough we can ma k e V arbitra rily small; in particular , we co uld have chosen a δ for which V < ǫ/ 2 holds, which w ould imply d 1 ( J P χ i K , 0 ) < ǫ/ 2. (9.6) Since C = P D i we ea s ily obtain by C o rollary 5.5 and (9 .4 ) d 1 ( C, J X i ∈K χ i K ) = d 1 ( X i ∈K D i , J X i ∈K χ i K ) ≤ X i ∈K d 1 ( D i , J χ i K ) ≤ X i ∈K ǫ/ 2 |K | = ǫ/ 2 , and co m bined with (9.6 ) this yie lds d 1 ( C, 0 ) < ǫ , a nd proves that C = 0 in this case. 9.4 A techn ical lemma The following somewhat technical lemma will b e us ed in the pro of of the main result of this section; it allows us to prov e , using Zor n’s Lemma, the existence of maximal families with certain prop erties. Lemma 9.5. L et { D α } α<γ b e a family of elements D α of ˆ H d \{ 0 } , indexe d by an or dinal γ , such that fo r every β < γ ther e holds P α ≤ β D α = L α ≤ β D α and P α ≤ β D α  C for some fixe d C ∈ ˆ H d with ℓ ( C ) < ∞ . Then ( D α ) α<γ is u nc onditionally summable and ther e ho lds P α<γ D α  C and P α<γ D α = L α<γ D α . Pr o of. Since D α 6 = 0 , Observ ation 5.3 implies that ℓ ( D α ) > 0 for every α < γ . As we ar e ass uming tha t ℓ ( P α ≤ β D α ) = P α ≤ β ℓ ( D α ) and that ℓ ( P α ≤ β D α ) ≤ ℓ ( C ) for every β ≤ γ , we hav e P α ≤ β ℓ ( D α ) ≤ ℓ ( C ) < ∞ for every β < γ , w hich implies that γ is countable and P α<γ ℓ ( D α ) ≤ ℓ ( C ) < ∞ . (9.7) Let a 1 , a 2 , . . . be a n enumeration of γ . T o see that ( D α ) α<γ is unconditiona lly summable, note that for e very ǫ > 0 there is an n ∈ N such tha t fo r every k > n there holds ℓ ( P n ℓ ( C ) − λ , which implies P i ∈I \I ′ ℓ ( D ′ i ) < λ , a nd hence ℓ ( P i ∈I \I ′ D ′ i ) < λ by Lemma 9.1. But then, extending { D ′ i } i ∈I ′ by o ne member, namely P i ∈I \I ′ D ′ i , we obtain a finite family { D ′ i } i ∈I ′′ which satisfies ℓ ( D ′ i ) < λ for every i ∈ I ′′ , and it is not hard to see that C = L i ∈I ′′ D ′ i holds; this con tra dic ts the fact that C is not λ -fragmentable. Now let { D α } α<γ , D α ∈ ˆ H d \ 0 be an unconditionally s ummable family , indexed by an ordina l nu m ber γ , that is maximal with the following prop erties: (i) P α<β D α  C for every β ≤ γ ; 23 (ii) P α<β D α = L α<β D α for every β ≤ γ , and (iii) P α<γ ℓ ( D α ) ≤ ℓ ( C ) − λ . T o see that a maximal s uc h family exists, apply Zorn’s Lemma on the set of a ll such families ordered b y the subfamily relation, using Lemma 9.5 in order to show that every chain has an upper bound. W e are not yet assuming that this maximal family is no n-trivial. Let D := P α<γ D α , a nd note that D  C by (i). It is not hard to see that either D or C − D (o r b oth) is still no t λ -fragmentable, for if both split into families with element s o f lengths less that λ , then so do es C ; more formally , suppose there are finite families { D i } i ∈M and { D i } i ∈N such that D = L i ∈M D i , C − D = L i ∈N D i , and ℓ ( D i ) < λ for ev er y i ∈ M ∪ N . W e claim that C = L i ∈M∪N D i . Easily , C = P i ∈M∪N D i . T o see that ℓ ( C ) = P i ∈M∪N ℓ ( D i ), recall that ℓ ( C ) = ℓ ( D ) + ℓ ( C − D ) by (i), that ℓ ( D ) = P i ∈M ℓ ( D i ), and that ℓ ( C − D ) = P i ∈N ℓ ( D i ). This pr o ves that either D or C − D is not λ -fragmentable. But if C − D is not λ -fragmentable, then it is primitive: for if there is an F 0 6 = 0 with F 0  C − D and F 0 6 = C − D , then either F 0 or F 1 := C − D − F 0 has length at least λ since C − D is not λ -fragmentable and, by (9.3 ) , C − D = F 0 L F 1 . Assume without loss of genera lit y tha t ℓ ( F 1 ) ≥ λ ; we can now enlarge the family { D α } α<γ by o ne member, namely F 0 , to obtain a new family { D α } α<γ + that contradicts the maximality of { D i } i ∈ I : to prov e that { D α } α<γ + also satisfies requir emen t (i) it suffices to chec k that P α<γ + D α  C . W e ha ve P α<γ + D α = D + F 0 by construction, and by asser tio n (ii) of Lemma 9.2 we obtain D + F 0  C , (9.12) which prov es that { D α } α<γ + satisfies (i). T o prov e tha t { D α } α<γ + also s atisfies requirement (ii), it suffices a gain to consider the case β = γ + ; in other words, to pr o ve that D + F 0 = L α<γ + D α . Th us we have to pr o ve tha t ℓ ( D + F 0 ) = P α<γ + ℓ ( D α ) = ℓ ( D ) + ℓ ( F 0 ), where for the last equality we used the fa ct that (ii) holds for β = γ and D γ = F 0 . But this follows from assertion (iii) of Lemma 9.2, and so { D α } α<γ + also satisfies (ii). Finally , to see that { D α } α<γ + satisfies (iii), note that P α ∈ γ + D α = D + F 0 , that ℓ ( C ) = ℓ ( D + F 0 ) + ℓ ( F 1 ) b y (9.12), and that ℓ ( F 1 ) ≥ λ . This completes the pro of that if C − D is not λ - fragmentable then C − D is pr imitiv e, for other wise the maximality o f { D α } α<γ is contradicted. Thu s, if C − D is not λ -fra gmen table then we are do ne , since D  C and so we a lso ha ve C − D  C by (8.1 ). So supp ose it is not, in which case it is D that is not λ -fra gmen table. Recall that ℓ ( D ) ≤ ℓ ( C ) − λ by (iii). T o sum up, having assumed that C is not λ -frag men table, we pr o ved that either there is a primitive B  C , in which case we are done, or ther e is a D  C that is a lso no t λ -fragmentable (for the sa me λ ) and satisfies ℓ ( D ) ≤ ℓ ( C ) − λ . In the latter c a se, we ca n rep eat the whole argument replacing C with C 1 := D ; this will ag ain yield either a primitive B  C 1 , or a C 2  C 1 that is als o no t λ -fragmentable a nd satisfies ℓ ( C 2 ) ≤ ℓ ( C 1 ) − λ ≤ ℓ ( C ) − 2 λ , and so on. But as ℓ ( C ) is finite and λ po sitiv e, this pro cedure m ust stop after finitely many steps, yielding a primitive B  C j  C j − 1 . . .  C . As  is tra nsitiv e (Lemma 9.2) we obtain B  C . This co mpletes the pr o of. 24 W e can now state and pr ove the main result of this section. Corollary 9.7. F or every C 6 = 0 ∈ ˆ H d ( X ) with ℓ ( C ) < ∞ ther e is a family { D i } i ∈ I of primitive elements of ˆ H d ( X ) such t ha t C = L i ∈ I D i . Pr o of. Using Zo rn’s Lemma we find a maxima l family { D α } α<γ of primitive D α ∈ ˆ H d such that (i) P α<β D α  C for every β ≤ γ ; a nd (ii) P α<β D α = L α<β D α for every β ≤ γ . Indeed, co nsider the set of all such families o rdered by the subfamily r e lation, and apply Lemma 9 .5 in o rder to show that every chain ha s an upper bo und. Let D := L i ∈ I D i . W e claim that C − D = 0. F or supp ose not. Then by Lemma 9.6 there is a primitive F  C − D . Now extend the family { D α } α<γ by one member D γ := F . By (ii) of Lemma 9.2 the new family still sa tisfies (i). T o prov e that it also satis fies (ii) we o nly have to show that ℓ ( P α ≤ γ D α ) = P α ≤ γ ℓ ( D α ) = ℓ ( D ) + ℓ ( F ) (where w e used the fact that the original family satisfies (ii)), but this follows from (iii) of Lemma 9.2 . Thus the extended family c on tradicts the maximalit y of { D α } α<γ , which prov es tha t C − D = 0 and establishes our a ssertion. 10 Pro of for primitiv e elemen ts By Cor ollary 9.7 every non-trivia l element C of ˆ H d can b e written as a sum of primitive elements D i so that ℓ ( C ) = P ℓ ( D i ). All that r emains to s how is that our main theorem holds for those elements D i : Lemma 1 0.1. If D ∈ ˆ H d is primitive then t her e is a cir clex z such that D = J z K and ℓ ( z ) = ℓ ( D ) . Pr o of. W e are go ing to obtain the desired clos ed simplex z as a limit, in a sense, of a sequence o f clo sed simplices σ 1 i related to D . Our proo f is organis e d in three steps. In the first s tep we construct this sequence ( σ 1 i ). In the second step we construc t z and, at the same time, homoto pies betw ee n z and the σ 1 i in a ppropriate area extensio ns of X , implying that J z K = lim J sig 1 i K . Finally , in a third step we show that D = J z K and that ℓ ( z ) = ℓ ( D ). W e then remar k that the closed simplex z we co nstructed must indeed be a circlex. Step I: t he sequen ce ( σ 1 i ) By Co rollary 5.2 there is a se q uence of 1-cycles ( χ i ) i ∈ N such that ( J χ i K ) i ∈ N is a Cauch y sequence in D and ℓ ( D ) = lim i ℓ ( χ i ). By conca tenating some of the simplices in χ i if necessary , we may assume without loss of g eneralit y that every simplex in χ i is c losed. F or every i enu- merate the (close d) simplices in χ i as σ 1 i , . . . , σ k i i in such a way that ℓ ( σ j i ) ≥ ℓ ( σ m i ) if j < m . (10.1) F or co n venience, if m > k i then we let σ m i denote a triv ia l 1-simplex in X (thus ℓ ( σ m i ) = 0 for m > k i ). 25 Let M ⊆ N be the set of sup erscripts m such that ( σ m i ) i ∈ N has no infinite subsequence ( σ m α i ) i ∈ N such that lim i ℓ ( σ m α i ) = 0. Note that, b y (10.1), if m ∈ M then { 1 , . . . , m − 1 } ⊂ M . (10.2) W e beg in with a simple a nd instructive fact indicating the significance of M : Claim. if M = ∅ then D = 0 . Indeed, if M = ∅ then there is an infinite subsequence ( σ α i ) i ∈ N of ( σ i ) i ∈ N such that l im i ℓ ( σ 1 α i ) = 0. W e will show that for every ǫ > 0 there holds d 1 ( D , 0 ) < ǫ . F or this, pick j = α k ∈ N large enoug h that (i) d 1 ( D , J χ j K ) < ǫ/ 2 ; (ii) ℓ ( χ j ) < ℓ ( D ) + ǫ , and (iii) ℓ ( σ 1 j ) < ℓ ( D ) λ , where λ = λ ( ǫ ) ∈ R + is some para meter that we will choo se later. By (10.1) we hav e ℓ ( σ m j ) < ℓ ( D ) λ for every m ∈ N . By Lemma 4.1 there is fo r every m an area extension X m of X of excess area at most U ℓ 2 ( σ m j ) in which σ m j is nu ll-homologo us. Combining all these a rea extensions we obtain a single area extension X ǫ of X of excess area at most v := P m ∈ N U ℓ 2 ( σ m j ) in which χ j is nu ll-homologo us. This means tha t d 1 ( J χ j K , 0 ) ≤ v . (10.3) Since P m ∈ N ℓ ( σ m j ) = ℓ ( χ j ) < ℓ ( D ) + ǫ , given ℓ ( D ) and ǫ we can, b y (4.2) and (iii), ch o ose λ small enough that v < ǫ/ 2. As d 1 ( D , 0 ) ≤ d 1 ( D , J χ j K ) + d 1 ( J χ j K , 0 ) < ǫ/ 2 + v b y (i) a nd (10.3), a nd ǫ was chosen arbitra rily , we hav e prov ed the Claim. As w e are assuming that D is primitive, the Claim proves that M 6 = ∅ , and th us 1 ∈ M by (10.2). W e ma y assume without loss of gener alit y that σ 1 i has constant sp eed for every i . (10.4) W e a r e going to construct z as a ‘limit’ o f the σ 1 i (it will turn out that M = { 1 } ). F or this, let ( χ a i ) i ∈ N be a subsequence of ( χ i ) i ∈ N such that lim i ℓ ( σ 1 a i ) =: r exists. Note that we hav e alrea dy prov e d that r > 0. Moreover, r < ∞ holds since C is primitiv e and thus, eas ily , ℓ ( C ) < ∞ . It is not har d to s ee that ther e is a subseq uence ( σ 1 b i ) i ∈ N of ( σ 1 a i ) i ∈ N such that the re s trictions σ 1 b i ↾ Q conv erg e p oin t wise. (10.5) Indeed, let q 1 , q 2 , . . . b e an en umera tion of Q . Find a subsequence ( τ 0 i ) i ∈ N of ( σ 1 a i ) i ∈ N such that the p oint s τ 0 i ( q 1 ) conv erg e. Then find a subseq uence ( τ 1 i ) i ∈ N of ( τ 0 i ) i ∈ N such tha t the p oints τ 1 i ( q 2 ) also conv erg e , and so on. Now letting σ 1 b i = τ i i satisfies 10 .5. (W e could have chosen any dense c o un table subs e t of [0 , 1] instead of Q .) 26 Step I I: Constr uct ion of z and h By (10.4 ) and (1 0.5) it follo ws ea sily that for e very δ there is a n n ∈ N such that σ 1 b i and σ 1 b j are δ -close for every i, j ≥ n . (10.6) Using Lemma 4.3 and (10.6) we can now cons truct a subsequence ( σ 1 c i ) i ∈ N of ( σ 1 b i ) i ∈ N such that for every i there is an ar ea extension X ′ i of X o f excess ar e a at most 2 − i in which σ 1 c i and σ 1 c i +1 are ho mo topic: for every i = 0 , 1 , . . . , use (10 .6) to obta in a c i such that σ 1 b i and σ 1 b j are f ( r, 2 − i )-close for every i, j ≥ c i , where the function f is that of Lemma 4.3. Cho osing c i larger if needed, we may also ensure that c i > c i − 1 (where we set c − 1 := 0), and that ℓ ( σ 1 b i ) < r + 2 − i for every i ≥ c i . The n, by Lemma 4.3, there is indeed a n extension X ′ i as desired. Let h i be a homotopy fro m σ 1 c i +1 to σ 1 c i in X ′ i as supplied by Le mma 4.3. Combining all h i together w e can obtain a contin uous function h ′ : (0 , 1] × [0 , 1] → X ′ , where X ′ := S X ′ i . W e a re later g oing to “co mplete” h ′ int o a ho motop y h : [0 , 1] × [0 , 1] → X ′ such that h (0 , x ) is o ur desired s implex z . T o define h ′ , supp ose that for every i w e had chosen the domain of h i to be [2 − i , 2 − ( i +1) ] × [0 , 1]. In tuitively , the int erv al [2 − i , 2 − i +1 ] here corr e sponds to ‘time’; think of time as running in the negative directio n if y o u prefer the homotopies to b e from σ 1 c i to σ 1 c i +1 rather than the o ther way round. Now let h ′ := S h i . Let R := { 2 − i | i ∈ N } . W e c laim that h ′ ↾ ( R × [0 , 1]) is unifor mly contin uo us. (10.7) F or supp ose not. Then, there is s ome ǫ ∈ R + and an infinite sequence of pairs P i = { p i , q i } of p oin ts in R × [0 , 1] s uch that d ( h ′ ( p i ) , h ′ ( q i )) > ǫ for every i and the dista nc e b et ween p i and q i conv erg es to 0 . Note that for every s ∈ R the subspace { s } × [0 , 1] is co mpact, thu s the function h ′ ↾ ( { s } ) × [0 , 1] is uniformly contin uous b y Lemma 2.1. This mea ns tha t { s } × [0 , 1 ] cannot contain a n infinite subsequence of ( P i ) i ∈ N for an y s ∈ R . Ev en mo r e, { s } × [0 , 1] cannot meet an infinite subsequence of ( P i ) i ∈ N , b ecause the distance betw een p i and q i conv erg es to 0. It follows tha t { 0 } × [0 , 1 ] co n tains an accumulation p oin t (0 , x ) o f ( P i ) i ∈ N , i.e. a p oin t (0 , x ) every ne ig h b ourho od of which co n tains infinitely many pairs P i . Now let δ be some (small) po sitiv e rea l n umber. Pick an x ′ ∈ ( Q ∩ [0 , 1]) such that | x ′ − x | < δ / 2, and consider the op en ball O := B δ ((0 , x ′ )) in [0 , 1 ] × [0 , 1]. Let R O := O ∩ ( R × { x ′ } ). Cho osing δ small eno ugh we can make sure that for every s ∈ R O there holds ℓ ( ρ s ) < r + ǫ , (10.8) where ρ s : [0 , 1 ] → X is defined by x 7→ h ′ ( s, x ); indeed, ρ s coincides by defini- tion with so me σ 1 i , and lim i ℓ ( σ 1 i ) = r . As O ∋ x , there is an infinite subsequence o f ( P i ) i ∈ N contained in O . More- ov er, by (10.5) h ′ ( R O ) has a unique accumulation point in X . Thus we can find a pair P j = { p j , q j } in O such that if s (resp ectively , s ′ ) is the ele - men t of R for whic h p j ∈ { s } × [0 , 1] (resp., q j ∈ { s ′ } × [0 , 1]) holds, then d ( h ′ ( s, x ′ ) , h ′ ( s ′ , x ′ )) < ǫ/ 2. 27 Since ρ s coincides with some σ 1 i , it has consta n t sp eed. As || p j , ( s, x ′ ) || < 2 δ , this together with (10.8) implies d ( h ′ (( s, x ′ )) , h ′ ( p j )) < 2 δ ( r + ǫ ); similar ly , we also hav e d ( h ′ (( s ′ , x ′ )) , h ′ ( q j )) < 2 δ ( r + ǫ ). Th us, by the tria ngle inequality applied to the four points h ′ ( p j ) , h ′ (( s, x ′ )) , h ′ (( s ′ , x ′ )) and h ′ ( q j ) w e obtain d ( h ′ ( p j ) , h ′ ( q j )) ≤ 2 δ ( r + ǫ ) + ǫ/ 2 + 2 δ ( r + ǫ ) . Since ǫ and r ar e fixed and we can choos e δ freely , we can force this dista nce to be smaller than ǫ c o n tradicting the choice of the P i . This proves (10 .7). The co mpletion of R × [0 , 1] is ( R ∪ { 0 } ) × [0 , 1]; th us, b y (2.1) and (10.7), h ′ ↾ ( R × [0 , 1 ]) can b e extended into a uniformly contin uous function h ′′ : ( R ∪ { 0 } ) × [0 , 1] → X . Next, w e prov e that h := h ′ ∪ h ′′ is contin uous . (10.9) Clearly , h is contin uo us at any po in t in (0 , 1 ] × [0 , 1]. So pick x ∈ { 0 } × [0 , 1] and ǫ ∈ R + . By the contin uity of h ′′ , ther e is a basic op en neighbour hoo d O ǫ of x in R × [0 , 1] that is mapp ed by h ′′ within the ball B ǫ/ 2 ( h ( x )). Let m ǫ ∈ N be lar ge enough that h i has width le s s that ǫ/ 2 for every i ≥ m ǫ ; such an m ǫ exists by the second sentence o f Lemma 4.3 and the choice of the h i . Assume without loss of g e neralit y that O ǫ do es not meet 2 − i × [0 , 1] for i < m ǫ . Extend O ǫ int o a se t O ′ ⊆ [0 , 1] × [0 , 1] a s follows. F or every i ≥ m ǫ and every p oin t p = (2 − i , y ) ∈ O , put in to O ′ the line segment L p connecting p to the po in t (2 − ( i +1) , y ). Note that for every point y ∈ L p we have d ( h ′ ( y ) , h ′ ( p )) ≤ ǫ / 2 since h ′ coincides with h i on L p by the definition of h ′ and h i has width less that ǫ/ 2. As O ∩ ( R × [0 , 1]) is mapp ed by h ′′ within the ball B ǫ/ 2 ( h ( x )), this implies that h ( O ′ ) ⊆ B ǫ ( h ( x )). But O ′ contains by construction an op en subse t of [0 , 1] × [0 , 1] containing x . This prov es (10.9 ), whic h means tha t h is a homotopy in X ′ betw een the closed 1-simplex h (0 , x ) and σ 1 m 0 = h (1 , x ). W e now define z ( x ) := h (0 , x ), w hich is going to b e the simplex we are lo oking for. Note that for every j the restr iction h ↾ ([0 , 2 − j ] × [0 , 1]) is a homotopy betw een z and σ 1 m j in X ′ , but this homotopy do es no t use the are a extensions X ′ 1 , . . . , X ′ j − 1 . Thus, as the area extension X ′ i has b y co nstruction excess area 2 − i for every i , we obtain d 1 ( J σ 1 m j K , J z K ) ≤ 2 − ( j − 1) for every j by the definition of d 1 , since σ m j and z are homotopic in the area extension S i ≥ j X ′ i of X . This prov es that ( J σ 1 m i K ) i ∈ N is a Cauch y seq uence with limit Z := J z K . (10.10) Step I I I Our next a im is to prov e that ℓ ( z ) ≤ r . (10.11) Recall that r was defined in Step I. Supp ose, to the contrary , there is a finite se- quence S = s 1 < s 2 < . . . < s k of p oint s in [0 , 1] with P 1 ≤ i r . Clearly , we may assume that s j ∈ Q for every j . Let ǫ := r ′ − r 2 k . By 28 (10.5) and the construction of h w e obtain that lim i σ 1 β i ( s j ) = h (0 , s j ) = z ( s j ) for every j . Th us, c ho osing i 0 ∈ N large enough we can mak e sure that d ( σ 1 β i ( s j ) , z ( s j )) < ǫ for ev ery j and every i > i 0 . But then, the sequence S witnesses the fact that ℓ ( σ 1 β i ) ≥ r ′ for ev er y i > i 0 , which contradicts the choice of ( σ 1 i ) i ∈ N and proves (10 .1 1). F ro m (10.11 ) we will now e asily yie ld ℓ ( Z ) = r . (10.12) Firstly , note that b y (10.10 ) and the definition of ℓ ( Z ) w e hav e ℓ ( Z ) ≤ r b y (10.11). Suppose that ℓ ( Z ) = r ′ < r , and let ( J σ ′ i K ) i ∈ N be a Cauch y sequence in Z with lim ℓ ( σ ′ i ) = r ′ . Replacing σ 1 c i in χ c i for ev ery i b y σ ′ i we obtain a new sequence ( χ ′ i ) i ∈ N from ( χ i ) i ∈ N , and it follows easily from (10.10) that ( J χ ′ i K ) i ∈ N ∈ D s ince ( J χ i K ) i ∈ N ∈ D . But lim i ℓ ( χ ′ c i ) = lim i ℓ ( χ c i ) − r + r ′ < lim i ℓ ( χ i ), which co n tradicts the choice of ( χ i ) i ∈ N . Thus ℓ ( Z ) = r a s claimed. Similarly to the pro of of (10.11) one can also easily prove that z has co nstan t sp eed. (10.13) W e now claim that Z  D . Indeed, w e have ℓ ( D − Z ) ≥ ℓ ( D ) − ℓ ( Z ) b y Corollar y 5 .5. Moreov er, by Lemma 3.3 we hav e D − Z = lim ( J χ a i − σ 1 a i K ), and th us ℓ ( D − Z ) ≤ lim ℓ ( χ a i − σ 1 a i ) = lim ℓ ( χ a i ) − lim ℓ ( σ 1 a i ) = ℓ ( D ) − r = ℓ ( D ) − ℓ ( Z ) , where we used (10.1 2) . Thus Z  D as cla imed, and as D is pr imitive we obtain Z = D . Finally , we claim that z is a cir clex. Easily , the simplex z is clo sed since all the σ 1 i are. Suppo se the image of z is not a circle. Then, there must b e p oints x 6 = y ∈ [0 , 1 ) such that z ( x ) = z ( y ). Now consider the t wo simplices z 1 and z 2 obtained by sub dividing z at these two p oin ts x, y , and define Z 1 := J z 1 K and Z 2 := J z 2 K . Easily , ℓ ( z ) = ℓ ( z 1 ) + ℓ ( z 2 ). W e will s ho w that Z 1  Z . F or this, note that Z − Z 1 = J z − z 1 K b y Lemma 3.3 , a nd so Z − Z 1 = J z 2 K = Z 2 . Thus ℓ ( Z − Z 1 ) = ℓ ( Z 2 ) ≤ ℓ ( z 2 ) = ℓ ( z ) − ℓ ( z 1 ) ≤ ℓ ( z ) − ℓ ( Z 1 ) = ℓ ( Z ) − ℓ ( Z 1 ) , and with Corollary 5.5 w e obtain ℓ ( Z − Z 1 ) = ℓ ( Z ) − ℓ ( Z 1 ), i.e. Z 1  Z as claimed. But as we hav e already shown that Z = D and D w as assumed to be pr imitiv e, w e obtain Z = Z 1 , and thus ℓ ( z 1 ) = ℓ ( z ) s ince ℓ ( z ) = ℓ ( Z ). This means that ℓ ( z 2 ) = 0, which cannot be the case b y (1 0.13). This contradiction prov es that z is a circlex. Thu s we have prov ed Lemma 10.1 , whic h combined with Coro llary 9.7 proves our main result Theorem 3 .4: Pr o of of The or em 3.4. Supp ose fir st that ℓ ( C ) < ∞ . Then we can apply Coro l- lary 9.7 to obta in C = L i ∈ I D i where the D i are primitive. Applying Lemma 10.1 to e a c h D i we obtain a circlex z i with D i = J z i K and ℓ ( z i ) = ℓ ( D i ). Note that we hav e ℓ ( C ) = P ℓ ( D i ) by the definition of L . Thus ℓ ( C ) = P ℓ ( z i ). It remains to 29 chec k that ( z i ) i ∈ N is a σ -repres en tative of C . Indeed, we have C = lim P j ≤ i D i by (9.1 ) , and substituting D i by J z i K w e obtain C = lim P j ≤ i J z i K , which means that ( z i ) i ∈ N is indeed a σ -repr esen tative of C by definition. This prov es the assertion in this case. The other ca se, when ℓ ( C ) = ∞ is easier. All w e need to show is the existence of a σ -repres e n tative of C . F or this, let ( C i ) i ∈ N with C i ∈ H d be a sequence in C , and for every C i pick a n 1-cycle c i such that J z i K ∈ C i . No w putting z i := c i − P j 0 of le ng ths to the edges of G . More pr ecisely , any such assignment na turally induces a distance d ℓ ( x, y ) be t ween any tw o points x, y , and we let | G | ℓ denote the completion o f the corr esponding metric s pa ce. F or more details see [19], where the space | G | ℓ is extensively s tudied. It turns out that choos ing an appro priate ass ignmen t ℓ o ne obtains a metric space homeo- morphic to | G | : Theorem 11. 2 (Georgakop oulos [19 ]) . If G is lo c al ly finite and P e ∈ E ( G ) ℓ ( e ) < ∞ then | G | ℓ ∼ = | G | . Pr o of of The or em 11.1 ( s ketch). Fix a no rmal spanning tree T of G . Choo s e ℓ : E ( G ) → R > 0 such that | G | ℓ ∼ = | G | and mor eo ver the sums of the squar es of the leng ths of the fundamental cycles with r espect to T is finite. F or exa mple, we could start with an assig nmen t ℓ ′ with P ℓ ′ ( e ) < ∞ , which guarantees | G | ℓ ∼ = | G | by Theorem 11.2, and then let ℓ ( e ) := ℓ ′ ( e ) /m ( e ) where m ( e ) is the nu m ber of fundament al cycles c o n taining e . W e now define a map f : C ( G ) → ˆ H d ( | G | ℓ ) which will turn out to be a canonical iso mo rphism. Given a C ∈ C ( G ), wr ite C as the sum of a family F of fundamental cycles with resp ect to T ; this is possible b y [6, Theorem 8.5.8]. W e will now construct a lo op σ in | G | ℓ whose class will b ecome the ima g e f ( C ) of C . W e b egin with a lo op τ in | G | ℓ that trav ers e s each edge of T once in each direction and tr averses no other edges of G . T o see that such a loop exists, r eplace each edge of T by a pa ir of pa rallel edges to obtain the auxilia ry m ultigraph T ′ , and apply [7 , Theor em 2.5] to o btain a top ological Euler tour τ ′ of T ′ . Now τ ′ clearly ‘pro jects’ to the desired lo op τ . W e then mo dify τ into σ by attaching to it the cycles in F . T o achieve this, assume that τ maps a non-trivial in ter v al I v to e ac h vertex v of G . No w for every fundament al cycle F ∈ F , let v F w F be the chord of F , and as sume without loss of g eneralit y that v F is closer to the ro ot of T than w F . Mo dify τ so a s to use the interv al 30 I v F , previously ma pped to v F , in or der to travel once around F , starting and ending a t v F . D o ing so for every F ∈ F we obtain the lo op σ fr om τ . One still has to check that σ is indeed co n tinuous, but this is not ha r d. W e let f ( C ) := J 1 σ K ∈ ˆ H d ( | G | ). The map f is well-defined since T and τ are fixed, and every C ∈ C ( G ) has a unique repr esen tation as a sum of fundamental cycles with respe c t to T . T o see that f is injective, let C 6 = D ∈ C ( G ). Then the re pr esen tatio ns of C and D as sums of fundamental cycles differ by at least one fundamen tal cycle , since there must be a chord e of T con tained in o ne of C, D but not in the other. No w following the lines of Theorem 6.1 one can prov e that f ( C ) 6 = f ( D ); indeed, d 1 ( f ( C ) , f ( D )) is b ounded from b elow by a function of the length o f e . It r e mains to show tha t f is onto. Pick an element B o f ˆ H d ( | G | ) for which we would like to find a preimage. Let ( B i ) i ∈ N be a Cauch y sequence in B . F or every B i choose an 1-cycle χ i such that J χ i K = B i . Using the lo op τ from our earlier construction, w e ca n join all the simplices in χ i int o one lo op ρ i which, as τ is n ull- homotopic, is homolog ous to χ i . Now let C i ∈ C ( G ) b e the sum P { a e F e | e ∈ E ( G ) \ E ( T ) } of fundamental cycles whose chords are trav er sed by ρ i (here F e denotes the fundamen tal cycle containing the chord e and a e is the multiplicit y of trav ersa ls of e by ρ i ). W e cla im that f ( C i ) is the equiv alence class of the c onstan t sequence ( J ρ i K ). T o b egin with, recall that f ( C i ) is by definition the equiv alence class of the constant seq uence ( J σ i K ) for some lo op σ i that trav erses the same chords of T as χ i do es. Ho wev er , the tw o lo ops will in g eneral not b e homolog ous, since the order in which these chords are trav er sed may differ at infinitely many po sitions. But ˆ H d has the a bility of ‘disentangling’ infinite pro ducts o f commutators, a nd indeed, we will show tha t d 1 ( J ρ i K , J σ i K ) = 0. F or this, recall that we chose the edge-lengths ℓ ( e ) s o that the sum of the squares of the lengths of all the fundamen tal cycles is finite. Applying Lemma 4.1 to each fundamental c ycle, we can c onstruct an are a extension of | G | ℓ with finite excess area in which every fundamen tal cycle is null-homologous. This means that for every ǫ > 0 there is an ar ea extension X ǫ of | G | ℓ of e x cess area a t most ǫ in which all but finitely many fundamental cycles are null-homologous. Note that in each suc h X ǫ the lo ops ρ i and σ i are homolog ous, since they tr a verse the same chords, and all but finitely many of these c hords do not matter in X ǫ ; th us the order in which they trav ers e the chords do es not matter (re call that H 1 is ab elian). This means that d 1 ( J ρ i K , J σ i K ) = 0 as cla imed. W e hav e th us found a sequence C i ∈ C ( G ) such that ( f ( C i )) conv erg es to B , but we would lik e to have an element C ∈ C ( G ) with f ( C ) = B . T o achieve this, we choos e a subseq uenc e ( C a i ) of ( C i ) that conv er ges, as a set, to a n element C of C ( G ); such a subsequence exists by compactness. It is now straightforward to chec k that f ( C ) = B as desired: we can b ound d 1 ( f ( C ) , f ( C a i )) fr o m ab ov e by any ǫ choosing i large enoug h. Indeed, ch o ose i so that the sum of the squares of the lengths of the fundamental cycles with resp ect to chords in the sy mmetric difference C − C a i is s ma ll compared to ǫ . Since the sequence ( f ( C i )) conv erg es to B this immediately y ields f ( C ) = B . This completes the pr o of that f is onto, which makes it an iso morphism, a nd by constructio n a ca nonical one. Using this, o ne now ea sily obtains Theorem 1.3 a s a coro llary of our ma in re- sult Theorem 3.4 . Indeed, given C ∈ C ( G ) w e apply Theorem 3.4 to f ( C ), where f is the cano nical isomorphism of Theorem 11.1 , to obtain a σ -representativ e 31 ( z i ) of f ( C ) with every z i being a circlex. Now if tw o of these circlexe s share an edge e , then we can r emo ve e from b oth and combine the rema ining a rcs in to a new closed simplex, th us obtaining a new σ -representative of smaller total length, contradicting Theorem 3.4. This pro ves tha t the z i are edge -disjoin t, and since f is canonica l the f − 1 ( z i ) corr espond to the same circle s of | G | and sum up to C . In fact, this way we get something s lig h tly strong er than Theor e m 1.3: for a given C ∈ C ( G ) there may b e several wa ys to decomp ose it as a sum of edge- disjoint circles; see [17, p. 6] for an interesting exa mple. Theor em 1.3 canno t distinguish b et ween any of those wa ys, but our Theorem 3.4 can: it returns one of minimal length. A s the total leng th of such a decomp osition do es not o nly depe nd o n the edge - set (see [19, Example 4.5.]), this fac t ca n b e used in or der to control the decompo sition we o btain by v arying the edge-leng ths. F urther mo re, with Theorem 1.4 w e g eneralise, in a se ns e, Theor em 1.3 to non-lo cally-finite gra phs. F or such graphs there a re many candidate top olo- gies on whic h C ( G ) can be ba sed, so there is no standard cycle spa ce theor y . Theorem 1 .4 helps to o vercome this difficulty by o ffer ing a g eneral result that, for each choice of a top ology , yields a coro llary similar to Theo rem 1.3. This approach is expla ined in [19, Section 5]. 12 Higher dimensions Our definition of ˆ H d can b e easily adapted to yield higher dimensional homolog y groups ˆ H d,n . One can then ask if an a nalogue of o ur main r e sult Theorem 3 .4 still holds in higher dimensio ns, but one should firs t choo se a notion of n -dimensional conten t v ol (), s ince there a re several w ays to genera lise ‘le ngth’ to higher dimen- sions. Having c hosen such a notion, e.g. the n -dimensional Hausdorff measure, one could then try to prov e the following. Problem 12 .1. F o r every c omp act metric sp ac e X and C ∈ ˆ H d,n ( X ) , ther e is a σ - r epr esentative ( z i ) i ∈ N of C with P i v ol ( z i ) = v ol ( C ) . Most parts of our pro of Theor e m 3.4, in particular Theorem 8.1, could still be used in an attempt to prov e Pr oblem 12 .1. T o b egin with, o ne would need to generalise the r e s ults of Section 4 for the chosen notio n o f conten t, which do es not seem to b e hard. The biggest difficult y though seems to b e a gener alisation of (10 .7) . Ac kno wledgemen t I am very g r ateful to Reinhard Diestel for fruitful discussio ns that led to the definition of ˆ H d . I would like to thank the anonymous referee for their helpful remarks . References [1] M.A. Ar mstrong. Ba sic T op o lo gy . Springer-V erlag, 19 8 3. [2] W. A. Bogley and A. J. Sier adski. Univ ers al path space s. Pr eprin t. 32 [3] O. Bog opolsk i and A. Zas tr o w. The word pr oblem for some uncountable groups given by c o un table words. T o app ear in T op olo gy and its Applic a- tions . [4] H. Bruhn and A. Georga k op oulos. Bas es a nd closed s paces with infinite sums. Line ar Algeb r a and its A pplic ations , 435:200 7 –2018, 201 1. [5] H. Br uhn and M. Stein. MacLane’s planarity criterion for lo cally finite graphs. J. Combin. The ory (S erie s B) , 96 :225–239 , 2006 . [6] R. Diestel. Gr aph The ory (3rd edition). Spring er-V erlag , 20 05. Electronic edition av ailable a t: http://www .math.uni-hamburg.d e/ home/diestel/books/graph.theory . [7] R. Diestel. Lo cally finite graphs w ith ends: A topolog ical approach, I. Basic theory. In Infinite Gr aphs: Intr o ductions, Conne ctions, Surveys. Sp e cial issu e of Discr ete Math. , volume 311 (15), pages 1 423–144 7 , 20 11. [8] R. Diestel and D. K¨ uhn. On infinite cycle s I. Co mbinatoric a , 24:6 8 –89, 2004. [9] R. Diestel and D. K¨ uhn. On infinite cy cles I I. Co m bi natoric a , 2 4:91–116 , 2004. [10] R. Diestel a nd P . Spr¨ uss e l. On the ho mo logy of lo cally compact spaces with ends. T o app ear in T op o lo gy and its Applic ations . [11] R. Diestel and P . Spr¨ usse l. The homology of a lo cally finite graph with ends. Combinatoric a , 30:6 8 1–714, 2 010. [12] R. Diestel and P . Spr ¨ ussel. The fundamental gr o up of a lo cally finite graph with ends. A dvanc es in Mathematics , 22 6:2643–26 75, 20 11. [13] J . Dydak. Co verings a nd fundamen tal groups: a new appro ac h. Preprint 2011. [14] K . E da and K. Sak ai. A F actor of Singular Homo logy. Tsukub a J. Math. , 15(2):351 –387, 19 91. [15] S. Eilenberg and N. Steenro d. F oundations of Algebr aic T op olo gy . P rinceton Univ. Pres s, 1952. [16] A. Geo rgakopoulos. Infinite hamilton cycle s in squa r es o f lo cally finite graphs. A dvanc es in Mathematics , 220:67 0–705, 2 009. [17] A. Georgakop oulos. T op ological circ le s and Euler tours in lo cally finite graphs. Ele ctr onic J. Comb. , 16:#R40, 20 09. [18] A. Georga k op oulos. Uniquenes s of elec tr ical cur ren ts in a netw o rk o f finite total resistance. J. L ondon Math. So c. , 82(1):2 56–272, 2010. [19] A. Geor gak o poulos. Graph topolo gies induced by edge lengths. In Infi- nite Gr aphs: Intr o ductions, Conne ctions, S urveys. Sp e cial issue of Discr ete Math. , volume 31 1 (15), pages 1 523–154 2, 2 011. 33 [20] A. Georga k op oulos and P . Spr ¨ ussel. Geo detic to pologica l cycles in lo cally finite graphs . Ele ctr onic J. Comb. , 1 6 :#R144, 2009 . [21] A. Hatc her . A lgebr aic Top olo gy . Cambrigde Univ. Press, 2002. [22] S. MacLane. A combinatorial c ondition for planar graphs. F und. Ma t h. , 28:22– 32, 19 37. [23] S. Mardesic and J. Segal. Shap e the ory. The inverse syst em appr o ach. North-Holland Mathematical Librar y , V ol. 2 6, 19 82. [24] R. E. Meg ginson. An Intr o duction t o Banach Sp ac e The ory . Gra duate T exts in Mathematics. Springer, 19 98. [25] R. Meise and D. V og t. Intr o duction to F unctional Analysis . O xford Uni- versit y Press , 199 7. [26] S. Willard. Gener al top olo gy . Do ver P ublications, 2 004. 34