A combinatorial model for the Menger curve
AA COMBINATORIAL MODEL FOR THE MENGER CURVE
ARISTOTELIS PANAGIOTOPOULOS AND S(cid:32)LAWOMIR SOLECKI
Abstract.
We represent the universal Menger curve as the topological real-ization | M | of the projective Fra¨ıss´e limit M of the class of all finite connectedgraphs. We show that M satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson pro-jective universality theorem. Our arguments involve only 0-dimensional topol-ogy and constructions on finite graphs. Using the topological realization M (cid:55)→ | M | , we transfer some of these properties to the Menger curve: we provethe approximate projective homogeneity theorem, recover Anderson’s finite ho-mogeneity theorem, and prove a variant of Anderson–Wilson’s theorem. Thefinite homogeneity theorem is the first instance of an “injective” homogeneitytheorem being proved using the projective Fra¨ıss´e method. We indicate howour approach to the Menger curve may extend to higher dimensions. Introduction
The Menger curve is a 1-dimensional Peano continuum that is classically ex-tracted from the cube in the same way that the Cantor space is extracted from theinterval: subdivide C = [0 , into 3 congruent subcubes; let C be the unionof these subcubes which intersect the one-skeleton of [0 , ; repeat this processon each subcube again and again to define C n ; the Menger curve is defined to bethe intersection (cid:84) n C n . With this construction Menger found the first example ofa universal space for the class of 1-dimensional continua, that is, a 1-dimensionalcontinuum in which every other 1-dimensional continuum embeds [14]. Figure 1.
From C to C .The Menger curve is a canonical continuum whose topological properties do notdepend on the various geometric parameters of the above iterative process. In fact,many other constructions of universal 1-dimensional continua (e.g., [12, 16]) whichappeared soon after [14] were later shown to produce the same space; see [1]. Mathematics Subject Classification.
Key words and phrases.
Projective Fra¨ıss´e limits, Menger curve, homogeneity, universality,homology Menger compactum.Research of Solecki supported by NSF grants DMS-1800680 and 1700426. a r X i v : . [ m a t h . L O ] J a n ARISTOTELIS PANAGIOTOPOULOS AND S(cid:32)LAWOMIR SOLECKI
In this paper, we develop a combinatorial model for the Menger curve usingan analogue of projective Fra¨ıss´e theory from [10]. The
Menger prespace M is acompact graph-structure on the Cantor space. In a sense, M is the generic inverselimit in the category C of all connected epimorphisms between finite connectedgraphs . The edge relation on M turns out to be an equivalence relation and theMenger curve is then defined to be the quotient | M | = M /R of M with respect tothis relation.This definition of the Menger curve as the topological realization | M | of thecombinatorial object M has certain technical and foundational advantages. On thefoundational side, the definition of M is canonical since it is constructed through C without making any ad-hoc choices for the bonding maps. Moreover, the definitionof | M | is intrinsic, in that it makes no reference to external spaces such as suchas [0 , . On the technical side, when proving results about the Menger curve,we can often replace various complications coming from 1-dimensional topologyof | M | with combinatorial problems about graphs in C . Moreover, like any otherprojective Fra¨ıss´e limit, M has the following projective extension property built inby the construction: for every g ∈ C and any connected epimorphism f as in thediagram, there is a connected epimorphism h with g ◦ h = f . M AB h f g Having this universal property of M as our point of departure, and expanding on itusing combinatorial properties of C , we can integrate various aspects of the Mengercurve into a unified theory as follows. • Anderson’s homogeneity theorem [1] states that any bijection between finitesubsets of | M | extends to a global homeomorphism of | M | . This theorem waslater generalized in [13] to the strongest possible homogeneity result for | M | ,namely, that every homeomorphism between locally non-separating, closedsubsets of | M | extends to a global homeomorphism of | M | . In Theorem 4.1,we prove a homogeneity result for M analogous to the homogeneity resultfor the Menger curve in [13]. From that we recover Anderson’s homogeneityresult for | M | . Our proof of Theorem 4.1 relies on C being closed under acertain mapping cylinder construction. • Anderson–Wilson’s projective universality theorem states that | M | admitsan open, continuous, and connected map onto any Peano continuum .Moreover, all preimages of points under this map can be taken to be home-omorphic to the Menger curve [2, 20]. In Theorem 5.1, we prove a combi-natorial analogue of the Anderson–Wilson theorem for M . In the process,we isolate a combinatorial property of C that is responsible for this strongform of projective universality. In Corollary 5.1, we establish a variant In this paper we use the newer term connected map in place of the synonymous term monotonemap used in [2, 20].
COMBINATORIAL MODEL FOR THE MENGER CURVE 3 of Anderson–Wilson’s theorem for | M | where the map produced is weaklylocally-connected instead of open. • In Theorem 6.1, we prove that | M | satisfies an approximate projective ho-mogeneity property that is analogous to the property satisfied by manyother continua presented as topological realizations of projective Fra¨ıss´elimits; see [3] and [10] for examples. Namely, we show that if γ , γ : | M | → X are continuous connected maps onto a Peano continuum X , then thereis a sequence ( h n ) of homeomorphisms of | M | so that ( γ ◦ h n ) convergesuniformly to γ .It is worth mentioning that throughout Section 4 one can find analogies with theabstract homotopy theory in the spirit of model categories.Finally, pursuing an extension of this approach to higher dimensional Mengercompacta, we define higher dimensional analogues of C , M , and | M | . For every n ∈{ , , . . . } ∪ {∞} , we define C n to be the class of all ( n − n -dimensional, ( n − C n is a projective Fra¨ıss´e class. Interestingly, the same is shown to hold for theclass (cid:101) C n , which is defined by replacing “( n − n − C n . As far as we are aware, these “homology n -Menger spaces”introduced here—and for n = ∞ this “homology Hilbert cube”—have not beenconsidered before. Contents
Introduction 11. The class C of finite connected graphs 32. Topological graphs and Peano continua 53. The Menger curve 84. The combinatorics of homogeneity 105. The combinatorics of universality 156. The approximate projective homogeneity property 187. The n -dimensional case 21References 251. The class C of finite connected graphs Let A be a set and let R be any subset of A . We say that R is a reflexive if R ( a, a ) holds for all a ∈ A . We say that R is symmetric if for every a, b ∈ A we havethat R ( a, b ) implies R ( b, a ). We finally say that R is transitive if the conjunctionof R ( a, b ) and R ( b, c ) implies R ( a, c ). By a graph ( A, R A ), simply denoted by A ,we mean a set A together with a specified subset R A of A that is both reflexive andsymmetric. Notice that reflexivity makes our definition of a graph non-standardbut it allows us to treat graphs as 1-dimensional simplicial complexes. A clique ofa graph ( A, R A ) is any subset C of A with the property that for all a, b ∈ C we havethat R A ( a, b ). A map f : B → A is a homomorphism between graphs if it maps ARISTOTELIS PANAGIOTOPOULOS AND S(cid:32)LAWOMIR SOLECKI edges to edges, that is, if R B ( b, b (cid:48) ) implies R B ( f ( b ) , f ( b (cid:48) )), for every b, b (cid:48) ∈ B . Ahomomorphism f is an epimorphism if it is moreover surjective on both verticesand edges. An isomorphism is an injective epimorphism. By a subgraph of agraph we understand an induced subgraph.We isolate a collection C of finite graphs together with special epimorphismsbetween them, the point being, that various topological and dynamical propertiesof the Menger curve are reflections of combinatorial properties of C . A subset X ofa finite graph A is connected if, for all non-empty U , U ⊆ X with X = U ∪ U ,there exist x ∈ U and x ∈ U such that R A ( x , x ). A graph A is connected ifthe domain of A is a connected subset. An epimorphism f : B → A is connected if the preimage of each connected subset of A is a connected subset of B . Definition 1.1.
Let C be the category of all finite connected graphs with morphismsin C being connected epimorphisms. Our first task is to establish that C is a projective Fra¨ıss´e class. Projective Fra¨ıss´etheory was developed in [10] in the more general setting of L -structures. For thesake of perspective, we recall from [10] the Fra¨ıss´e class axioms in this more generalsetup. For the unfamiliar reader, we point out that a graph is just an example ofan L -structure where the language L consists of a binary relation symbol R . Animportant difference between the definition below and the one in [10] is that here,a Fra¨ıss´e class is allowed to consist of a strict subcollection of epimorphisms, e.g.only the epimorphisms which are connected.Let F be a class of finite L -structures with a fixed family of morphisms amongthe structures in F . We assume that each morphism is an epimorphism with respectto L . We say that F is a projective Fra¨ıss´e class if(1) F is countable up to isomorphism, that is, any sub-collection of pairwisenon-isomorphic structures of F is countable;(2) morphisms are closed under composition and each identity map is a mor-phism;(3) for B, C ∈ F ; there exist D ∈ F and morphisms f : D → B and g : D → C ;and(4) for every two morphisms f : B → A and g : C → A , there exist morphisms f (cid:48) : D → B and g (cid:48) : D → C such that f ◦ f (cid:48) = g ◦ g (cid:48) .We will refer to the last property above as the projective amalgamationproperty . We have the following theorem. Theorem 1.1. C is a projective Fra¨ıss´e family.Proof. We check here that C satisfies the projective amalgamation property. Therest of the properties follow easily. Let f : B → A and g : C → A be connectedepimorphisms and let D be the subgraph of the product graph B × C , induced ondomain { ( b, c ) ∈ B × C : f ( b ) = g ( c ) } . Recall that in the product graph B × C there is an edge between ( b, c ) and ( b (cid:48) , c (cid:48) ) ifand only if R B ( b, b (cid:48) ) and R C ( c, c (cid:48) ). Let also f (cid:48) = p B : D → B , g (cid:48) = p C : D → C be COMBINATORIAL MODEL FOR THE MENGER CURVE 5 the canonical projections. By the definition of B × C it is immediate that π B , π C are homomorphisms.We show that p B is a connected epimorphism. By symmetry, the same argumentapplies for p C . The fact that g is surjective on vertexes implies that p B is surjectiveon vertexes since for every b there is a c b with f ( b ) = g ( c b ), and hence there is d = ( b, c b ) with π B ( d ) = b . By the same argument, and since g is surjective onedges, we have that p B is surjective on edges as well. So p B is an epimorphism.Moreover, since g is connected, g − ( f ( b )) is connected for every b ∈ B . Hence thepoint fibers p − B ( b ) = { b } × g − ( f ( b ))of π B are connected for every b ∈ B . The following general lemma implies thereforethat π B is connected. (cid:3) Lemma 1.1.
A function between two graphs of C is a connected epimorphism ifand only if it is an epimorphism and preimages of points are connected.Proof. Only the direction ⇐ needs to be checked. Let f : B → A be an epimorphismsuch that preimages of points are connected. It suffices to show that preimages ofedges are connected. Let b , b ∈ B be such that R A ( f ( b ) , f ( b )). Since f is anepimorphism, there are b (cid:48) , b (cid:48) ∈ B that form an edge and are such that f ( b (cid:48) ) = f ( b )and f ( b (cid:48) ) = f ( b ). Since the preimages of f ( b ) and f ( b ) are connected, there isa path connecting b with b (cid:48) and b with b (cid:48) . Since b (cid:48) and b (cid:48) are connected by anedge, b and b are connected by a path, as required. (cid:3) Topological graphs and Peano continua
We import some notions from [10] and we apply them here in the special caseof graphs. A topological graph K is a graph ( K, R K ), whose domain K is a0-dimensional, compact, metrizable topological space and R K is a closed subset of K . All types of morphisms we consider between topological graphs are assumedto be continuous. Moreover, we automatically view all finite graphs as topologicalstructures endowed with the discrete topology.We extend C to the class C ω of all topological graphs and epimorphisms whichare “approximable” within C . A concrete description of C ω is given in Proposition2.1. The rest of the paragraph defines C ω in abstract terms. Let ( K n , f nm , N ) be aninverse system of finite connected graphs with bonding maps f nm : K n → K m from C . It is easy to check that the inverse limit K = lim ←− ( K n , f nm ) ∈ C ω is a topologicalgraph, where ( x , x , . . . ) is R -connected with ( y , y , . . . ) in K if for every n , x n is R -connected with y n in K n ; see for example the proof of Proposition 2.1. Wecollect in C ω all topological graphs K which are inverse limits of sequences withbonding maps from C . Notice that every finite connected graph is in C ω . If A ∈ C and K = lim ←− ( K n , f nm ) ∈ C ω , then an epimorphism h : K → A is in C ω if and only ifthere exists m , and a morphism h (cid:48) : K m → A in C , such that h is the compositionof h (cid:48) with the canonical projection f m from K to B m . For two topological graphs K, L ∈ C ω , an epimorphism h : L → K is said to be in C ω if for each A ∈ C and each g : K → A in C ω , the composition g ◦ h is in C ω . Finally, an epimorphism h : L → K ARISTOTELIS PANAGIOTOPOULOS AND S(cid:32)LAWOMIR SOLECKI is an isomorphism if it is injective and both h, h − are in C ω . Notice that h is anisomorphism between K = lim ←− ( K n , f nm ) ∈ C ω and L = lim ←− ( L n , g nm ) ∈ C ω if and onlyif there is a sequence ( h i ) of morphisms in C and two strictly increasing sequences( k i ) and ( l i ) of natural numbers such that for each ih i ◦ h i +1 = f k i +1 k i and h i +1 ◦ h i +2 = g l i +1 l i . We now give a more concrete description of the graphs and morphisms of C ω .Let K be a topological graph. We say a subset X of K is connected if, forall open U , U ⊆ K with X ∩ U (cid:54) = ∅ (cid:54) = X ∩ U and X ⊆ U ∪ U , there exist x ∈ X ∩ U and x ∈ X ∩ U such that R K ( x , x ). We say that a topological graph( K, R K ) is connected if K is connected as a subset of the graph. We say that itis locally-connected if it admits a basis of its topology consisting of connectedsets in then above sense. Let K, L be topological graphs and let f : L → K bean epimorphism. We say that f is a connected epimorphism if the preimageof each closed connected subset of K is connected. Note that the above notionscoincide with the analogous notions introduced for finite graphs. Proposition 2.1. C ω is the class of all connected epimorphisms between connected,locally-connected, topological graphs.Proof. Let K = lim ←− ( K n , f nm ) ∈ C ω with f nm ∈ C . The underlying space of the graph K is 0-dimensional, compact, and metrizable, since it is a countable inverse limitof discrete finite spaces. The set R K is closed and contains the diagonal as anintersection of closed relations containing the diagonal. This proves that that K isa topological graph. We see now that K is also connected. Let also f n : K → K n be the projection induced by the inverse system. Assume that U , U are non-empty open subsets of K with K ⊆ U ∪ U . Since K is connected, we can pick x ∈ f ( U ) and y ∈ f ( U ) with R K ( x , y ). Assume by induction that wepicked x n ∈ f n ( U ) and y n ∈ f n ( U ), with R K n ( x n , y n ), so that f nn − ( x n ) = x n − and f nn − ( y n ) = y n − . Using the fact that f − n ( { x n , y n } ) is connected and that f n +1 n is an epimorphism we can pick x n +1 ∈ f n +1 ( U ) and y n +1 ∈ f n +1 ( U ) with R K n +1 ( x n +1 , y n +1 ), and so that f n +1 n ( x n +1 ) = x n and f n +1 n ( y n +1 ) = y n . Hence,( x , x , . . . ) ∈ U and ( y , y , . . . ) ∈ U are such that R K (( x , x , . . . ) , ( y , y , . . . )).The exact same argument can be applied to show that every clopen set of K of theform f − n ( x ), where x ∈ K n , is connected. Hence K is locally-connected as well.Let now L = lim ←− ( L n , g nm ) ∈ C ω as well and let h : L → K be a morphism in C ω .By definition, for every m there is an n and a connected epimorphism h (cid:48) : L n → K m ,so that h (cid:48) ◦ g n = f m ◦ h , where g n : K → K n is the canonical projection. Sinceevery connected clopen subset ∆ of K is of the form f − m ( X ) for large enough m and some connected subset X of K m , we have that h − (∆) = ( h (cid:48) ◦ g n ) − ( X ) isa connected clopen subset of L . The rest follows from the fact that every closedconnected subsets of K and L are interstions of a decreasing sequence of connectedclopen subsets of the same spaces.We turn to the converse statements first for graphs and then for morphisms.Let K be a connected, locally-connected, topological graph. It is not difficult tosee that K admits a basis U of connected clopen sets. Using compactness of K as COMBINATORIAL MODEL FOR THE MENGER CURVE 7 well as of every element of U , we can find a sequence U n of finite covers of K sothat U n ⊂ U , U n refines U n − , if U, V ∈ U n then U ∩ V = ∅ , and (cid:83) n U n separatespoints of K . One can define a graph structure on U n by putting an R -edge between U and V if there is x ∈ U and y ∈ V with R K ( x, y ). It is easy to see now that f nm : U n → U m is a connected epimorphism between finite connected graphs andthat K = lim ←− ( U n , f nm ).Let now h : L → K be a connected epimorphism between connected, locally-connected, topological graphs. By the previous paragraph K = lim ←− ( K n , f nm ) and L = lim ←− ( L n , g nm ), where f nm , g nm ∈ C . It suffices to show that for every m there is n ,and a map h (cid:48) : L n → K m with h (cid:48) ∈ C and g m ◦ h = h (cid:48) ◦ f n . Fix some m and let n large enough so that { g − n ( y ) : y ∈ L n } refines { ( f m ◦ h ) − ( x ) : x ∈ K m } . Let also h (cid:48) : L n → K m be the unique map that witnesses this refinement. Using that f m ◦ h and g n are connected epimorphisms it is easy to see that h (cid:48) is in C as well. (cid:3) Next we illustrate the relationship between topological graphs and Peano con-tinua. Recall that a continuum is a connected, compact, metrizable space. A
Peano continuum is a continuum that is locally-connected. A map φ : Y → X between topological spaces is connected if φ − ( Z ) is connected for every closedconnected subset Z of X . Here connected and locally-connected refer to the stan-dard topological notion. We also adopt the convention that the empty space is notconnected. We will always accompany ambiguous terminology such as “connected”with further specification such as “graph” or “space” to distinguish between ourcombinatorial and the standard topological notion of connectedness.A topological graph K ∈ C ω is a prespace if the edge relation R is also transitive.In other words, if K is a collection of cliques. This makes R an equivalence relationand we denote by [ x ] the equivalence class of x ∈ K . Similarly, for every subset F of K we denote by [ F ] the set of all y ∈ K which lie in some equivalence class [ x ]with x ∈ F . The topological realization | K | of a prespace K is defined to be thequotient K/R K = { [ x ] : x ∈ K } , endowed with the quotient topology. We denote by π the quotient map K (cid:55)→ | K | .Since R K is compact, | K | is compact and metrizable. In fact, we have the followingtheorem. Theorem 2.1.
For a topological space X the following are equivalent: (1) X is a Peano continuum; (2) X is homeomorphic to | K | for some prespace K ∈ C ω . We start with a lemma.
Lemma 2.1.
Let K = lim ←− ( K n , g nm ) ∈ C ω be a prespace, let x ∈ K and let g n : K → K n be the natural canonical projections. Consider the following families: • P x = { g − ( a ) : g ∈ C ω , g ([ x ]) = a } , where g ranges over all maps g : K → A in C ω , with A ∈ C , and a ∈ A ; • P x = { g − ( Q ) : g ∈ C ω , g ([ x ]) = Q } , where g ranges over all maps g : K → A in C ω , with A ∈ C , and Q ⊆ A ; ARISTOTELIS PANAGIOTOPOULOS AND S(cid:32)LAWOMIR SOLECKI • P x = { g − ( Q ) : g ∈ C ω , g ([ x ]) = Q } , where everything is as in P x , but g ranges only over { g n : n ∈ N } .If P x is either of the above families, then P xπ = { π ( P ) : P ∈ P x } is a neighborhoodbasis of [ x ] in | K | consisting of closed sets.Proof. Let P ∈ P xi and set U = [ P c ] c ⊆ K . Notice that [ P c ] is the projection ofthe closed set { ( x, y ) ∈ K × K | ( x, y ) ∈ (cid:0) R K (cid:92) ( K × P c ) (cid:1) } , along the compact second coordinate and therefore U is open. Since R K is anequivalence relation and [ P c ] is R K invariant, then so is U . Hence, π ( U ) is an opensubset of | K | , and it clearly holds that [ x ] ∈ π ( U ) ⊆ π ( P ). Since π : K → | K | is continuous and P clopen we have that π ( P ) is a closed neighborhood of [ x ].Compactness of K implies that any open cover of K can be refined by a partitionof the form { g − n ( b ) : b ∈ K n } , for large enough n . Hence P x projects through π toa neighborhood basis of [ x ]. It is not difficult now to see that P x = P x ⊇ P x . (cid:3) We turn now back to the proof of Theorem 2.1.
Proof of Theorem 2.1.
First we show that (2) = ⇒ (1). Let P be the collection ofclopen subsets of K of the form f − ( a ), where f ranges over all f : K → A in C ω and a ∈ A . By Lemma 2.1, P projects via π to a neighborhood basis of | K | . Itsuffices to show that π ( P ) is connected for every P ∈ P ; see Theorem 2.5 [8], forexample. Since every P ∈ P is itself an element of C ω , it suffice to show that | K | is a connected space for every prespace K ∈ C ω . But any clopen partition of | K | pulls back through π to a clopen partition { U , U } of K which is invariant, that is,[ U ] = U and [ U ] = U . By Theorem 2.1, U is either empty or the whole space.For (1) = ⇒ (2), let X be a Peano continuum. By Bing’s Partition theorem (see[5]) there is a sequence ( O n ) of finite collections of disjoint open subsets of X , sothat for all n ∈ N we have that:(1) (cid:83) O n is dense in X ;(2) O is connected, for all O ∈ O n ;(3) O n +1 refines O n ;(4) any open cover of X is refined by O m , for large enough m .We turn each finite set O n into a graph by putting an edge between O and O (cid:48) , ifand only if, O ∩ O (cid:48) (cid:54) = ∅ . Let f nm : O n → O m be the uniquely defined refinement map.Since every O ∈ (cid:83) n O n is connected, it follows that f nm ∈ C . Let K = lim ←− ( O n , f nm ).Let ρ : K → X , mapping each point x = ( O , O , . . . ) ∈ K to the unique—byproperty (4) above—point ρ ( x ) with { ρ ( x ) } = (cid:84) n O n . It is easy to see that R K isthe pullback of equality on X under ρ , and hence, K is a prespace with X (cid:39) | K | . (cid:3) The Menger curve
The next theorem is proved using the methods of [10]. For completeness wesummarize the construction of F below. COMBINATORIAL MODEL FOR THE MENGER CURVE 9
Theorem 3.1. If F is a projective Fra¨ıss´e family, then there exists a unique topo-logical structure F ∈ F ω such that: (1) for each A ∈ F , there exists a morphism in F ω from F to A ; (2) for A, B ∈ F and morphisms f : F → A and g : B → A in F ω there existsa morphism h : F → B in F ω such that f = g ◦ h . We say that F is the projective Fra¨ıss´e limit of F . The second property inthe above statement is called projective extension property . We briefly sketchhere the construction of F out of F . For more details, see [10]. Construction of a generic sequence.
We build F as an inverse limit of a genericsequence ( L n , t nm ) of morphisms t nm ∈ F . By property (1) in the definition of aFra¨ıss´e class we can make two countable lists ( A n : n ≥ e n : C n → B n : n ≥ F . Moreover wemake sure that every morphism type contained in F appears infinitely often in ( e n )above. Let L = A . Assume that L n has been defined together with all maps t ni ,for all i < n . Using property (3) in the definition of a Fra¨ıss´e class we get H ∈ F together with maps f : H → L n , g : H → A n +1 . Notice now that since H is finite,there is a finite list s , . . . , s k of morphism types from H to B n +1 in F . Using k -many times projective amalgamation we get f (cid:48) : H (cid:48) → H and d j : H (cid:48) → C n +1 in F with s j ◦ f (cid:48) = e n +1 ◦ d j for all j ≤ k . Set L n +1 = H (cid:48) and t n +1 i = t ni ◦ f ◦ f (cid:48) . Itis not difficult to see that the way “saturated” ( L n , t nm ) with respect to ( A n ) and( e n ) endows F with properties (1) and (2) of Theorem 3.1 above.As a consequence of Theorems 1.1,3.1, we can now consider projective Fra¨ıss´elimit M of C . We call M the Menger prespace . Theorem 3.2.
The Menger prespace M is a prespace containing cliques of size atmost . Its topological realization | M | is the Menger curve.Proof. The Menger curve is the unique 1-dimensional, Peano continuum with thedisjoint arcs property ([4], see also [1, 13]). Recall that a space X has the dis-joint arcs property if every continuous map { , } × [0 , (cid:55)→ X can be uniformlyapproximated by maps which send { } × [0 ,
1] and { } × [0 ,
1] to disjoint sets.By Theorem 2.1, we know that | M | is a Peano continuum. To show that | M | is 1-dimensional we find for every open cover a refinement whose nerve is one-dimensional. Let V be any open cover of | M | and let f : M → A be any f ∈ C ω with A ∈ C , so that V f = { π ( f − ( a )) : a ∈ A } refines V . Let g : B → A be in C , sothat B has no cliques of size 3. For example one can barycentrically subdivide A and map the new vertexes to either of its two neighbors. The projective extensionproperty of M provides us with a further refinement V h = { π ( h − ( b )) : b ∈ B } of V f . Notice that since B has no cliques of size 3, the nerve of V h is isomorphic to B . Since | M | is a regular topological space and V h is finite, we can find for every W ∈ V h an open U W ⊇ W , so that { U W : W ∈ V h } has the same nerve as V h andstill refines V .For the disjoint arcs property, let γ , γ : [0 , → | M | be two maps and let V bean open cover of | M | . We will find disjoint γ (cid:48) , γ (cid:48) : [0 , → | M | which are V -close to γ and γ , that is, for every x ∈ [0 , V ∈ V , so that both γ i ( x ) , γ (cid:48) i ( x ) lie in V . As in the previous paragraph, let V f = { π ( f − ( a )) : a ∈ A } be a refinement of V and consider an open cover U f = { U a : a ∈ A } refining of V , with U a ⊇ π ( f − ( a )),having the same nerve as V f . Notice that for every i ∈ { , } there is a finite cover V i of [0 ,
1] with connected open intervals, and an assignment α i : V i → A , so that γ i ( V ) ⊆ U a , for every V ∈ V i with α i ( V ) = a . Let J be the unique graph on domain { , , } so that R J ( j, j (cid:48) ) if and only if | j − j (cid:48) | ≤ , and notice that the canonicalprojection ρ : J × A → A is in C . Hence by the projective extension property of M we have a connected epimorphism h : M → J × A so that f = ρ ◦ h . Using thefact that π ( h − ( X )) is path-connected for every connected subset X of J × A , itis easy now to construct a paths γ (cid:48) and γ (cid:48) which are V -close to the original pathsand that moreover, γ i ([0 , ⊂ π ( h − ( { i } × A )). (cid:3) The combinatorics of homogeneity
In Theorem 4.1 below, we prove an injective homogeneity result for M analo-gous to the main result for | M | in [13]. In Corollary 4.1, we recover Anderson’shomogeneity result for the Menger curve | M | . We note that, as in Section 6, anappropriate version of projective homogeneity can always be obtained naturally andwithout much difficulty for any continuum which has been presented as a topolog-ical realization of some projective Fra¨ıss´e limit; see [3] and [10] for example. Herewe provide the first example of an injective homogeneity property that is obtainedusing projective Fra¨ıss´e theoretic methods.Let K be a closed subgraph of M . We say that K is locally non-separating iffor each clopen connected W , the set W \ K is connected. Theorem 4.1. If K = [ K ] and L = [ L ] are locally non-separating subgraphs of M ,then each isomorphism from K to L extends to an automorphism of M . For the proof of Theorem 4.1 we run a standard “back and forth” argumentbased on a lifting property for inclusions
K (cid:44) → M of locally non-separating sets; seepage 13. This lifting property strengthens the projective extension property of M .Viewed from an abstract homotopy theoretic standpoint, the lifting propertysuggests that maps in C relate to the above inclusion K (cid:44) → M in the same way thattrivial fibrations relate to cofibrations within a model category. The analogy withmodel categories is also reflected in the way we prove the lifting property: we definea combinatorial analogue of the mapping cylinder construction for homomorphismsbetween graphs and we show that for any f : M → A in C ω , the induced map from K to A factors through a map of the form r ◦ i , where i is an inclusion and r amapping cylinder retraction. Before we describe the mapping cylinder constructionwe start with two general lemmas. The next result is probably known, but we couldnot find a reference for it. Lemma 4.1.
A closed subset K of M is locally non-separating if and only if foreach clopen connected set W ⊇ K and each clopen set V with K ⊆ V ⊆ W thereexists a clopen set U such that K ⊆ U ⊆ V and W \ U is connected. COMBINATORIAL MODEL FOR THE MENGER CURVE 11
Proof.
Only the direction from left to right needs a proof. Fix a connected clopenset W . Since W \ K is open, we have W \ K = (cid:83) k ∈ N V k for some V k clopen andconnected. Let k (0) = 0 and define U = V . Given U n , let k ( n + 1) be the smallestnatural number such that V k ( n +1) (cid:54)⊆ U n and U n ∪ V k ( n +1) is connected, if such k ( n + 1) exists. Otherwise, let k ( n + 1) = k ( n ). Let U n +1 = U n ∪ V k ( n +1) .Since U n ⊆ U n +1 for each n , by compactness, it will suffice to show that(1) W \ K = (cid:91) n ∈ N U n . This follows as in the last part of the proof of Lemma 6.1: assume that x ∈ W \ K and x (cid:54)∈ (cid:83) n ∈ N U n ; let k ( x ) be such that x ∈ V k ( x ) ; check that [ V k ( x ) ] ∩ (cid:83) n ∈ N U n = ∅ ;and derive a contradiction from the fact that W \ K is connected. (cid:3) Lemma 4.2. If K ∈ C ω is a prespace, Z ⊆ V ⊆ K , Z = [ Z ] , and V is open, thenthere is W ⊆ K open with Z ⊆ W and [ W ] ⊆ V . If moreover Z is closed, then W can be additionally chosen to clopen.Proof. It suffices to show that for every z ∈ Z we can find W z clopen with z ∈ W z and [ W z ] ⊆ V . If such W z doesn’t exist then one can find sequences ( x n )and ( y n )so that y n ∈ [ x n ], x n converging to z , and y n ∈ V c . By compactness of V c wecan assume that ( y n ) converges to y ∈ Y . But since R K is closed this implies that y ∈ [ x ], contradicting that Z = [ Z ] ⊆ V . (cid:3) Let X be any finite (reflexive) graph and let α : X → A be a graph homo-morphism with A ∈ C . We assume that dom( A ) ∩ dom( X ) = ∅ . The mappingcylinder C α of α is the unique graph on domain dom( A ) ∪ dom( X ) with:(1) C α (cid:22) dom( A ) = A and C α (cid:22) dom( X ) = X ;(2) for each x ∈ X and a ∈ A , there is an edge in C α between x and a if andonly if a = α ( x (cid:48) ) for some x (cid:48) ∈ X with R X ( x, x (cid:48) ).The mapping cylinder C α comes together with two natural graph inclusions A, X (cid:44) → C α and a canonical retraction r α : C α → A given by: r α ( x ) = α ( x ), if x ∈ X ;and r α ( x ) = x , otherwise. It is easy to check that both C α , r α are in C . Lemma 4.3.
Let K = [ K ] be a locally non-separating subgraph of M ; let X bea finite graph; let α : X → A be graph homomorphism, with A ∈ C . For every f : M → A in C ω and every graph homomorphism q : K → X with α ◦ q = f (cid:22) K ,there is ˜ f : M → C α in C ω , with r α ◦ ˜ f = f and ˜ f (cid:22) K = q . K XC α M A q αr α f ˜ f Proof.
Let K , X , A , α , f , q be the data provided in the statement of Lemma 4.3and set K x = q − ( x ), for every x ∈ X . Claim.
There is g : B → A in C and h : M → B in C ω , with g ◦ h = f , togetherwith collections { D x : x ∈ X } and { D a : a ∈ A } of subgraphs of B , so that if we set B a = g − ( a ) for all a ∈ A , we have:(1) { dom( D a ) } ∪ { dom( D x ) : x ∈ X, α ( x ) = a } is a partition of dom( B a );(2) the image of K x under h is contained in D x ;(3) R X ( x, x (cid:48) ) if and only if there is b ∈ D x and b (cid:48) ∈ D x (cid:48) with R B ( b, b (cid:48) );(4) R X ( x, x (cid:48) ) for some x (cid:48) with α ( x (cid:48) ) = a if and only if there is b ∈ D x and b (cid:48) ∈ B a with R B ( b, b (cid:48) );(5) for every connected component C of D x there is c ∈ C and b ∈ D α ( x ) with R ( b, c );(6) if (cid:101) D is the subgraph of B on domain (cid:83) a ∈ A dom( D a ), then g (cid:22) (cid:101) D : (cid:101) D → A is in C and as a consequence D a is connected. Proof of Claim.
Since { K x : x ∈ X } is a finite collection of closed subsets of a 0-dimensional, metrizable topological space we can find for each x a clopen subset W x of M containing K x so that W x ∩ W x (cid:54) = ∅ if and only if x = x (cid:48) . By Lemma 4.2 wecan find for each x a clopen subset W x of M containing [ K x ] so that [ W x ] ∩ [ W x (cid:48) ] (cid:54) = ∅ if and only if R X ( x, x (cid:48) ) and [ W x ] ∩ f − ( a ) (cid:54) = ∅ if and only if there is x (cid:48) ∈ X with R X ( x, x (cid:48) ) and α ( x (cid:48) ) = a . Finally, since K is locally non-separating, we can chosefor every—possibly trivial—edge e = { a, a (cid:48) } of A , a clopen subset W e of M \ K sothat f ( W e ) = e .Let now h (cid:48) : M → B (cid:48) be any map in C ω which refines f as well as the partitiongenerated by all the sets W x , W x , W e collected above. Let also g (cid:48) : B (cid:48) → A be theunique map with h (cid:48) ◦ g (cid:48) = f and set D (cid:48) x be the subgraph of B (cid:48) a generated on domain h (cid:48) ( K x ) and D (cid:48) a be the subgraph of B (cid:48) a generated on dom( B (cid:48) a ) \ ( (cid:83) x dom( D (cid:48) x )). Itis easy to see that g (cid:48) ∈ C and the resulting h (cid:48) , g (cid:48) , { D (cid:48) a } , { D (cid:48) x } satisfy properties (1),(2), (3), (4) above. Moreover if (cid:101) D (cid:48) is the subgraph of B (cid:48) on domain (cid:83) a ∈ A dom( D (cid:48) a ),then g (cid:48) (cid:22) (cid:101) D (cid:48) : (cid:101) D (cid:48) → A is an epimorphism.Since locally non-separating sets are nowhere dense, for every a ∈ A we can chosea clopen set V (cid:48) a of f − ( a ) \ K so that h (cid:48) ( V (cid:48) a ) intersects every connected componentof every graph D (cid:48) x with a = α ( x ). For every a ∈ A , set W a = f − ( a ), K a = W a ∩ K , V a = (cid:0) ( h (cid:48) ) − ( (cid:83) x : α ( x )= a D (cid:48) x ) (cid:1) \ V (cid:48) a . By Lemma 4.1 we get a clopen subset U a of M with K a ⊆ U a ⊆ V a so that W a \ U a is connected. As above we can find h : M → B in C ω and g (cid:48)(cid:48) : B → B (cid:48) in C with g (cid:48)(cid:48) ◦ h = h (cid:48) , and so that h refines the partitiongenerated by { U a : a ∈ A } . Set g = g (cid:48) ◦ g (cid:48)(cid:48) and B a = g − ( a ). Let also D a be thesubgraph of B a on domain h ( W a \ U a ) and let D x be the subgraph of B a on domain( g (cid:48)(cid:48) ) − ( D (cid:48) x ) ∩ h ( U a ). Notice that all properties we established for g (cid:48) are preservedunder refinements and that g additionally satisfies properties (5) and (6). (cid:3) Given the configuration of the above claim, let E x be the subgraph of B generatedon dom( D x ) ∪ dom( D α ( x ) ). Properties (5) and (6) above imply that E x is connected.Let E (cid:48) x be an isomorphic copy of E x and let i x : E (cid:48) x → B be an embedding witnessingthis isomorphism. Let G be the mapping cylinder with respect to the map i : (cid:116) x ∈ X E (cid:48) x → B , where i = (cid:116) x ∈ X i x , and let r : G → B be the associated retraction.By projective extension property we get f : M → G with ( g ◦ r ) ◦ f = f . By COMBINATORIAL MODEL FOR THE MENGER CURVE 13 properties (3), (4), (5), (6) above, and the fact that ( f ) − ( x ) is connected forall x ∈ X , we have the map f : G → C α that maps E (cid:48) x ∪ D x to x and D a to a is in C . It is also immediate that r α ◦ f = g ◦ r . To finish the proof we set˜ f = f ◦ f . As a consequence we have r α ◦ ˜ f = r α ◦ f ◦ f = g ◦ r ◦ f = f and˜ f ( K x ) = f ◦ f ( K x ) ⊆ f (dom( E (cid:48) x ) ∪ dom( D x )) = { x } . (cid:3) We can turn now to the proof of the main theorem of this section.
Proof of Theorem 4.1.
The proof of Theorem 4.1 is a standard “back and forth”argument based on the following lifting property. Notice that the content of thelower commuting triangle is our usual projective extension property.
Lifting property for M . Let K = [ K ] be a locally non-separating subgraphof M . Let also g : B → A in C and f : M → A in C ω . Then for every graphhomomorphism p : K → B , with g ◦ p = f (cid:22) K , there is h : M → B in C ω with g ◦ h = f and h (cid:22) K = p . K B M A p ghf We are left to show that the above lifting property holds. Notice first that if g : B → A is in C and β : X → B , α : X → A , are graph homomorphisms with g ◦ β = α , then there is a unique extension g ∗ : C β → C α of g which makes the rightdiagram below commute. It is easy to check that g ∗ is in C . XB A β αg (cid:32) XC β C α B A r β g ∗ r α g Let now f, g, p be as in the statement of the Lifting Property for M and let X be a graph isomorphic to the graph that is the image of K in B under p . Letalso β : X → B be this isomorphism and let q : K → X be the unique map with β ◦ q = p . Notice that β is not an embedding—in general—but it is always aninjective homomorphism. Set α : X → A be the homomorphism g ◦ β .By Lemma 4.3 we have ˜ f : M → C α in C ω , with r α ◦ ˜ f = f and ˜ f (cid:22) K = q .Let g ∗ : C β → C α be the extension of g to C β described above. By the projectiveextension property of M we get a map ˜ h : M → C β with g ∗ ◦ ˜ h = ˜ f . It follows thatthe map h : M → B defined by r β ◦ ˜ h is the desired map. To see this notice that f = r α ◦ ˜ f = r α ◦ g ∗ ◦ ˜ h = g ◦ r β ◦ ˜ h = g ◦ h . By a similar diagram chasing, usingthat g ∗ (cid:22) X = id X and ( g ∗ ) − ( X ) = X we get that p = h (cid:22) K . (cid:3) We finish this section by showing how one can derive Anderson’s homogeneityfor the Menger curve [1] from Theorem 4.1.
Corollary 4.1 (Anderson [1]) . Any bijection between finite subsets of | M | extendsto a homeomorphism of | M | .Proof. Let φ : F → F (cid:48) be a bijection between finite subsets of | M | . If φ liftsthrough π : M → | M | to a bijection φ π between ∪ F and ∪ F (cid:48) then by Theorem4.1, φ π extends to a global automorphism φ π : M → M , and φ = π ◦ φ π ◦ π − isthe required homeomorphism extending φ . Here we used that finite subsets of M are locally non-separating, which easily follows from Lemma 4.1 and the projectiveextension property of M . Hence the proof reduces to the following claim. Claim.
For every finite subset F of | M | , there exists a homeomorphism ψ : | M | →| M | so that every element [ y ] in ψ ( F ) = { ψ ([ x ]) : [ x ] ∈ F } is a singleton (as a subsetof M ). Proof of Claim.
Let E be the equivalence relation on M defined by xEx (cid:48) if either x = x (cid:48) ; or if x (cid:48) ∈ [ x ] and [ x ] ∈ F . Let M (cid:48) = M /E , let ρ : M → M (cid:48) be the quotientmap, and let R (cid:48) be the equivalence relation on M (cid:48) , that is the push-forward of R under ρ . Since ρ is R -invariant, R (cid:48) is well defined. Notice that ρ is continuous since E is compact and hence the induced map | ρ | : | M | → M (cid:48) /R (cid:48) on the quotients is ahomeomorphism. It suffices to show that there exists an isomorphism φ : M → M (cid:48) in C ω . If so, the map π ◦ φ − ◦ ( π (cid:48) ) − ◦ | ρ | , where π (cid:48) : M (cid:48) → M (cid:48) /R (cid:48) is the quotientmap, is the desired homeomorphism ψ . Hence, by Theorem 3.1, we have to checkthat M (cid:48) (with the relation R (cid:48) ) is in C ω and that it satisfies properties (1) and (2)therein.To see that M (cid:48) is in C ω notice first that the union of any two R -connected clopensubsets of M is clopen and R -connected. Since F is finite one can easily generate abasis for the topology of M (cid:48) consisting of clopen R (cid:48) -connected sets. The rest followsfrom Proposition 2.1.We now check that M (cid:48) satisfies property (1) from Theorem 3.1. Let A ∈ C andlet n be a number strictly larger than the cardinality of F . Consider the graph δ n A which is attained by subdividing every edge of A n -times, that is, each non-trivialedge ( v, v (cid:48) ) of A is replaced a chain ( v, v ) , ( v , v ) , . . . , ( v n , v (cid:48) ) of n -many edges.Notice that for every map ( v, v (cid:48) ) (cid:55)→ γ { , . . . n } which assigns to each edge ( v, v (cid:48) )of A a number less or equal to n we define a map d γ : δ n A → A collapsing everyvertex v m with m > γ (( v, v (cid:48) )) to v (cid:48) and every vertex v m with m ≤ γ (( v, v (cid:48) )) to v .Let f : M → δ n A be any C ω map. By the choice of n , there is an assignment γ asabove so that for every edge ( v, v (cid:48) ) there is no [ x ] ∈ F with f ([ x ]) = ( v k , v k +1 ),where k = γ (( v, v (cid:48) )). The map g : M → A with g = d γ ◦ f is easily shown to pushforward through ρ to a C ω map g ρ : M (cid:48) → A .Property (2) from Theorem 3.1 is proved for M (cid:48) in a similar fashion. Let f : M (cid:48) → A in C ω and g : B → A in C . Notice that f : ρ : M → A is in C ω . We can nowconstruct the desired map h : M (cid:48) → B by relativizing the argument of the previousparagraph with respect to the constrains f and g . The claim and, therefore, alsothe corollary follow. (cid:3) COMBINATORIAL MODEL FOR THE MENGER CURVE 15 The combinatorics of universality
In Theorem 5.1 we prove for M a combinatorial analogue of a strengthened ver-sion of Anderson–Wilson’s theorem. We use this to establish a variant of Anderson–Wilson’s theorem for the Menger curve | M | ; see Corollary 5.1. Notice that the fol-lowing weak version of Corollary 5.1 already follows from the projective extensionproperty of M and Theorem 2.1. Proposition 5.1.
Every Peano curve X is the continuous surjective image of theMenger curve | M | under a continuous and connected map | h | : | M | → X .Proof. By Theorem 2.1, the space X is homeomorphic to | K | for some prespace K = lim ←− ( K n , g nm ) ∈ C ω . By the first property of Theorem 3.1 we get a connectedepimorphism h : M → K . We lift h to a connected epimorphism h : M → K byrepeated application of the second property of Theorem 3.1. Since h is a graphhomomorphism cliques in M map to cliques in K . As a consequence h induces amap | h | : M → | K | between the quotients which is easy to see that it is continuousand connected. (cid:3) To strengthen the features of the map h in Proposition 5.1 we will isolate certaincombinatorial properties of C and incorporate them in the construction of the map h above. Our arguments can be adapted to other Fra¨ıss´e classes F which satisfythe analogous properties. Definition 5.1.
Let F be a projective Fra¨ıss´e class. The projective amalgam f (cid:48) , g (cid:48) of f, g below is called structurally exact (with respect to f ), if for every B ⊆ B with f (cid:22) B in F , if we set D = ( f (cid:48) ) − ( B ) , then we have D ∈ F and g (cid:48) (cid:22) D ∈ F . D BC A g (cid:48) f (cid:48) fg We say that F has structurally exact amalgamation if every f, g as aboveadmit structurally exact amalgam. We say that F has two–sided structurallyexact amalgamation if every f, g as above admit an amalgam that is structurallyexact with respect to both f and g . Structural exactness is a natural generalization of the well studied notion of exactness . Recall that an amalgamation diagram, as in Definition 5.1, is exact iffor every b ∈ B, c ∈ C with f ( b ) = g ( c ) there is d ∈ D so that f (cid:48) ( d ) = b and g (cid:48) ( d ) = c ; see [7]. In the context of Proposition 5.1, structural exactness of C will allow us to strengthen the connectedness properties of the map h . Two–sidedstructural exactness together with the next property will additionally allow us tocontrol isomorphism type of the fibers of h . Definition 5.2.
Let F be a projective Fra¨ıss´e class. We say that F admits localrefinements if for every f : B → A in F and every embedding i : A → A , thereis g : B → A in F and an embedding j : B → B so that g ◦ j = i ◦ f . Lemma 5.1.
The class C has two–sided structurally exact amalgams and localrefinements.Proof. The amalgam provided in the proof of Theorem 1.1 is structurally exactwith respect to both f and g as well. It is also easy to check that C admits localrefinements. (cid:3) We can now prove the main theorem of this section.
Theorem 5.1.
For every K ∈ C ω there exists a connected epimorphism h : M → K which is open and satisfies the following properties: (1) for every x ∈ M there exists a collection N of clopen subsets of M , with (cid:84) N = [ x ] , so that for every N ∈ N and for every closed connected subgraph F of h ( N ) ⊂ K the subgraph h − ( F ) ∩ N of M is connected; (2) for every closed subgraph Q of K that is a clique, the subgraph h − ( Q ) of M is isomorphic to M .Proof. Fix sequences ( M n , f nm ) and ( K n , g nm ) in C with M = lim ←− ( M n , f nm ) and K =lim ←− ( K n , g nm ). We denote by f n and g n the induced maps M (cid:55)→ M n and K (cid:55)→ K n .We will first use the fact that C has structurally exact amalgams to produce map h : M → K in C ω which is open and satisfies the property (1) in the statement of thetheorem. Then, we will illustrate how to adjust the construction to additionallyfulfill property (2) of the statement. We point out that part of the argumentbelow—deriving from exactness that the map h is open—can also be found in [7].We build h as an inverse limit of a coherent sequence of maps h i : M n ( i ) → K i from C where ( n ( i ) : i ∈ N ) is some increasing sequence of natural numbers.By the first property of Theorem 3.1 we get n (0) and a connected epimorphism h : M n (0) → K . Assume now that we have defined n ( i ) and h i . Setting f = h i and g = g i +1 i in initial diagram of Definition 5.1 we get a structurally exact amalgam D , f (cid:48) : D → M n ( i ) , g (cid:48) : D → K i +1 . Using the extension property of Theorem 3.1 wefind n ( i +1) and a map p : M n ( i +1) → D such that f (cid:48) ◦ p = f n ( i +1) n ( i ) . Set h i +1 = p ◦ g (cid:48) .This finishes the induction and we therefore get a map h = lim ←− ( h i ) in C ω from M to K . Claim.
For every i ∈ N and a ∈ M n ( i ) we have that h ( f − n ( i ) ( a )) = g − i ( h i ( a )). Proof of Claim.
The non-trivial direction, h ( f − n ( i ) ( a )) ⊇ g − i ( h i ( a )), follows fromexactness of D in the inductive step above. In particular, let x = ( x , x , . . . ) ∈ K with x i = h i ( a ) and let y i = a . Then, since D is exact, there is d ∈ D with f (cid:48) ( d ) = y i and g (cid:48) ( d ) = x i +1 . Let y i +1 = d (cid:48) for any d (cid:48) ∈ p − ( d ). Continuing thisway we build inductively y = ( y , y , . . . ) ∈ M with h ( y ) = x . (cid:3) By the above claim, the fact that h i is open, and since the family of all sets ofthe form f − n ( i ) ( a ) forms a basis for the topology of M , it follows that h is open.Next we show that h satisfies Property (1) in the statement of the Theorem. Let x ∈ M and notice that for every n , the subgraph Q x,n = f n ([ x ]) of M n is a clique (ofsize at most 2). Set N = { f − n ( Q n ) : n ∈ N } and notice that by Lemma 2.1 it follows COMBINATORIAL MODEL FOR THE MENGER CURVE 17 that N is indeed a collection of clopen subsets of M with (cid:84) N = [ x ]. Let N ∈ N and let F be a closed connected subgraph of h ( N ) ⊂ K . By reparametrizing thesequences ( M n ) and ( K n ) above we can assume that N = f − ( Q ) for some clique Q in M and that n = n ( i ) = i in the definition of the sequence h i above. Set Q n = ( f n ) − ( Q ) and let F n = g n ( F ). It is immediate that F n is a connectedsubgraph of K n included in h n ( Q n ), for every n ∈ N . Let E n = h − n ( F n ) ∩ Q n .While f nm (cid:22) E n could fail to be a connected epimorphism, the following claim istrue: Claim. E n is a connected subgraph of Q n . Proof.
We prove this inductively. To run the induction we will actually need thestronger statement that h n (cid:22) E n : E n → F n is in C . Let E = h − ( F ) ∩ Q . Since Q is a clique, h (cid:22) E is a connected epimorphism from E onto F .Assume now that h n (cid:22) E n : E n → F n is in C . Since the structural exactness of D, f (cid:48) , g (cid:48) at the stage n of the construction above is stable under precomposing with p : M n +1 → D , we have that h n +1 (cid:22) ( f n +1 n ) − ( E n ) is a connected epimorphismfrom ( f n +1 n ) − ( E n ) to ( g n +1 n ) − ( F n ). Notice now that E n +1 , that was defined as h − n +1 ( F n +1 ) ∩ Q n +1 , equals h − n +1 ( F n +1 ) ∩ ( f n +1 n ) − ( E n ). Since E n +1 is the preimageof the connected set F n +1 under the connected epimorphism h n +1 (cid:22) ( f n +1 n ) − ( E n ),the map h n +1 (cid:22) E n +1 : E n +1 → F n +1 is connected as well. (cid:3) Since f nm ( E n ) = E m , the above claim implies that the inverse limit E = lim ←− ( E n , f nm (cid:22) E n )is a closed and connected subgraph of M (although not in general locally-connected),with h − ( F ) ∩ N = E . Hence indeed h satisfies the property (1) above.We finish by describing how the above construction can be modified so that h additionally satisfies property (2) of the statement. Recall from Section 3 that anytopological graph is isomorphic to M if it can be expressed as an inverse limit ofa generic sequence ( L n , t nm ). Recall also that a sequence ( L n , t nm ) is generic if is“saturated” with respect to ( A n ) and ( e n ); see the construction in Section 3.Let Q be the collection of all closed subsgraphs of K which are cliques. Fix Q ∈ Q and for each i set Q i = g i ( Q ) and L Qi = h − i ( Q i ). Notice that Q i is a cliquein K i and as a consequence g i +1 i (cid:22) Q i +1 is a connected epimorphism from Q i +1 to Q i . Hence, by assuming during the construction of h i +1 in the above that theamalgam f (cid:48) : D → M n ( i ) , g (cid:48) : D → K i +1 is two–sided structurally exact, we havethat f (cid:48) (cid:22) ( f (cid:48) ) − ( Q i +1 ) is in C , and therefore, f n ( i +1) n ( i ) (cid:22) L Qi +1 : L Qi +1 → L Qi is in C .Therefore, for every Q ∈ Q we already have that h − ( Q ) = lim ←− ( L Qn , f nm (cid:22) L Qn ) ∈ C ω .In order to arrange for h to have property (2) we need to make sure that for every Q ∈ Q the sequence ( L Qn , f nm (cid:22) L Qn ) is generic. This is done by modifying slightlythe definition of h i above. In particular, let ( A n ) and ( e n : C n → B n ) be as in theconstruction described in Section 3 and assume that for every Q ∈ Q the finitesequence ( L Qn , f nm (cid:22) L Qn ; n, m ≤ i ) has been saturated with respect to ( A n ; n ≤ i )and ( e n ; n ≤ i ). In the process of defining h i +1 , after we construct D, f (cid:48) , g (cid:48) as thetwo–sided structurally exact amalgam of h i and g i +1 i , we further refine it via a map r : D (cid:48) → D in C which makes sure that if r (cid:48) = f (cid:48) ◦ r is the map from D (cid:48) to M n ( i ) then for every Q ∈ Q we have that:(i) there exists a map in C from ( r (cid:48) ) − ( Q i +1 ) to A i +1 ;(ii) if s ∈ C is any map from ( f (cid:48) ) − ( Q i +1 ) to B i +1 then there exists d ∈ C from( r (cid:48) ) − ( Q i +1 ) to C i +1 so that s ◦ (cid:0) r (cid:22) ( r (cid:48) ) − ( Q i +1 ) (cid:1) = e i +1 ◦ d .This is easily done since the “local problems” (i) and (ii) can be turned into “globalproblems” given that C has the local refinement property, and then get solved usingfinitely many application of the amalgamation property of C .Going back to the construction of h i +1 above, we can now use the extensionproperty of M to find n ( i + 1) and a map p : M n ( i +1) → D (cid:48) such that f (cid:48) ◦ r ◦ p = f n ( i +1) n ( i ) and set h i +1 = p ◦ r ◦ g (cid:48) . (cid:3) As a corollary we get the following variant of Anderson–Wilson’s projective uni-versality theorem [2, 20]. Notice that the corresponding map in [2, 20] is shown tobe monotone , that is, preimages of points are connected. Since we are working withcompact spaces, a map is monotone if and and only if it is connected [11, p.131].Moreover, as pointed out by Gianluca Basso, the map we construct is not open.Instead we get that it is weakly locally-connected : a continuous φ : Y → X betweentopological spaces is called weakly locally-connected if Y admits a collection N of neighborhoods so that { int( N ) : N ∈ N } generates the topology of Y and for ev-ery N ∈ N , and for every closed subset Z of φ (int( N )) we have that φ − ( Z ) ∩ N isconnected. This property seems rather technical but is very useful for constructingnice sections for the map φ ; see [15]. Corollary 5.1 (see also Anderson [2], Wilson [20]) . If X is a Peano continuum,then there exists a continuous surjective map | h | : | M | → X which is connected,weakly locally-connected, and | h | − ( x ) is homeomorphic to | M | , for every x ∈ X .Proof. By Theorem 2.1, the space X is homeomorphic to | K | for some prespace K = lim ←− ( K n , g nm ) in C ω . Let h : M → K be the map provided by Theorem 5.1.Since h is an R -homomorphism, the map h induces a map | h | : | M | → | K | betweenthe quotients. It is easy to check that | h | continuous and surjective and connected.The rest follow from properties (1) and (2) of Theorem 2.1. (cid:3) The approximate projective homogeneity property
The Menger prespace M , being the projective Fra¨ıss´e limit of C , automaticallyenjoys the projective homogeneity property : for every f, g : M → A , with A ∈ C and f, g ∈ C ω , there is φ ∈ Aut( M ) with f ◦ φ = g . From this property we can naturallyderive the following approximate projective homogeneity property for theMenger curve | M | . Theorem 6.1. If γ , γ : | M | → X are continuous and connected maps from theMenger curve onto some Peano continuum X , then for every open cover V of X there is h ∈ Homeo( | M | ) so that ( γ ◦ h ) and γ are V -close, that is, ∀ y ∈ | M | ∃ V ∈ V ( γ ◦ h )( y ) , γ ( y ) ∈ V. COMBINATORIAL MODEL FOR THE MENGER CURVE 19
In other words, if we endow the space Maps ( | M | , X ), of all continuous andconnected maps from | M | onto the Peano continuum X with the compact opentopology, then the orbit of each γ ∈ Maps ( | M | , X ) under the natural action ofHomeo( | M | ) on Maps ( | M | , X ) is dense in Maps ( | M | , X ). We start with a lemma. Lemma 6.1.
Let A ∈ C and let U = { U a : a ∈ dom( A ) } be an open cover of M consisting of connected subgraphs. If U a ∩ U b (cid:54) = ∅ ⇐⇒ R A ( a, b ) , then there is u : M → A in C ω so that u − ( a ) ⊆ U a , for all a ∈ dom( A ) .Proof. First we pick for each each a ∈ dom( A ) a clopen connected subgraph W a of M , with dom( W a ) ⊆ dom( V a ), so that(2) W a ∩ W b (cid:54) = ∅ if and only if R A ( a, b ) . This can always be arranged as follows. Let f : M → B be any map in C ω , with B ∈ C , so that { f − ( b ) : b ∈ dom( B ) } refines U . Let B × Q A ∈ C be the product—see proof of Theorem 1.1—of B with the clique Q A on domain dom( A ), and let p : B × Q A → B the natural projection. Let C ∈ C be the graph attained bysubdividing every non-trivial edge of the graph B × Q A , and let r : C → B × Q A be any map which maps every vertex that came from a subdivision to either ofits two neighbors; and every vertex already in dom( B × Q A ) to itself. Clearly themap s : C → B with s = p ◦ r is in C . By the projective extension property of M —see; Theorem 3.1—we can replace f with a map f : M → C from C ω . Noticethat for the map f we can choose: for every a ∈ dom( A ), a vertex v a ∈ dom( C )with f − ( v a ) ⊆ U a , so that v a (cid:54) = v b if a (cid:54) = b ; and for every a, b ∈ dom( A ) with R A ( a, b ), a path P := P ( a, b ) in C from v a to v b , with f − ( P ) ⊆ U a (cid:83) U b , so thatthe collections of all these paths forms a “strongly pairwise disjoint” system, i.e.,if the paths P, P (cid:48) are distinct and v ∈ P , v (cid:48) ∈ P (cid:48) , with R C ( v, v (cid:48) ), then either v isan endpoint of P or v (cid:48) is an endpoint of P (cid:48) . Using this “strongly pairwise disjoint”system of paths it is easy to define the collection { W a : a ∈ dom( A ) } .Next we find clopen, connected subgraphs (cid:102) W a of M with(3) dom( W a ) ⊆ dom( (cid:102) W a ) ⊆ U a , (cid:102) W a ∩ (cid:102) W (cid:48) a = ∅ if a (cid:54) = a (cid:48) , dom( M ) = (cid:91) a dom( (cid:102) W a ) , and define the map u : M → A with u − ( a ) = (cid:102) W a . Properties (2), (3), and the factthat U a ∩ U b (cid:54) = ∅ ⇐⇒ R A ( a, b ) will then imply that this is indeed the desired map.We define (cid:102) W a as the union (cid:83) n W na of an increasing sequence of clopen subgraphsof U a . Let ( O k ) be an enumeration of a basis for the topology of dom( M ) consistingof clopen connected graphs with the property that O k ∩ U a (cid:54) = ∅ implies O k ⊆ U a forall a ∈ dom( A ) and k ∈ N . We set W a = W a , for every a ∈ dom( A ). Assume that W na has been defined for all a ∈ dom( A ), and let k ( n + 1) be the smallest naturalnumber so that O k ( n +1) ∪ ( (cid:83) a W na ) is a connected graph strictly expanding (cid:83) a W na ,if such k number exists; otherwise, let k ( n + 1) = ∞ . If k ( n + 1) ∈ N then O k ( n +1) is compact and locally-connected. Hence, O k ( n +1) \ ( (cid:83) a W na ) is the union of finitelymany clopen connected subgraphs R , . . . , R m of M . It is easy to see that for each i ≤ m there is some a ( i ) so that W na ( i ) ∪ R i is connected. Let W n +1 a be the unionof W na together with all R i with a ( i ) = a , if k ( n + 1) ∈ N ; and let W n +1 a = W na , if k ( n + 1) = ∞ . This finishes the definition of { W na : a ∈ dom( A ) } for each n ∈ N and an easy induction shows that { W na : a ∈ dom( A ) } is a disjoint collection ofclopen connected graphs with W na ⊆ U a . We are left to show that M = (cid:91) n (cid:91) a W na , since then, by compactness of dom( M ), the union along N will stabilize at somefinite n , and (cid:102) W a = (cid:83) n W na will therefore be clopen. Assume towards contradictionthat some x ∈ dom( M ) is not in the domain of the above union and let k ( x ) besuch that x ∈ O k ( x ) . It follows that(4) [ O k ( x ) ] ∩ (cid:91) n (cid:91) a W na = ∅ , since otherwise [ V k ( x ) ] ∩ (cid:83) n ≤ l (cid:83) a W na (cid:54) = ∅ for some l , implying that for each n > l , k ( n ) (cid:54) = k ( n + 1) and k ( n ) < k ( x ), which is contradictory. But then, setting X =dom( M ) \ (cid:83) n (cid:83) a dom( W na ), we have by (4) that: M = [ (cid:91) x ∈ X O k ( x ) ] (cid:91) (cid:0) (cid:91) n (cid:91) a W na (cid:1) , with [ (cid:91) x ∈ X O k ( x ) ] (cid:92) (cid:0) (cid:91) n (cid:91) a W na (cid:1) = ∅ , Contradicting that M is a connected graph. (cid:3) We can now finish the proof of Theorem 6.1.
Proof of Theorem 6.1.
By Theorem 2.1, X is homeomorphic to | K | = π K ( K ) forsome prespace K ∈ C ω . Let g : K → A be a map in C ω with A ∈ C so that { π K ( g − ( a )) | a ∈ dom( A ) } refines V . Since each π K ( g − ( a )) is a compact andconnected subset of a locally-connected space we can find connected open subsets V a ⊇ π ( g − ( a )) of | K | , with V a ∩ V b (cid:54) = ∅ if and only if R A ( a, b ), so that { V a | a ∈ dom( A ) } refines V . Let U a := ( γ ◦ π M ) − ( V a ) , U a := ( γ ◦ π M ) − ( V a ), and set U := { U a | a ∈ dom( A ) } , U := { U a | a ∈ dom( A ) } . Then U and U are opencovers of M consisting of connected graphs of M so that:(5) U a ∩ U b (cid:54) = ∅ ⇐⇒ R A ( a, b ) ⇐⇒ U a ∩ U b (cid:54) = ∅ To see that U a and U a are connected graphs, notice that, since X is a Peanocontinuum, V a is the increasing union of compact connected sets, and since γ is aconnected map, γ − ( V a ) is also the increasing union of compact connected sets.Let u and u be the maps given by applying Lemma 6.1 to the covers U and U ,respectively. By the projective homogeneity property of M there is ϕ ∈ Aut( M ) sothat u ◦ ϕ = u . Let h : | M | → | M | with h ([ x ]) = ( π M ◦ ϕ )( x ). Since ϕ ∈ Aut( M ),it follows that h is a well-defined homeomorphism of | M | . To check that this isthe desired homeomorphism, let y ∈ | M | and fix x ∈ dom( M ) with π M ( x ) = y .Set a := u ( x ) and notice that since x ∈ u − ( a ) ⊆ ( γ ◦ π M ) − ( V a ), we havethat γ ( y ) = γ ◦ π M ( x ) ∈ V a . On the other hand, h ( y ) = π M ( ϕ ( x )), and since u − ( a ) ⊆ ( γ ◦ π M ) − ( V a ), we have that ϕ ( x ) ∈ u − ( a ) ⊆ ( γ ◦ π ) − ( V a ). Hence, h ( y ) = γ ◦ π M ◦ ϕ ( x ) ∈ V a . Since { V a : a ∈ dom( A ) } refines V , we are done. (cid:3) COMBINATORIAL MODEL FOR THE MENGER CURVE 21 The n -dimensional case In this section, we consider simplicial complexes that are more general thangraphs. A simplicial complex C is a family of finite sets that is closed downwards,that is, if σ ∈ C and τ ⊂ σ then τ ∈ C . The elements σ of C are called faces ofthe simplicial complex. We set dom( C ) = ∪ C to be the domain of the simplicialcomplex. A subcomplex D of C is a simplicial complex with D ⊆ C . A simplicialmap f : B → A is a map from dom( B ) to dom( A ) with f σ ∈ A whenever σ ∈ B ,where f σ stands for the set { f ( v ) : v ∈ σ } .Let C be simplicial complex and let σ ∈ C . The dimension dim( σ ) of σ is n ≥ ( −
1) if the cardinality of σ is n + 1. We say that C is n -dimensional ifdim( σ ) ≤ n for every σ ∈ C . We briefly recall some definitions from algebraictopology. For more details see Definition 7.2 and the discussion after the proof ofTheorem 7.1. We say that C is n -connected if all homotopy groups π k ( C ) of C ,with k ≤ n , vanish. We say that it is n -acyclic if all (reduced) homology groups (cid:101) H k ( C ) of C , with k ≤ n , vanish. Similarly, a simplicial map f : B → A is called n -connected if the preimage of every n -connected subcomplex of A under f is n -connected, and it is called n -acyclic if the preimage of every n -acyclic subcomplexof A under f is n -acyclic. Since a simplicial complex A is ( − A ) (cid:54) = ∅ , a simplicial map is ( − Definition 7.1.
For every n ∈ { , , . . . } ∪ {∞} , let C n be the class of all ( n − -connected simplicial maps between finite, n -dimensional, ( n − -connected simpli-cial complexes. Similarly let (cid:101) C n be the class of all ( n − -acyclic simplicial mapsbetween finite, n -dimensional, ( n − -acyclic simplicial complexes. Theorem 7.1.
For all n as above, both C n and (cid:101) C n are projective Fra¨ıss´e. For the proof of Theorem 7.1 will need the next lemma. Let ρ be a finite set.The simplex ∆( ρ ) on ρ is the simplicial complex { σ : σ ⊆ ρ } . If C is a simplicialcomplex and ρ ∈ C then ∆( ρ ) is a subcomplex of C . Lemma 7.1. If f : B → A is a simplicial map between two finite simplicial com-plexes, then we have that: (1) f is ( n − -connected if and only if f − (∆( ρ )) is ( n − -connected forevery ρ ∈ A with dim( ρ ) ≤ n . (2) f is ( n − -acyclic if and only if f − (∆( ρ )) is ( n − -acyclic for every ρ ∈ A with dim( ρ ) ≤ n . Before we discuss the proof of Lemma 7.1 we show how it implies Theorem 7.1.
Proof of Theorem 7.1.
We just check here the projective amalgamation property.Fix n and let f : B → A and g : C → A be maps in C n . We will define the projectiveamalgam D, f (cid:48) , g (cid:48) as the n –skeleton Sk n ( B × A C ) of the simplicial pullback B × A C ,together with the canonical projection maps π B , π C . Recall that the simplicialpullback B × A C is defined on domain dom( B ) × dom( A ) dom( C ) as the simplicial complex whose faces are precicely all sets of the form σ × A τ = { ( b, c ) : b ∈ σ, c ∈ τ, f σ = gτ } , where σ ∈ B and τ ∈ C . We let D be the simplicial complex attained by B × A C after we omit all faces of dimension strictly larger than n . Let f (cid:48) = π B and g (cid:48) = π C be the projection maps ( b, c ) (cid:55)→ b and ( b, c ) (cid:55)→ c from D to B and C respectively. It is easy to check that both f (cid:48) , g (cid:48) are simplicial epimorphisms(surjective on faces). We now check that f (cid:48) : D → B is ( n − D is ( n − g (cid:48) .Let B be a ( n − B . To show that D = ( f (cid:48) ) − ( B ) is( n − f (cid:48) ) − (∆( ρ )) is ( n − ρ ∈ B . Let τ be the image of ρ under f and let ∆( τ ) thecorresponding simplex, that is a subcomplex of A . Let also C = g − (∆ τ ) andnotice that, since g ∈ C n , C is a ( n − C . Noticethat C is isomorphic to the subcomplex K of Sk n (∆( τ ) × ∆( τ ) C ) spanned bythe vertexes in graph ∗ ( g (cid:22) dom( C )) := { ( w, v ) ∈ τ × dom( C ) : g ( v ) = w } , where∆( τ ) × ∆( τ ) C is formed with respect to id : ∆( τ ) → ∆( τ ) and g (cid:22) dom( C ) : C → ∆( τ ). Now, again by Lemma 7.1(1), it is easy to see that the function ( f (cid:22) ρ ) × idfrom Sk n (∆( ρ ) × ∆( τ ) C ) to Sk n (∆( τ ) × ∆( τ ) C ) is ( n − D is simply the preimage of K under this map and K is isomorphic to C which is( n − (cid:101) C n satisfies the projective amalgamation property. (cid:3) Lemma 7.1 (1) and (2) are special cases of [19, Proposition 7.6] and [6, Corollary4.3], respectively. However, since we are dealing with finite combinatorial objects,one can provide a direct proof of Lemma 7.1. In the rest of this section we sketch thesteps for a hands-on proof Lemma 7.1 (2). The interested reader can fill the missingdetails. For Lemma 7.1 (1) recall that, by the Hurewicz Theorem, a simplicialcomplex is ( n − n ≥
2, if it is ( n − combinatorial paths and combinatorial homotopy from [9].We now recall from [18] basic notions from homology and the proceed to sketcha direct proof of Lemma 7.1 (2). Let C be a simplicial complex and let σ ∈ C .An orientation for σ is an equivalence class of expressions (cid:15) ( v , . . . , v n ), where σ = { v , . . . , v n } and (cid:15) ∈ {− , } . For n = − (cid:15) ( v , . . . , v n ) and (cid:15) (cid:48) ( v (cid:48) , . . . , v (cid:48) n ) are equivalent if for the uniquepermutation π with v i = v (cid:48) π ( i ) , we have that sgn( π ) = (cid:15)(cid:15) (cid:48) . There are preciselytwo orientations associated with each face. An oriented face (cid:42) σ in C is just anorientation for σ with σ ∈ C .The chain group C ( C ) of a complex C is the abelian group generated by ori-ented faces of C , with the relations (cid:42) σ + (cid:42) τ = 0, for any two distinct oriented faces (cid:42) σ and (cid:42) τ with σ = τ . Elements of C ( C ) are called chains . Each chain is uniquelyrepresented as a finite sum (cid:80) i (cid:42) σ i , where each (cid:42) σ i is an oriented face and, for all i, j , if σ i = σ j , then (cid:42) σ i = (cid:42) σ j . We say that (cid:42) σ i is in the chain (cid:80) i (cid:42) σ i . The empty COMBINATORIAL MODEL FOR THE MENGER CURVE 23 sum represents the identity element 0 ∈ C ( C ). An n -chain is a chain consistingentirely of n -dimensional oriented faces. A ( ≤ n ) -chain consists of oriented faceswhose dimension is less that or equal to n . The chain group is equipped with anendomorphism ∂ which is defined on the generators of C ( C ) by the following pro-cedure. If (cid:42) σ is one of the two ( − ∂ (cid:42) σ = 0. If (cid:42) σ isthe equivalence class of (cid:15) ( v , . . . , v n ) with n ≥
0, let(6) ∂ (cid:42) σ = n (cid:88) i =0 (cid:42) σ i , where (cid:42) σ i is the equivalence class of ( − i (cid:15) ( v , . . . , v i − , v i +1 , . . . , v n ). Let finally f : B → A be a simplicial map. This map induces a function f : C ( B ) → C ( A )given by the following rules. Let (cid:42) σ be an oriented face in B . If f σ has dimensionstrictly smaller than that of σ , let f ( (cid:42) σ ) = 0. If the dimensions of f σ and σ are equal and (cid:42) σ is the equivalence class of (cid:15) ( v , . . . , v n ), define f ( (cid:42) σ ) to be theequivalence class of (cid:15) ( f ( v ) , . . . , f ( v n )). One checks that f ◦ ∂ = ∂ ◦ f . We have now developed all homological prerequisites for the main definition.
Definition 7.2.
Let n ≥ ( − . A complex C will be called n -acyclic if for each ( ≤ n ) -chain ζ with ∂ζ = 0 there is a chain η with ζ = ∂η . The non-trivial direction of Lemma 7.1 (2) reduces to the following more generalstatement whose proof relies on Lemma 7.3 and Lemma 7.4
Lemma 7.2. If f : B → A is a simplicial map between finite simplicial complexesand for some l, n ∈ N we have that: (1) f is still simplicial when viewed as a map from Sk n ( B ) to Sk l ( A ) ; (2) f − (∆( σ )) is ( n − -acyclic for every σ ∈ Sk l ( A ) ;then B is ( n − -acyclic if A is ( l − -acyclic. For any simplex ∆( ρ ) on a set ρ we define the boundary Bd(∆( ρ )) of ∆( ρ ) tobe the simplicial complex ∆( ρ ) \ { ρ } . Lemma 7.3.
Let f : B → A be a simplicial map such that f − (∆( σ )) is n -acyclic,for every σ ∈ A . Let ζ be an ( ≤ n ) -chain in B such that each (cid:42) σ in ζ we have that dim( σ ) > dim( f σ ) . If ∂ζ = 0 , then there is a chain η such that ζ = ∂η .Sketch of Proof. The proof is by induction on l = max { dim( f σ ) : (cid:42) σ in ζ } . Let ζ = (cid:80) ρ ζ ρ + ζ − , where ρ varies over all l -dimensional faces of A for which there isa (cid:42) σ in ζ with f σ = ρ , and with ζ ρ collecting all such (cid:42) σ . Since0 = ∂ζ = (cid:88) ρ ∂ζ ρ + ∂ζ − and each ∂ζ ρ is a chain in f − (∆( ρ )) it follows actually that each ∂ζ ρ is a chain in f − (Bd(∆( ρ ))). By inductive hypothesis, and since ∂∂ζ ρ = 0, there exists a chain ξ ρ in f − (Bd(∆( ρ ))) with ∂ξ ρ = ∂ζ ρ . Since f − (∆( ρ )) is n -acyclic we get a chain η ρ in f − (∆( ρ )) with η ρ − ξ ρ = ∂η ρ . We have that ζ − ∂ ( (cid:88) ρ ξ ρ ) = (cid:88) ρ ξ ρ + ζ − . Since (cid:80) ρ ξ ρ + ζ − is a ( ≤ n )-chain in f − (Sk l − ( A )) with ∂ ( (cid:80) ρ ξ ρ + ζ − ) = ∂ζ − ∂∂ ( (cid:80) ρ ξ ρ ) = 0 we have, by inductive hypothesis, a chain η − with ∂η − = (cid:80) ρ ξ ρ + ζ − .Set η = (cid:80) ρ η ρ + η − . (cid:3) Lemma 7.4.
Let f : B → A be simplicial such that f − (∆( σ )) is l -acyclic for every σ ∈ A . Let (cid:42) σ and (cid:42) τ be oriented faces of B with f ( (cid:42) σ ) = f ( (cid:42) τ ) = (cid:42) ρ . If (cid:42) σ , (cid:42) τ , (cid:42) ρ havedimension l and f ( (cid:42) σ ) + f ( (cid:42) τ ) = 0 , then there is an l + 1 -chain (cid:15) in f − (∆( ρ )) and an l -chain γ in f − (Bd(∆( ρ ))) so that (cid:42) σ + (cid:42) τ = ∂(cid:15) + γ. Sketch of Proof.
The proof is by induction on l . By (6), we have that ∂ (cid:42) σ = (cid:88) ν (cid:42) σ ν and ∂ (cid:42) τ = (cid:88) ν (cid:42) τ ν , where ν varies over all ( l − ν ⊆ ρ and f ( σ ν ) = f ( τ ν ) = ν . It follows that f ( σ ν ) + f ( τ ν ) = 0 and therefore, by inductive assumption,we have that σ ν + τ ν = ∂(cid:15) ν + γ ν , for an l -chain (cid:15) ν in f − (∆( ν )) and an l − γ ν in f − (Bd(∆( ν ))). One can check now that Lemma 7.4 applies to thechain (cid:80) ν γ ν , producing an l -chain γ in f − (Bd(∆( ρ ))) with (cid:80) ν γ ν = ∂γ . Since ∂ ( (cid:42) σ + (cid:42) τ − ( (cid:80) ν (cid:15) ν + γ )) = 0 and f − (∆( ρ )) is l -acyclic, there exists an l + 1-chain (cid:15) in f − (∆( ρ )) such that (cid:42) σ + (cid:42) τ − ( (cid:88) ν (cid:15) ν + γ ) = ∂(cid:15). It follows that (cid:42) σ + (cid:42) τ = ∂(cid:15) + ( (cid:80) ν (cid:15) ν + γ ), where (cid:80) ν (cid:15) ν + γ is an l -chain in f − (Bd(∆( ρ ))), as required. (cid:3) Proof Sketch of Lemma 7.2.
Assume without loss of generality that l ≤ n and no-tice that (2) implies that for every τ ∈ Sk l ( A ), there is σ ∈ A , with f σ = τ . Let ζ B be a ( ≤ n − B with ∂ζ B . We will find a chain η B with ∂η B = ζ B . Claim.
We can assume without loss of generality that f ( ζ B ) = 0. Proof of claim.
Set ζ = f ( ζ B ). Since A is ( l − ≤ l )-chain η in A with ∂η = ζ . Set η = (cid:80) i (cid:42) τ i . Since τ i ∈ Sk l ( A ), we can find a chain η (cid:48) = (cid:80) i (cid:42) σ i in B , with dim( σ i ) = dim( τ i ) and f ( (cid:42) σ i ) = (cid:42) τ i . One can now replace ζ B with ζ B − ∂η (cid:48) which satisfies all the desired properties. Moreover, if ζ B − ∂η (cid:48) = ∂η (cid:48)(cid:48) for somecycle η (cid:48)(cid:48) , then ζ B = ∂ ( η (cid:48) + η (cid:48)(cid:48) ). (cid:3) By Lemma 7.3 we can further assume that ζ B is in fact a ( ≤ l − ζ B = (cid:88) i ( (cid:42) σ i + (cid:42) τ i ) + ζ (cid:48) , COMBINATORIAL MODEL FOR THE MENGER CURVE 25 where f ( (cid:42) σ i ) + f ( (cid:42) τ i ) = 0, dim( f σ i ) = dim( σ i ) = dim( τ i ) = dim( f τ i ) = l −
1, and ζ (cid:48) is an ( ≤ l − ρ i = f σ i = f τ i . By Lemma 7.4, for each i , there is achain (cid:15) i and a chain γ i in f − (Bd(∆( ρ i ))) such that (cid:42) σ i + (cid:42) τ i = ∂(cid:15) i + γ i . Thus, ζ B = ∂ ( (cid:88) i (cid:15) i ) + ( (cid:88) i γ i + ζ (cid:48) ) . One checks now that f ( (cid:80) i γ i + ζ (cid:48) ) is a chain in Sk l − ( A ) and the above equationimplies that ∂ ( (cid:80) i γ i + ζ (cid:48) ) = 0. By inductive assumption we can find η with ∂η =( (cid:80) i γ i + ζ (cid:48) ) and set η B = ( (cid:80) i (cid:15) i ) + η to be the required chain. (cid:3) As in Section 3, we can now construct generic sequences for C n and (cid:101) C n whoseinverse limits we denote by M n and (cid:101) M n respectively. Both M n and (cid:101) M n are compact n -dimensional simplicial complexes and as in Theorem 3.2 it is easy to see that therelation R , where xRy iff there is a face σ with x, y ∈ σ , is an equivalence relation.We let | M n | = M n /R and (cid:101) M n = (cid:101) M n /R . It follows that | M | and | (cid:101) M | are bothhomeomorphic to the Cantor space 2 N ; both | M | and | (cid:101) M | are homeomorphic tothe Menger curve | M | ; and as in Theorem 3.2 one can see that both | M n | and (cid:101) M n are Peano continua. While one expects | M n | to be the usual Menger compactum ofdimension n (see [4]), we observe that for n >
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