SSIMPLICIAL APPROXIMATION AND
COMPLEXITY GROWTH
DANIEL J. PONS
To Antonia
Abstract:
This work is motivated by two problems: 1) The approach of manifolds and spacesby triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problemsas related, new criteria and methods for approximating smooth manifolds are deduced. When thesequences of polyhedra are obtained by the action of a discrete group or semigroup, further controlis given by geometric, topologic and complexity observables. We give a set of relevant examples toillustrate the results, both in infinite and finite dimensions.
Analysis situs, an ancestor of modern topology, arose as a clandestine area of mathe-matics in the nineteenth century. Gradually it became more accepted, thanks to thework of H. Poincaré, P. Alexandrov, O. Veblen, H. Hopf, J. Alexander, A. Kolmogorov,H. Weyl, L. Brouwer, H. Whitney, W. Hodge and S. Lefschetz, among others.One of its driving forces, the approximation of shapes (and spaces) through thejuxtaposition of prisms or polyhedra, permeated to science and art, becoming essen-tial in our view of the world. From P. Picasso’s cubism to quantum gravity, humanperception seemed to accept simplices as elementary blocks to approach forms andspace.It is standard, from a mathematical perspective, to infer estimates of error, com-plexity, and changes in both complexity and error in a process of approximation, alsoto estabilish quantitative and qualitative criteria for convergence, and infer bounds forthe speed at which such a convergence (if any) occurs.This paper is motivated by those problems; we obtain, using suitable tools, resultsof this kind for evolving polyhedra on manifolds. We describe sequences of complexesassociated to coverings of spaces by open sets. Sequences of this type, consideredby P. Alexandrov (see [Ale-Pon]) under the name of projective spectra, yield, undersuitable convergence assumptions, approximations of a paracompact Hausdorff spaceup to homeomorphism.We regard the number of simplices and the dimension of each complex in thesequence as a measure of its complexity, and control its growth not only in the limit, butalso at every stage. This delivers sequences of irreducible complexes, those for whichthe excess of complexity is eliminated, say. If the space in question is a differentiable1 a r X i v : . [ m a t h . DG ] M a y anifold endowed with a Riemannian metric, those irreducible complexes, togetherwith available tools from geometric measure theory, yield a quantitative approximationas well.To perform those constructions in a systematic way, we consider actions of discretegroups and semigroups, say Γ , on complexes associated to coverings by open sets.We describe Γ -representations/actions that yield convergent sequences of complexes,to make a connection with expansive systems, or e-systems; in those systems theconvergent sequence of complexes is obtained by iteration of a suitable initial simplicialcomplex, a generator, say.If the space where Γ acts expansively is a closed Riemannian manifold, estimatesfor the minimal complexity of the generating complex are achieved. This is possiblethanks to comparison results in differential geometry.We briefly mention the contents of this work.Sections 2.1, 2.2 and 2.3 provide some notation and framework.In Section 2.4 complexity functions for simplicial complexes are proposed, and wemention their main and useful properties.Section 2.5 deals with concrete realizations of complexes in Euclidean space; thisis needed, together with the functions introduced in Section 2.4, to obtain betterapproximations of spaces when compared with those achieved by arbitrary convergentsequences (Section 2.6); this is developed in Section 2.8 both from a qualitative andquantitative perspective.In Section 2.7 increasing sequences of numbers control the complexity growth insequences of complexes constructed from finer and finer coverings, as measured by thefunctions introduced in Section 2.4, yielding a quantitative description of the processin the limit. Those growths are measured by what we call the simplicial growth up todimension k , denoted by ent k , and by the dimension growth, denoted by Dim. In factent k is a generalization of what is known as topological entropy (see [Wal]), meanwhileDim is a relative of mean dimension (see [Gro3]).Section 3 begins with a natural framework for groups and semigroups actions onspaces, usually known as Γ -spaces. We mention the natural morphisms between objectsof this type, some advantages of this perspective, to define the evolution of simplicialcomplexes in Γ -spaces, where the growths of complexity can be measured.In Section 3.1 the exponential growth of the -simplices is studied under assump-tions on Γ , to infer some quantitative control at every stage.In Section 3.2 we describe a particular type of Γ -spaces, namely Γ -spaces withproperty-e. The first remarkable issue of the expansive property, or property-e, is thatit can be characterized using either topological (set theoretic) or geometric tools. Theset theoretic characterization leads to the concept of a generator, an open cover thathas a good response to the action of Γ , say. It could be seen as a complex that underthe action of Γ evolves towards an acceptable approximation of the space. We describein which sense the evolving nerves of generating covers approximate the space, andrecall fundamental results in geometry and topology that suit our developments. Allthe results from previous Sections can be used in this scenario, and the adaptation ofthem is left to the reader. 2n Section 3.3, assuming that the space is of Riemannian type, we provide estimatesto have a better control of the generating process. Those estimates find concreteapplications in Section 5.Finally, in Section 4 and Section 5 we present some examples. Section 4 deals withinfinite dimensional examples where estimates for the simplicial growth, as measuredby the family { ent k | k ∈ N } , and the dimension growth, as measured by Dim,appear. Section 5 describes finite dimensional closed manifolds for which an expansiveaction can be constructed. Some of the examples in finite dimension are not new, andthe list of examples is far from being exhaustive nor definitive; their (not so detailed)description is included for many purposes:1. To ensure that the results of Sections 3.2 and 3.3 are non-void, enabling concreteconstructions and estimates.2. To have an idea of the methods used to construct them.3. To allow the construction of new examples from known ones.Sections 4 and 5 are not entirely independent: all the examples in Section 5 can beused in Section 4.1 to construct infinite dimensional closed manifolds with property-e. We state properties of the canonical simplicial complex associated to the covering ofa space by open sets, known as the nerve of the covering. Some statements can befound in [Ale-Pon], [Hur-Wall], [Lef], and the references therein. Other properties arenew (at least for the author), and all of them will be used in this article. If V is a compact Hausdorff space we denote by C V the set of covers of V by opensets: one calls the members of C V open covers . Remark 2.1.
Since V is compact, it suffices to identify C V with the totality of all finite covers by open sets of V to simplify. If α and β belong to C V , one says that α is finer than β if whenever A is an elementin α there exists some B in β such that A ⊆ B , and writes α (cid:31) β if that is the case.This notion induces a partial order on C V .If { α, β } ⊂ C V , one denotes by α ∩ β the refinement of α by β (or equivalentlythe refinement of β by α ): its elements are intersections of one element from α andanother from β . One can write α ∩ β := l.u.b. { γ | γ (cid:31) α, γ (cid:31) β } , where l.u.b. denotes the supremum (or least upper bound) in C V induced by (cid:31) . Some of the constructions and results are valid in more general spaces, but our intention is toprovide examples in compact manifolds, usually without a boundary. emark 2.2. One can play further with those notions and use the language of lattices,something that we give for granted.
Let α be given as { A i | i ∈ I } , where I is an indexing set (finite since V iscompact). Associated to α is a simplicial complex, known as the nerve of α , that wedenote by K ( α ) , uniquely defined up to homotopy, and whose simplices are constructedas follows: for every k in N the set of k -dimensional simplices of K ( α ) , denoted by (cid:52) k ( α ) , is given by { [ a i (0) , ..., a i ( k ) ] | k (cid:92) r =0 A i ( r ) (cid:54) = ∅ } , where for each i in I we identify the open set A i with the -simplex [ a i ] . Given α in C V , for every k we denote by |(cid:52) k ( α ) | the cardinality of (cid:52) k ( α ) , i.e. thenumber of k -simplices in K ( α ) . By those means one introduces the dimension of K ( α ) , denoted by dim K ( α ) , as the maximal k for which |(cid:52) k ( α ) | is different fromzero.For α and β in C V with α (cid:31) β there exists a simplicial map from K ( α ) to K ( β ) ,say T αβ : K ( α ) → K ( β ) , defined up to homotopy, satisfying the following properties:1. If A i ⊆ B j , then T αβ [ a i ] = [ b j ] .2. Whenever k > and σ is in (cid:52) k ( α ) , then the image of σ under T αβ is completelydetermined by the image of the -simplices making up σ : this allows the possi-bility that T αβ σ is in (cid:52) l ( β ) for some l ≤ k (for example when different verticesof σ are mapped to the same -simplex in K ( β ) ).One says that T αβ is compatible with (cid:31) . It is important to note:1. Such a map need not be unique.2. If α (cid:31) β (cid:31) γ and we have constructed two simplicial maps T αβ : K ( α ) → K ( β ) and T βγ : K ( β ) → K ( γ ) compatible with (cid:31) , then we have a simplicial map T αγ : K ( α ) → K ( γ ) given by T αγ = T βγ · T αβ that is also compatible with (cid:31) .There are open covers we distinguish for later purposes. Definition 2.3.
One says that α in C V is irreducible if no open refinement of α hasa nerve isomorphic with a proper sub-complex of K ( α ) , i.e. if there is no β finer than α that admits a strict simplicial embedding from its nerve to the nerve of α . Lemma 2.4.
Irreducible covers have the following properties:1. If α is irreducible then every member of it contains a point in V that is notcontained in other member. . If α is irreducible then whenever β (cid:31) α all the simplicial maps from K ( β ) to K ( α ) compatible with (cid:31) are surjective.3. If V is compact, then every α in C V has an irreducible refinement (one says thatirreducible covers are cofinal in ( C V , (cid:31) )) .4. If V is a manifold whose (real) dimension is equal to n , then for every irreducible α in C V one has dim K ( α ) ≤ n .Proof.
1: If α = { A i | i ∈ I } has some element, say A j , that contains no point thatis not contained in the rest of the A i ’s, then α (cid:48) = { A i | i ∈ I (cid:114) { j } } is a refinementof α and K ( α (cid:48) ) is a proper subcomplex of K ( α ) .2: This is clear from the definition.3: Let α in C V be given, and consider a sequence { α n } n ∈ N in C V so that α ≺ α ≺ α ≺ ... , where all the α n ’s are reducible, and such that the corresponding nerves forma sequence of complexes, with K ( α n +1 ) being a proper subcomplex of K ( α n ) , where α = α . By compacity of V the sequence must stop, and the last term is an irreduciblerefinement of α .4: Being V a manifold of dimension n , it suffices to prove the result on some openset homeomorphic to R n , where the statement is obviously true. To handle a better notation, given an open cover α = { A i | i ∈ I } , we denote by ∧ k ( α ) the set of injective mappings (cid:126)i : { , ..., k } → I k +1 such that ∩ kr =0 A i ( r ) (cid:54) = ∅ ,modulo permutations. By those means we identify the k -simplex [ a i (0) , ..., a i ( k ) ] with σ k(cid:126)i whenever (cid:126)i is in ∧ k ( α ) . Therefore we have a bijection between (cid:52) k ( α ) and ∧ k ( α ) .If ( G, +) is an Abelian group one identifies C k ( α, G ) with the (Abelian) group of k -chains in K ( α ) with coefficients in G , so that C k ( α, G ) := { (cid:88) (cid:126)i ∈∧ k ( α ) g (cid:126)i σ k(cid:126)i | g (cid:126)i ∈ G } . Remark 2.5.
Given a permutation of ( k + 1) letters, say ξ , we are identifying σ k(cid:126)i with σ kξ(cid:126)i = [ a ξi (0) , ..., a ξi ( k ) ] in (cid:52) k ( α ) , although in C k ( α, G ) we have σ kξ(cid:126)i = sgn ( ξ ) σ k(cid:126)i , wheresgn ( ξ ) denotes the sign of ξ . Introduce the boundary operator, denoted by ∂ , as the map that sends k -chainsto ( k − -chains in a G -linear way. Since C k ( α, G ) is generated by the elements in (cid:52) k ( α ) , it suffices to define the action of ∂ on the elements of (cid:52) k ( α ) .Thus given (cid:126)i in ∧ k ( α ) we set ∂σ k(cid:126)i := k (cid:88) r =0 ( − r σ k − (cid:126)i (cid:114) i ( r ) , σ k − (cid:126)i (cid:114) i ( r ) = [ a i (0) , ..., (cid:100) a i ( r ) , ..., a i ( k ) ] provided that σ k(cid:126)i = [ a i (0) , ..., a i ( k ) ] , where (cid:98) a meansthat a is deleted.One verifies that for every c in C k ( α, G ) one has ∂ c = ∂∂c = 0 in C k − ( α, G ) , i.e.the boundary of the boundary of every k -chain is equal to zero.Using the boundary operator one defines two subgroups of C k ( α, G ) :1. The subgroup of k -cycles , denoted by Z k ( α, G ) , and defined through Z k ( α, G ) := C k ( α, G ) ∩ { c | ∂c = 0 } .
2. The subgroup of k -boundaries , denoted by B k ( α, G ) , and defined through B k ( α, G ) := C k ( α, G ) ∩ ∂C k +1 ( α, G ) . By those means the k -th homology group of K ( α ) with coefficients in G is defined,namely H k ( α, G ) := Z k ( α, G ) B k ( α, G ) . Remark 2.6.
An algebraist would say that H k measures the inexactness of the se-quence .... → C k +1 ∂ → C k ∂ → C k − → ... A geometer/topologist would say that H k measures the amount of closed k -chainsthat are not filled in the space in question (i.e. that are not boundaries) up to bordism. Let H ∗ ( α, G ) = (cid:76) dim K ( α ) i =0 H i ( α, G ) be the graded G -module associated to the ho-mology of K ( α ) with coefficients in G. In particular if G is taken as R , one denotes by B i ( α ) := dim R H i ( α, R ) the i -th Betti number of K ( α ) . Regarding the structure of thecomplex ( C ∗ ( α, R ) , ∂ ) , one has the isomorphism C i ( α, R ) = Z i ( α, R ) (cid:76) B i − ( α, R ) .If no confussion arises we identify c i ( α ) , z i ( α ) and b i ( α ) with the real dimension of C i ( α, R ) , Z i ( α, R ) and B i ( α, R ) respectively, whence in particular B i ( α ) = z i ( α ) − b i ( α ) and c i ( α ) = z i ( α ) + b i − ( α ) follow.Using the previous nomenclature one defines χ t ( α ) := (cid:80) dim K ( α ) i =0 t i B i ( α ) , so that χ − ( α ) is the Euler-Poincaré characteristic of K ( α ) . The equalities for B i ( α ) and c i ( α ) entail that χ − ( α ) is equal to the sum (cid:80) dim K ( α ) i =0 ( − i c i ( α ) .From the definitions/constructions one has the equality c i ( α ) = |(cid:52) i ( α ) | for every i in N , therefore: Lemma 2.7.
Whenever α is in C V one has the identity χ − ( α ) = dim K ( α ) (cid:88) i =0 ( − i |(cid:52) i ( α ) | . .4 Complexity functions In this Section we define complexity functions for the simplices of an open cover on V. We infer some properties of their minimizers and some estimates for them. The nextobservation is fundamental.
Lemma 2.8.
For every k in N and α in C V the minimum of |(cid:52) k ( β ) | among those β ’s finer than α is obtained for irreducible β ’s. In particular, if α is irreducible, thenthe minimum mentioned above is obtained for α itself. The same is true for the sum (cid:80) ki =0 |(cid:52) i ( β ) | and for the dimension dim K ( β ) .Proof. Follows from 3 and 2 in Lemma 2.4, namely that irreducible covers are cofinalin the directed set ( C V , (cid:31) ) : hence if α is irreducible then for every β finer than α all the simplicial maps from the nerve of β to the nerve of α compatible with (cid:31) aresurjective.To quantify the complexity of K ( α ) , that we measure in terms of its dimension andits number of simplices, also by similar quantities in K ( β ) whenever β is finer than α ,we introduce the functions Dim K ( · ) , G k ( · ) and S k ( · ) from C V to N through:Dim K ( α ) := min β (cid:31) α dim K ( β ) , G k ( α ) := k (cid:88) i =0 |(cid:52) i ( α ) | and S k ( α ) := min β (cid:31) α G k ( β ) . From the definitions, Lemma 2.8, and the identity in Lemma 2.7 we observe:
Lemma 2.9.
For every α in C V we have1. If k is larger than zero G ( α ) ≤ G k − ( α ) ≤ G k ( α ) , max l ∈{ ,...,k } |(cid:52) l ( α ) | ≤ G k ( α ) ≤ ( k + 1) max l ∈{ ,...,k } |(cid:52) l ( α ) | , with |(cid:52) l ( α ) | ≤ (cid:18) G ( α ) l + 1 (cid:19) ≤ G ( α ) l +1 ( l + 1)! .
2. Dim K ( α ) is equal to dim K ( β ) for some irreducible β finer than α .3. S k ( α ) is equal to G k ( β ) for some irreducible β finer than α .4. The identity χ − ( α ) = 2 dim K ( α ) − (cid:88) i =0 ( − i G i ( α ) + ( − dim K ( α ) G dim K ( α ) ( α ) . .5 Euclidean realization of nerves Let α = { A i | i ∈ I } be an open cover for V . We say that a partition of unity for V is compatible with α if it satisfies the following conditions:1. (cid:80) i ∈ I x i ( v ) = 1 for every v in V .2. For every i in I we have that x i ( v ) = 0 whenever v is not in A i .Identify the -simplex [ a i ] of K ( α ) corresponding to A i with the unit vector in R | I | along the i -th direction, to denote the image of the map x : V −→ R | I | v (cid:55)→ x ( v ) = (cid:88) i ∈ I x i ( v )[ a i ] by | K ( α ) | , and call it an Euclidean realization of K ( α ) . Observe that different par-titions of unit on V compatible with α induce maps from V to R | I | that are homotopic.Sometimes we identify | K ( α ) | with a polyhedral current in R | I | . We do this asfollows: for every i in I we have the -current [ a i ] that corresponds to the pure pointmeasure supported at distance one from the origin along the i -th axis. Using theconvention of Section 2.3, for (cid:126)i in ∧ k ( α ) we identify σ k(cid:126)i with the polyhedral k -current (cid:107) σ k(cid:126)i (cid:107) ∧ −→ σ k(cid:126)i , where (cid:107) σ k(cid:126)i (cid:107) = H k (cid:120) spt σ k(cid:126)i is the k -dimensional Hausdorff measure on R | I | whose support is the convex hull of { [ a i (0) ] , ..., [ a i ( k ) ] } , meanwhile −→ σ k(cid:126)i is a k -vectorfieldof unit length tangent to such a plane (see [Fed] for all the details).Observe that the chain complex associated to | K ( α ) | is isomorphic with that de-fined in Section 2.3 for K ( α ) . The results in this Section are a simplified version, suitable for the applications in thiswork, of general results attributed to P. Alexandrov, S. Lefschetz and V. Ponomarev(see [Ale-Pon]-[Lef]). In the literature the nomenclature is not uniform: we try tounify some notions as well.Consider a sequence { α n } n ∈ N in C V with α n +1 (cid:31) α n . If K ( α ) is identified withthe simplicial complex that corresponds to the nerve of α , then for every n we have asimplicial map T n : K ( α n +1 ) → K ( α n ) compatible with (cid:31) and defined up to homotopy(see Section 2.2). Those simplicial maps can be composed inductively to get a map T mn from K ( α m ) to K ( α n ) whenever m > n in the usual way, where T mn := T n · T n +1 ··· T m − .We have an infinite sequence of simplicial complexes and mappings making up adirected set ( K ( α n ) , T n ) n ∈ N .One says that the sequence ( K ( α n ) , T n ) n ∈ N is convergent if every member of α n consists at most of a point when n goes to infinity.8s m tends to infinity we have a surjective simplicial map from K ( α m ) to K ( α n ) for every n . We naturally identify the inverse or projective limit of the directed set ( K ( α n ) , T n ) n ∈ N with the nerve of α n when n tends to infinity, that we denote by lim ← ( K ( α n ) , T n ) , to state: Proposition 2.10. (Alexandrov-Ponomarev [Ale-Pon]) Assume that the sequence ( K ( α n ) , T n ) n ∈ N is convergent. Then when n goes to infinity the nerve of α n and V are homeomorphic.Proof. We describe the projective limit of the directed set ( K ( α n ) , T n ) n ∈ N , to see thatthere is a homeomorphism between such a limit and V .Let σ := { σ n } n ∈ N be a sequence of simplices, with σ n in K ( α n ) for every n. Wesay that σ is an admissible sequence or a thread for ( K ( α n ) , T n ) n ∈ N if σ n = T mn σ m whenever m is larger than n , and say that an admissible sequence σ (cid:48) is an extension of σ if for every n the simplex σ n is a face (not necessarily a proper one) of σ (cid:48) n . If theadmissible sequence σ has no extensions other than itself, we say that it is a maximaladmissible sequence (or a maximal thread ).Provide ( K ( α n ) , T n ) n ∈ N with the following topology: given a simplex σ n in K ( α n ) for some n , a basic open set around σ n consists of all maximal admissible sequences σ (cid:48) such that σ (cid:48) n is a face of σ n . In such a way one generates a topology for the limitspace, namely the set of all maximal admissible sequences.Whenever v is a point in V we have a simplex σ n ( v ) in K ( α n ) that correspondsto all the open sets in α n to which v belongs; due to the convergence assumption wenote that σ ( v ) = { σ n ( v ) } n ∈ N is a maximal admissible sequence, and conversely, everymaximal admissible sequence in ( K ( α n ) , T n ) n ∈ N is of the form σ ( v ) for some v in V .Therefore V is isomorphic to the inverse limit of ( K ( α n ) , T n ) n ∈ N , and at this stageit is easy to see that they are homeomorphic. Remark 2.11.
Neither a metric nor a differentiable structure on V are required inProposition 2.10. Remark 2.12.
Proposition 2.10 can be refined sometimes: it might happen that forsome finite n all the elements in α n together with their intersections are contractible(see Figure 1 in Section 5). Then α n is said to be a ‘good cover’, and it is known thatin such a case K ( α n ) is homotopically equivalent to V (see [Hat] for example). One is led to consider convergent sequences of coverings to reconstruct and/orapproximate a given space up to homeomorphism in the limit. On every paracompactHausdorff space a convergent sequence can be constructed in an arbitrary way. It is ofinterest, however, to create them under some quantitative and qualitative control. Wewill see in Section 2.7 that the family of complexity functions introduced in Section2.4 are of much use for those purposes. Moreover, if we endow V with a Riemannianmetric, one can consider subsequences of complexes associated to those complexityfunctions, and have a better approximation of V in the limit (Section 2.8).9 .7 Controlling sequences Let ( K ( α n ) , T n ) n ∈ N be a sequence of nerves and simplicial mappings built up from asequence { α n } n ∈ N of open covers for V , with α n +1 (cid:31) α n . From Lemma 2.9 we knowthat if we consider the sequence { S k ( α n ) } n ∈ N of positive integers there exists, for every n in N , at least one irreducible β k,n finer than α n so that S k ( α n ) is equal to G k ( β k,n ) .Fix k and let { β k,n } n ∈ N be a sequence of irreducible covers that achieve, for each n in N , the minimum of S k ( α n ) . Since β k,n is finer than α n , then when n goes to infinitywe have, under the hypothesis of Proposition 2.10, that K ( β k,n ) is homeomorphic to V ; since β k,n is irreducible the dimension of K ( β k,n ) is equal to the dimension of V .For every i ∈ { , ..., dim V } consider the increasing sequence of positive integers {G i ( β k,n ) } n ∈ N ; each sequence goes to infinity as n increases. The next Propositionprovides a correlation between those sequences thanks to Lemma 2.9. Proposition 2.13.
Let V be a compact Hausdorff space, without a boundary, whosetopological dimension is uniform and finite. For a fixed k let { β k,n } n ∈ N be a sequenceof irreducible open covers associated to a convergent sequence ( K ( α n ) , T n ) n ∈ N . Thenas n goes to infinity we have the equality χ − ( V ) = 2 dim V − (cid:88) i =0 ( − i G i ( β k,n ) + ( − dim V G dim V ( β k,n ) . To have more control on a sequence { K ( α n ) } n ∈ N we consider strictly increasingsequences of positive real numbers, say { c ( n ) } n ∈ N , going to infinity and such that < lim inf n log S k ( α n ) c ( n ) =: ent ↓ k ( α n , c ( n )) ≤ ent ↑ k ( α n , c ( n )) := lim sup n log S k ( α n ) c ( n ) < ∞ . If the sequence { c ( n ) } n ∈ N satisfies those estimates, we say that it controls thesimplicial growth of { K ( α n ) } n ∈ N up to dimension k . If lim n log S k ( α n ) /c ( n ) exists,then ent ↓ k ( α n , c ( n )) = ent ↑ k ( α n , c ( n )) =: ent k ( α n , c ( n )) . Similarly, if < lim inf n Dim K ( α n ) c ( n ) =: Dim ↓ ( α n , c ( n )) ≤ Dim ↑ ( α n , c ( n )) := lim sup n Dim K ( α n ) c ( n ) < ∞ , we say that { c ( n ) } n ∈ N controls the dimension growth of { K ( α n ) } n ∈ N .Of course if lim n Dim K ( α n ) /c ( n ) exists, thenDim ↓ ( α n , c ( n )) = Dim ↑ ( α n , c ( n )) =: Dim ( α n , c ( n )) . Using Lemma 2.9 we deduce:
Theorem 1.
Assume that { c ( n ) } n ∈ N controls the simplicial growth of { K ( α n ) } n ∈ N upto dimension k for some finite k . Then { c ( n ) } n ∈ N is a controlling sequence for thegrowth of simplices of { K ( α n ) } n ∈ N up to dimension k for every finite k . roof. From Lemma 2.9 we see that G ( α n ) ≤ G k ( α n ) ≤ ( k + 1) max l ∈{ ,...,k } G ( α n ) l +1 ( l + 1)! , therefore we haveent ↓ ( α n , c ( n )) ≤ ent ↓ k ( α n , c ( n )) ≤ ( k + 1) ent ↓ ( α n , c ( n )) , and similarly ent ↑ ( α n , c ( n )) ≤ ent ↑ k ( α n , c ( n )) ≤ ( k + 1) ent ↑ ( α n , c ( n )) . By a simple interpolation we deduce that ent ↓ l ( α n , c ( n )) and ent ↓ k ( α n , c ( n )) arecomparable if both k and l are finite, and similarly for ent ↑ l ( α n , c ( n )) and ent ↑ k ( α n , c ( n )) .Natural choices for controlling sequences are:1. c ( n ) = n , and then (in the strict sense)(a) The simplicial growth is of exponential type.(b) The dimension growth is of linear type.2. c ( n ) = log n , and then(a) The simplicial growth is of polynomial type.(b) The dimension growth is of logarithmic type.Of course there are other possibilities.Theorem 1 says that the order of the simplicial growth up to dimension k in asequence ( K ( α n ) , T n ) n ∈ N is comparable to the order of growth of -simplices if k isfinite or V is finite dimensional.On the other hand, observe that a necessary condition for { K ( α n ) } n ∈ N to have asequence controlling its dimension growth is that the underlying space V must haveinfinite topological dimension, i.e. the supremum of Dim K ( α ) as α varies in C V mustbe unbounded; this condition is also sufficient if { K ( α n ) } n ∈ N is convergent. Remark 2.14.
We understand that { n } n ∈ N is the standard sequence; due to that wewill omit c ( n ) from the expressions whenever such a sequence is used. Now V is a smooth closed manifold, provided with a distance function arising fromsome Riemannian metric g , say d ≡ d g . Then the convergence of ( K ( α n ) , T n ) n ∈ N isequivalent to the statement that all the members of α n have a diameter that goes tozero as n goes to infinity, where we assume that α n +1 (cid:31) α n for every n .11s in Section 2.7 consider, for every k and n , an irreducible cover β k,n finer than α n so that S k ( α n ) is equal to G k ( β k,n ) . Then for each n we have a simplicial embeddingfrom the nerve of β k,n to the nerve of α n , but there is no guarantee that β k,n +1 (cid:31) β k,n ,nor that the members of β k,n are contractible.But we can do better; since the diameter of the members of α n are decreasing as n increases, then for every n there exists some m large enough such that every memberof the irreducible cover β k,n + m has a diameter smaller than some Lebesgue number of β k,n , yielding a surjective simplicial map T ( k ) n + mn : K ( β k,n + m ) → K ( β k,n ) .Assume that k is fixed: we have a subsequence { β k,φ ( n ) } n ∈ N ≡ { β φ ( n ) } n ∈ N of { β k,n } n ∈ N ≡ { β n } n ∈ N made up of irreducible covers and endowed with surjective simpli-cial maps T φ ( n ) : K ( β φ ( n +1) ) → K ( β φ ( n ) ) making up a directed set { K ( β φ ( n ) ) , T φ ( n ) } n ∈ N .We recover Proposition 2.10 for the projective limit lim ← ( K ( β φ ( n ) ) , T φ ( n ) ) , although with better quantitative control. This is due to the results in Section 2.7,and because the dimension of K ( β φ ( n ) ) is bounded by the dimension of V at everystage (Lemma 2.4).If (cid:15) ( φ ( n )) is the largest diameter of a member in β φ ( n ) = { B i | i ∈ I ( φ ( n )) } , weassume that n is large enough so that (cid:15) ( φ ( n )) is smaller than the injectivity radius of ( V, g ) . Then for every i in I ( φ ( n )) we can choose some b i in B i such that d ( b i , b j ) ≤ (cid:15) ( φ ( n )) whenever B i ∩ B j (cid:54) = ∅ , and identify b i with the -simplex [ b i ] that correspondsto B i .Let x φ ( n ) : V → | K ( β φ ( n ) ) | be an Euclidean realization of K ( β φ ( n ) ) (Section 2.5).Embed the -simplices of | K ( β φ ( n ) ) | in V using a Lipschitz map y φ ( n ) so that theimage of the -simplex [ b i , b j ] corresponds to the distance minimizing path or geodesicbetween the points b i and b j , that we regard as a rectifiable path (or current) in V .We observe (see [Gro2]): Lemma 2.15.
Endow the set of -simplices in | K ( β φ ( n ) ) | , namely (cid:52) ( | K ( β φ ( n ) ) | ) , withthe distance induced by the embedding y φ ( n ) of the -simplices in ( V, d ) , extending itto all (cid:52) ( | K ( β φ ( n ) ) | ) in the natural way; denote such a distance by d φ ( n ) . Then, as n goes to infinity, the metric space ( (cid:52) ( | K ( β φ ( n ) ) | ) , d φ ( n ) ) converges to ( V, d ) in theGromov-Hausdorff sense. Consider D ∗ ( V ) , the graded Z -module of general currents on V , and for every α in C V let P ∗ ( | K ( α ) | ) denote the graded Z -module of polyhedral currents on theEuclidean realization of K ( α ) . Hence if y : | K ( α ) | → V is of Lipschitz type, then weget a linear map y (cid:93) : P ∗ ( | K ( α ) | ) → D ∗ ( V ) . Let M ≡ M g be the mass norm on D ∗ ( V ) induced by the Riemannian metric g . Afundamental fact, to be found in [Fed], asserts that the closure in D ∗ ( V ) with respectto M of pushforwarded polyhedral currents by Lipschitz maps into V is R ∗ ( V ) , the Z -module of rectifiable currents on V . An important sub-module of R ∗ ( V ) , denotedby I ∗ ( V ) , is the Z -module of integral currents; it consists of rectifiable currents whose See [Fed] for more details. V and elementary constructionsfrom geometric measure theory give: Proposition 2.16. R ∗ ( V ) and I ∗ ( V ) depend on the Lipschitz structure on V chosen. Embed now the -simplices of | K ( β φ ( n ) ) | in V by means of a Lipschitz map y φ ( n ) using the geodesics that correspond to the -simplices as their boundary, straighteningthem as much as possible, so that in the process their mass (with respect to g ) tendsto minimize.Then proceed inductively to get, for every n and each j not bigger than the di-mension of V , a Lipschitz embedding y jφ ( n ) : (cid:52) j ( | K ( β φ ( n ) ) | ) (cid:44) → V of the j -simplices inthe Euclidean realization of K ( β φ ( v ) ) inside V , all whose images have a diameter notlarger than (cid:15) ( φ ( n )) ; we obtain a map at the level of currents y φ ( n ) (cid:93) : P ∗ ( | K ( β φ ( n ) ) | ) (cid:44) → I ∗ ( V ) ⊂ R ∗ ( V ) . More precisely , if (cid:126)i is in ∧ j ( β φ ( n ) ) , then y φ ( n ) (cid:93) σ j(cid:126)i is almost minimal among thoserectifiable currents whose boundary is ∂y φ ( n ) (cid:93) σ j(cid:126)i = y φ ( n ) (cid:93) ∂σ j(cid:126)i , for every j ≤ dim V . Byalmost minimal we mean that the minimum of mass might not occur, however thereis a sequence of Lipschitz maps leading to an infimum.This process of approximation gives: Theorem 2.
Let ( V, g ) be a smooth closed Riemannian manifold. Let { α n } n ∈ N be asequence in C V , with α n +1 (cid:31) α n , and such that the diameter of each member of α n goesto zero as n goes to infinity. Then for every k and every positive (cid:15) there exists some m ≡ m ( (cid:15) ) , a cover β k,m minimizing S k ( α m ) , and a Lipschitz map y m : | K ( β k,m ) | → V such that M g ( y m(cid:93) | K ( β k,m ) | − V ) < (cid:15), where V = (cid:107)V(cid:107) ∧ (cid:126) V is the current representing V , and | K ( β k,m ) | is the polyhedralcurrent that corresponds to an Euclidean realization of K ( β k,m ) . If the dimension of V is not , such a map is independent of the Lipschitz structure on V . Remark 2.17.
We know, thanks to the work of E. Moise, S. Donaldson and D. Sulli-van, that only in dimension there are smooth manifolds that are homeomorphic butnot Lipschitz equivalent. Γ -spaces We denote by Γ a countable or discrete group or semigroup whose cardinality is ℵ .Let ρ : Γ → Map ( V, V ) be a representation of Γ on the set of mappings of V , wherewe understand, if Γ is a group, that ρ (Γ) is a sub-group of Homeo ( V ) , the group ofhomeomorphisms on V . If Γ is a semigroup, then ρ (Γ) is a sub-semigroup of End ( V ) ,the semigroup of endomorphisms on V . We denote a structure of this type by a tuple ( V, Γ , ρ ) , and speak of a system, a representation of Γ , or a Γ -space. Visit Section 2.5, if needed, for the nomenclature used. emark 3.1. We restrict to discrete groups and semigroups since:1.
Some of the results are not valid for arbitrary Γ ’s.2. All the examples in this article belong to this class.
We identify the totality of those structures with the objects in the category of Γ -spaces. The standard morphisms between objects in this category are:1- Conjugations: Two systems ( V, Γ , ρ ) and ( W, Γ , ρ (cid:48) ) are said to be conjugated if there exists a homeomorphism x : V → W that interwinds the action of Γ , i.e. ahomeomorphism between V and W that is Γ -equivariant. The notion of being conju-gated is a strong equivalence relation; not only the underlying (topological-geometric)spaces in question are homeomorphic, moreover the dynamics induced by the mapsare, up to a continuous change of coordinates say, equivalent.2- Factors/Extensions: ( V, Γ , ρ ) is said to be an extension of ( W, Γ , ρ (cid:48) ) , or ( W, Γ , ρ (cid:48) ) is said to be a factor of ( V, Γ , ρ ) , if there is a continuous surjection x : V → W , sothat whenever γ is in Γ we have x · ρ ( γ ) = ρ (cid:48) ( γ ) · x . Remark 3.2.
Of course if ( V, Γ , ρ ) is both a factor and an extension of ( W, Γ , ρ (cid:48) ) ,then both systems are conjugated. Remark 3.3.
The advantage of using the language of categories in this context is thatsome operations of algebraic topology (for example the loop functor, the suspensionfunctor and the smash product) can be used as self-functors. By those means oneobtains new systems from known ones (see Section 5.3.3).
Consider the action of ρ (Γ) ’s inverses on elements of C V . If γ is in Γ we have amap ρ ( γ ) : C V → C V and also an induced map ρ ( γ ) − : C V → C V , where in the case ofsemigroups we understand that ρ ( γ ) − A is given by V ∩ { v | ρ ( γ ) v ∈ A } for everysubset A of V . Whenever F is a finite subset of Γ and α is in C V we set α F := (cid:92) γ ∈ F ρ ( γ ) − α. In what follows we describe K ( α F ) as F increases both from a quantitative and aqualitative perspective. Assume that Γ is generated by a finite subset of elements, say H , where we assume, if Γ is a group, that H contains all its inverses, i.e. that H = H − . Then whenever F isa finite subset of Γ we define its boundary with respect to H , that we denote by ∂ H F ,as the subset of F made up of those γ ’s such that hγ is not in F for some h in H .Consider an increasing sequence of subsets exhausting Γ , say { F ( n ) } n ∈ N . Such asequence is said to be of Følner type if the quotient between | ∂ H F ( n ) | and | F ( n ) | goesto zero as n goes to infinty. If such a sequence exists, then Γ is said to be amenable (see [Gro2] for more about this). 14 emark 3.4. Since all the results in Section 2.7 are valid in this context, we willnot repeat analogous statements unless this is relevant; one should replace c ( n ) by c ( | F ( n ) | ) , and α n by (cid:92) γ ∈ F ( n ) ρ ( γ ) − α, for example. We will write ent k ( α, Γ , ρ, c ) instead of ent k ( α n , c ( n )) in what follows,and similarly for Dim ( α, Γ , ρ, c ) . Consider the standard sequence { c ( n ) } n ∈ N = { n } n ∈ N (see Remarks 2.14 and 3.4).The next Lemma asserts that the upper and lower limits in Theorem 1 for ent coincide. Lemma 3.5.
Let { F ( n ) } n ∈ N be a Følner sequence for Γ . Then the following limitexists, and is independent of the Følner sequence:For every open cover α lim n log S ( α F ( n ) ) | F ( n ) | =: ent ( α, Γ , ρ ) . Proof.
The proof follows from a general convergence result for subadditive functions,known as the Orstein-Weiss Lemma; a proof can be found in [Gro3]. To use such aresult it suffices to note that (see [Wal], for example) |(cid:52) ( α ∩ β ) | = G ( α ∩ β ) ≤ G ( α ) G ( β ) , hence S ( α ∩ β ) ≤ S ( α ) S ( β ) . Remark 3.6.
If the growth of the number of simplices in { K ( α F ( n ) ) } n ∈ N is strictlyexponential, then ( V, Γ , ρ ) is said to have non zero finite topological entropy with respectto the cover α . The supremum of ent ( α, Γ , ρ ) among all α ’s in C V , that might not befinite, is known as the topological entropy for ( V, Γ , ρ ) , and denoted by ent ( V, Γ , ρ ) . Recall some constructions/results in Section 2.7: for a fixed k there exists a se-quence { β k,F ( n ) } n ∈ N of irreducible covers that achieve, for each n , the minimum of S k ( α F ( n ) ) . In particular, by Lemma 3.5 we have that ent ( α, Γ , ρ ) is given by lim n log S ( α F ( n ) ) | F ( n ) | , hence ent ( α, Γ , ρ ) = lim n log G ( β ,F ( n ) ) | F ( n ) | . On the other hand, since S ( α F ( n ) ) = S ( α F ( n − ∩ (cid:92) γ ∈ F ( n ) \ F ( n − ρ ( γ ) − α )
15e observe that G ( β ,F ( n ) ) = S ( α F ( n ) ) ≤ G ( β ,F ( n − ) S ( α ) | F ( n ) \ F ( n − | , where we recall that S ( α ) is the best lower bound for G ( β ) among those β ’s finerthan α , to infer by recursion: Proposition 3.7.
For every α the number of 0-simplices in the sequence { K ( β ,F ( n ) ) } n ∈ N is bounded at every stage by G ( β ,F ( n ) ) ≤ S ( α ) | F ( n ) | . In particular, Theorem 1 ensures that for each k we have the boundent ↑ k ( α, Γ , ρ ) ≤ ( k + 1) log S ( α ) . Remark 3.8.
The sensibility of the simplicial growth with respect to the initial con-dition α can be grasped thanks to Proposition 3.7. Indeed, for every n and k we have S k ( α F ( n ) ) ≤ S ( α ) ( k +1) | F ( n ) | . Let ρ : Γ → Map ( V, V ) be a representation of the group (or semigroup) Γ acting on V ,and choose some metric d V on V . We say ( V, Γ , ρ ) is expansive , or has property-e ,if there exists a constant (cid:15) strictly larger than zero, such that for every u differentfrom v there exists some γ in Γ with ρ ( γ ) ∗ d V ( u, v ) := d V ( ρ ( γ ) u, ρ ( γ ) v ) larger than (cid:15) . Remark 3.9.
Every (cid:15) that satisfies the condition given before is called an e-constantfor ( V, Γ , ρ ) . The e-constants depend on the given metric d V , but the existence of thoseconstants does not (see Lemma 3.10), and therefore one can omit the metric and saythat (Γ , ρ ) acts expansively on V, or that ( V, Γ , ρ ) has property-e. Given α = { A i | i ∈ I } in C V we say that it is a generator for ( V, Γ , ρ ) , or agenerating cover, if for every array { i ( γ ) | γ ∈ Γ } in I the intersection (cid:92) γ ∈ Γ ρ ( γ ) − A i ( γ ) contains at most one point. Thus if α is a generator for ( V, Γ , ρ ) then the maximum ofthe diameter of all the open sets making up α F decreases as F increases, and it goesto zero as F exhausts Γ .The next Lemma provides a rough relation between the e-constant and the diameterof a generator (see [Wal]). Lemma 3.10.
Assume that ( V, Γ , ρ ) is expansive. Let (cid:15) be some e-constant for somemetric on V , say d V . If α = { A i | i ∈ I } ∈ C V is such that the diameter of each A i is at most (cid:15) , then α is a generator for ( V, Γ , ρ ) . roof. Assume that we have a pair of points ( u, v ) in V that belong to (cid:84) γ ∈ F ρ ( γ ) − A i for some i in I, this for every subset F of Γ . Then for every γ in F the distancebetween ρ ( γ ) u and ρ ( γ ) v is at most (cid:15) , the diameter of A i ; since (cid:15) is an e-constant then u and v coincide, and α is a generator. Remark 3.11.
In Theorem 4 we will see that it is natural to estimate, for a fixedmetric d V on V , the largest e-constant. In Corollary 3.17, an upper bound will beprovided if d = d g for a Riemannian metric g that is regular enough. Our next aim is to observe that a generator should be considered as being a goodinitial condition to reconstruct the skeleton of V using (Γ , ρ ) ; in other words, a gen-erator generates a simplicial complex that is an acceptable approximation of thespace.For those purposes let { F ( n ) } n ∈ N be an increasing sequence exhausting Γ , notnecessarily amenable. If α is a generator for ( V, Γ , ρ ) then we have an infinite sequenceof simplicial complexes and mappings making up a directed set ( K ( α F ( n ) ) , T n ) n ∈ N thatis convergent in the sense of Section 2.6. From Proposition 2.10 and Theorem 2 weinfer: Theorem 3.
Assume that ( V, Γ , ρ ) has property-e, that α is a generator for ( V, Γ , ρ ) .Then1. When F exhausts Γ the complex K ( α F ) and V are homeomorphic.2. If ( V, g ) is Riemannian, closed and smooth, then for every positive (cid:15) and everyinteger k we can find a subset F ≡ F ( (cid:15) ) of Γ such that the minimizer β k,F of S k ( α F ) has the following property: there exists a Lipschitz map sending the poly-hedral current corresponding to an Euclidean realization of K ( β k,F ) into D ∗ ( V ) ,leaving it at a distance not bigger than (cid:15) , with respect to M g , from the current V associated to V . If the dimension of V is not , then the map is independentof the Lipschitz structure on V . As mentioned in Section 3, the notion of conjugacy is an equivalence relation inthe category of Γ -spaces. It is natural to expect that property-e, having both a metricand a set-theoretic characterization, will be invariant under conjugation. The nextLemma confirms this is the case: Lemma 3.12.
Assume that ( V, Γ , ρ ) has property-e, and let x : V → W be a homeo-morphism between V and W. Then ( W, Γ , ρ (cid:48) ) is also expansive, where ρ (cid:48) , the inducedrepresentation of Γ in W , is given by ρ (cid:48) ( γ ) = x · ρ ( γ ) · x − for every γ in Γ .Proof. We use the set theoretic characterization of property-e. Assume that α = { A i | i ∈ I } is a generator for ( V, Γ , ρ ) , and let α (cid:48) = { A (cid:48) i = x ( A i ) | i ∈ I } be its imagein C W . If { i ( γ ) | γ ∈ Γ } is an array in I indexed by Γ , then (cid:84) γ ∈ Γ ρ ( γ ) − A i ( γ ) consistsat most of one point, hence so does its image under x . Since x is a homeomorphism x ( (cid:92) γ ∈ Γ ρ ( γ ) − A i ( γ ) ) = (cid:92) γ ∈ Γ x · ρ ( γ ) − · x − · x ( A i ( γ ) ) = (cid:92) γ ∈ Γ ρ (cid:48) ( γ ) − A (cid:48) i ( γ ) , and we conclude that α (cid:48) is a generator for ( W, Γ , ρ (cid:48) ) .17 emark 3.13. From Theorem 3 and Lemma 3.12 we infer that if ( V, Γ , ρ ) and ( W, Γ , ρ (cid:48) ) are conjugated and if we know that one of them is expansive, then the simplicial com-plexes associated to the nerve of any of their generators evolve to complexes that arehomeomorphic. This cannot be strengthened to differentiable mappings in full gener-ality (see also Remark 2.17).One counterexample is provided by expansive actions of N d in S d (see Corollary5.2). If d ≥ the work of M. Kervaire and J. Milnor yields a finite number ofhomeomorphic but not diffeomorphic S d ’s.Other counterexamples are obtained if one glues these spheres, by connected sum,on manifolds that admit an expansive action (see Section 5); to the author’s knowledge,this was first done by T. Farrel and L. Jones for Anosov diffeomorphisms on T d (see[Fa-Jo]). In what follows we consider a Riemannian manifold, compact and without a boundary,whose dimension is d , and whose Riemannian metric g is at least of type C . If Rc ( g ) denotes the Ricci tensor of g , we denote by λ the biggest real number so that Rc ( g ) ≥ λ ( d − g all over V . Let d g be the distance function induced by g ; we denote by (cid:107) B R ( v ) (cid:107) g themass (or volume) of the ball of radius R centered at v whenever R is a positive realnumber and v is some point in V .Let S d ( λ ) be the simply connected space of dimension d whose sectional curvatureis everywhere equal to λ ; then (cid:107) (cid:102) B R (cid:107) λ denotes the mass of any ball of radius R in S d ( λ ) . If D denotes the diameter of ( V, g ) then by Bishop’s comparison (see [Be] or[Gro2] for example) whenever R ≤ D and t ≤ one has, for every point v in V , theestimate (cid:107) B R ( v ) (cid:107) g / (cid:107) B tR ( v ) (cid:107) g ≤ (cid:107) (cid:102) B R (cid:107) λ / (cid:107) (cid:103) B tR (cid:107) λ . Let C VR denote the minimal number of balls of radius R needed to cover ( V, g ) ; itis not difficult to see that such a number of balls is not bigger than the largest valueof (cid:107) B D ( v ) (cid:107) g / (cid:107) B R/ ( v ) (cid:107) g as v varies in V . Recalling Lemma 3.10 we conclude: Theorem 4.
Assume that ( V, Γ , ρ ) has property-e, and let (cid:15) = (cid:15) ( g ) be an expansivityconstant for the distance function d g induced by g . If Rc ( g ) ≥ λ ( d − g, where d isthe dimension of V , and the diameter of ( V, g ) is D , then there exists a generator for ( V, Γ , ρ ) whose cardinality is at most (cid:107) (cid:102) B D (cid:107) λ / (cid:107) (cid:103) B (cid:15)/ (cid:107) λ . Abbreviate (cid:107) (cid:102) B D (cid:107) λ / (cid:107) (cid:103) B (cid:15)/ (cid:107) λ by Θ( λ, D, (cid:15) )( g ) ≡ Θ( λ ( g ) , D ( g ) , (cid:15) ( g )) , and observethat Θ( λ, D, (cid:15) )( g ) is invariant under scalings of g . In particular, if λ is larger thanzero, then Θ( λ, D, (cid:15) ) is not bigger than Θ( λ, (cid:113) π ( d − λ , (cid:15) ) by Bonnet-Myers’ comparison(see [Be] or [Gro2] again).It becomes clear that, for a fixed g , better estimates for the expansivity constant (cid:15) ( g ) will improve the upper estimate for the minimal number of zero simplices that the18erve associated to a generator for ( V, Γ , ρ ) can have; we denote that minimal numberby |(cid:52) ( V, Γ , ρ ) | . The next definition provides an upper bound for |(cid:52) ( V, Γ , ρ ) | . Definition 3.14.
The real number Θ( V, Γ , ρ ) is defined as the best lower bound for Θ( λ, D, (cid:15) )( g ) as g varies within the Riemannian metrics on V of type C . To obtain a lower bound for |(cid:52) ( V, Γ , ρ ) | , consider the Følner sequence { F ( n ) } n ∈ N exhausting Γ . Let ζ ∈ C V have δ = δ ( g ) as a Lebesgue number (for some metric d g on V ); if α is a generator for ( V, Γ , ρ ) then for t large enough the open cover α F ( t ) has adiameter not bigger than δ , hence α F ( t ) is finer than ζ , therefore by the definition of S (see Section 2.4) the estimateent ( ζ, Γ , ρ ) ≤ ent ( α F ( t ) , Γ , ρ ) = ent ( α, Γ , ρ ) follows, hence: Lemma 3.15. If ( V, Γ , ρ ) has property-e, then the supremum of ent ( ζ, Γ , ρ ) as ζ variesin C V is a maximum, denoted by ent ( V, Γ , ρ ) , and is attained when ζ is a generatingcover. From Theorem 4, Definition 3.14, Proposition 3.7 and Lemma 3.5 we get a relationbetween Θ( V, Γ , ρ ) , |(cid:52) ( V, Γ , ρ ) | and ent ( V, Γ , ρ ) . Theorem 5.
Assume that ( V, Γ , ρ ) has property-e. Then e ent ( V, Γ ,ρ ) ≤ |(cid:52) ( V, Γ , ρ ) | ≤ Θ( V, Γ , ρ ) . Remark 3.16.
The sense and value of the Theorems for e-systems depend on the tasteof the reader:1. If she/he is interested in constructing V starting from a simplicial complex oflower complexity, it has been proved in Theorem 3 that a method to achieve sucha task is to find a group or semigroup Γ together with a representation ρ suchthat ( V, Γ , ρ ) has the expansive property.2. Once Γ and ρ have been found, bounds on the -simplices of the initial complexare provided by Theorem 5, assuming that ent ( V, Γ , ρ ) and/or Θ( V, Γ , ρ ) havebeen computed.3. From another perspective, if upper bounds for Θ( V, Γ , ρ ) are available, she/hehas upper bounds for ent ( V, Γ , ρ ) , the rate at which the approximation of V isachieved, and conversely. Recall that the e-constant is the minimal distance that any two points in V becomeseparated under the action of (Γ , ρ ) for a given distance function (not necessarily arisingfrom a Riemannian metric) on V , and it is of interest to know its order of magnitude(how large it is). To achieve that, observe that if λ and d ≥ are fixed, the function (cid:107) (cid:102) B R (cid:107) λ is a strictly increasing function of R .Using Theorems 4 and 5 we can state:19 orollary 3.17. Let g be a metric of type C , and let λ be the biggest real numbersatisfying Rc ( g ) ≥ λ ( d − g all over V . Then every (cid:15) such that (cid:107) (cid:103) B (cid:15)/ (cid:107) λ ≤ (cid:107) (cid:102) B D (cid:107) λ exp( ent ( V, Γ , ρ )) is an e-constant for the system ( V, Γ , ρ ) with respect to the distance d g . Remark 3.18.
If some e-constant (cid:15) for a distance function d = d g has been found,from Lemma 3.10 every cover whose components have a diameter at most (cid:15) is a gen-erator. An advantage of choosing the biggest e-constant allowed by Corollary 3.17 isthat the the intersection pattern of covers with larger diameter becomes simpler, hencethe associated simplicial complex is of lower complexity. This gives better upper boundsfor the simplicial growth in every dimension at every stage, as predicted by Proposition3.7. Remark 3.19.
Being property-e invariant under homeomorphisms (or conjugation)of Γ -systems, then so are the numbers associated to ent ( V, Γ , ρ ) and the minimalcomplexity that a generator can have. In contrast, if we intersect Remark 3.13 withN. Hitchin’s result asserting that in every dimension bigger than and equal to either mod or mod there are exotic spheres not admitting metrics of positive scalarcurvature (see [Law-Mi] ), then the estimate in Corollary 3.17 becomes more intriguing(after normalization of volume or diameter, say), despite its simplicity. Γ -shifths Consider a compact finite dimensional and connected manifold, say V , and infinitelymany copies of it indexed by a discrete amenable group Γ . Hence the total space is V := V Γ . Endow V with the weakest topology that makes the projection in all thecopies of V continuous; then by Tychonov’s Lemma the space V is compact for thistopology. Moreover, the set of all open covers for V , namely C V , coincides with ( C V ) Γ .Elements in V are maps v : Γ → V , thus the natural representation of Γ on Aut ( V ) is given, for each v in V and every pair { γ, γ (cid:48) } in Γ , by ( ρ ( γ (cid:48) ) v )( γ ) = v ( γ (cid:48) γ ) . A systematic study of Γ -spaces of this type, of Γ -invariant subsets of them (called Γ -subshifts), was presented in [Gro3]. Estimates on the complexity growth of thosesystems in the spirit of Dim ( V , Γ , ρ ) are provided in [Gro3], but with an emphasis onmetric and (co)homological observables. Instead of explaining results from [Gro3] (aninteresting task indeed !) we obtain, for some (cid:126)α in C V , estimates for Dim ( (cid:126)α, Γ , ρ ) andfor the family { ent k ( (cid:126)α, Γ , ρ ) | k ∈ N } . Thanks to Professor T. Friedrich for some key points on the subject. (cid:126)α = (cid:81) Γ α ( γ ) in C V = ( C V ) Γ . Then (cid:126)α ( γ ) denotes the component of (cid:126)α indexed by the coordinate γ, namely α ( γ ) . Natural constructions attributed toKünneth for Cartesian products of simplicial complexes (see [Hat] or [Lef], for example)ensure that for every k the set (cid:52) k ( (cid:126)α ) of k -simplices in K ( (cid:126)α ) is decomposed in productsof simplices in each of the K ( α ( γ )) as γ varies, i.e. (cid:52) k ( (cid:126)α ) = (cid:97) { (cid:126)k | | (cid:126)k | = k } (cid:89) γ ∈ Γ (cid:52) k ( γ ) ( α ( γ )) , where the (disjoint) union is over all the vectors (cid:126)k in N Γ such that | (cid:126)k | = (cid:88) γ ∈ Γ k ( γ ) is equal to k . That decomposition extends naturally to the Abelian group C k ( (cid:126)α, G ) of k chains on K ( (cid:126)α ) with coefficients in ( G, +) (see Section 2.3), and the boundaryoperator is compatible with such a decomposition (with the obvious plus or minus signs).The representation ρ : Γ → Aut ( V ) induces (see Section 3) an action on C V . If (cid:126)α = (cid:81) γ ∈ Γ α ( γ ) then for every δ in Γ we have ρ ( δ ) (cid:126)α = (cid:81) γ ∈ Γ α ( δγ ) , i.e. the componentsof (cid:126)α are translated by δ , so that ( ρ ( δ ) (cid:126)α )( γ ) = α ( δγ ) .Thus whenever F is a finite subset of Γ the refinement of (cid:126)α under the action of theinverse of the elements in F is given by (cid:126)α F := (cid:92) δ ∈ F ρ ( δ ) − (cid:126)α = (cid:89) γ ∈ Γ ( (cid:92) δ ∈ F α ( δγ ) ) , i.e. the component of (cid:126)α F in the coordinate γ is given by the common refinement ofthe subset { α ( δγ ) | δ ∈ F } of components of (cid:126)α , so that (cid:126)α F ( γ ) = (cid:84) δ ∈ F α ( δγ ) .Choose some γ (cid:48) in Γ and consider (cid:126)α as being: • (cid:126)α ( γ ) = α p if γ = γ (cid:48) , where α p is an irreducible cover whose nerve K ( α p ) hasdimension p , where ≤ p ≤ d , and d is the dimension of V . • (cid:126)α ( γ ) = { V } if γ (cid:54) = γ (cid:48) , where { V } is the trivial cover for V .Hence for every subset F of Γ the open cover (cid:126)α F is given by: • (cid:126)α F ( γ ) = α p if γ = δγ (cid:48) for some δ in F . • (cid:126)α F ( γ ) = { V } otherwise.Making use of Künneth’s relations we check that Dim K ( (cid:126)α F ) is given by | F | dim K ( α p ) = p | F | for every subset F of Γ , whence Dim ( (cid:126)α, Γ , ρ ) is equal to p .21o estimate the simplicial growth we observe that G ( (cid:126)α F ) = G ( α p ) | F | , and a little bit of algebra along the lines of Lemma 2.9 ensures, making use of Künneth’srelations, that ent k ( (cid:126)α, Γ , ρ ) = ent ( (cid:126)α, Γ , ρ ) = log |(cid:52) ( α p ) | for every k .We conclude that { c ( n ) } n ∈ N = { n } n ∈ N is a controlling sequence both for the sim-plicial and the dimension growth of { K ( (cid:126)α n ) } n ∈ N ≡ { K ( (cid:126)α F ( n ) ) } n ∈ N . Consider on V irreducible covers all of whose members have at least one point incommon; if α is an open cover with this property we say that α is a prismatic coverfor V . Prismatic covers satisfy:1. dim K ( α ) = G ( α ) − .2. If k is less or equal than dim K ( α ) then G k ( α ) = (cid:18) G ( α ) k + 1 (cid:19) + G k − ( α ) . Observe that if V is connected, then a prismatic cover whose nerve has dimensionsmaller or equal than the dimension of V always exists (indeed, prismatic covers getrid of all the topology on V , if any).Here ( R Γ , (cid:107) (cid:107) ) ≡ ( X, (cid:107) (cid:107) X ) is the Banach space of arrays of real numbersindexed by elements of a discrete group Γ with the l -norm. Elements in X arefunctions x : Γ → R such that (cid:107) x (cid:107) := (cid:80) γ ∈ Γ | x ( γ ) | < ∞ , and a basis for this linearspace is given by the maps { e γ : Γ → { , } | γ ∈ Γ } satisfying e γ ( γ (cid:48) ) = 1 if γ = γ (cid:48) ,and zero otherwise, so every element x in X can also be written as a sum (cid:80) γ ∈ Γ x ( γ ) e γ .The predual of ( R Γ , (cid:107) (cid:107) ) is the Banach space ( R Γ , (cid:107) (cid:107) ∞ ) = ( Y, (cid:107) (cid:107) Y ) offunctions y : Γ → R such that (cid:107) y (cid:107) ∞ := sup γ ∈ Γ | y ( γ ) | < ∞ . A basis for this linearspace is given by the maps { e γ : Γ → { , } | γ ∈ Γ } such that (cid:104) e γ , e γ (cid:48) (cid:105) = 1 if γ = γ (cid:48) ,and zero if γ (cid:54) = γ (cid:48) , therefore (cid:104) e γ , x (cid:105) = x ( γ ) for every x in X .On X one can also consider families of seminorms given by p C ( x ) := max { |(cid:104) y, x (cid:105)| := | (cid:88) γ ∈ Γ y ( γ ) x ( γ ) | | y ∈ C } , where C ranges over arbitrary finite subsets on Y . The open sets associated to thatfamily of seminorms generate a Hausdorff topology on X ; an application of Tychonov’sLemma ensures that bounded sets in X are compact in this topology.Let V := X ∩ { x | (cid:107) x (cid:107) ≤ , x ( γ ) ≥ for every γ } be the part of the unit ballin ( R Γ , (cid:107) (cid:107) ) all of whose coordinates are non-negative, and consider the topology22nduced by the family of seminorms { p C } . Then V is closed, convex, Hausdorff andcompact, and can be spanned by convex linear combinations of its extremal points,say E ( V ) , that consists of the set { e γ | γ ∈ Γ } together with the origin in X , that wedenote by e if no confussion arises. We regard e as the apex of the pyramid V , andsay that the convex set spanned by the rest of the extreme points is the base of V .Note that the base of V consists of maps v : Γ → [0 , such that (cid:107) v (cid:107) = (cid:80) γ ∈ Γ v ( γ ) = 1 .We pick the family { p C } as follows: if F is a finite subset of Γ we denote by p F theseminorm on X given by p F ( x ) := max { | x ( γ ) | | γ ∈ F } .The family { p F | F is a finite subset of Γ } of seminorms on X provide the desiredproperties on V . Note that given T bigger than zero: • On one side p F ( x ) < T if and only if for every γ in F we have | x ( γ ) | < T , hence { x | p F ( x ) < T } = (cid:92) γ ∈ F { x | p γ ( x ) < T } . • On the other side p F ( x ) > T if and only if there exists some γ in F with | x ( γ ) | > T , therefore { x | p F ( x ) > T } = (cid:91) γ ∈ F { x | p γ ( x ) > T } . Choose a positive number a small enough (in fact smaller than . ), and for every γ in Γ define an open cover α [ γ ] with two members for V as follows: • The open set A γ is given by those v in V such that p γ ( v ) > . − a . Thus A γ consists of those elements in V whose distance from the vertex e γ is smaller than . a , the distance being the one induced by the norm (cid:107) (cid:107) on X . • The open set A (cid:48) γ is given by those v in V such that p γ ( v ) < . a , i.e. A (cid:48) γ contains the apex of V and those points whose distance from the vertex e γ islarger than . − a .We construct two examples in this setup, one that does not use the group structureof Γ at all, meanwhile the other uses such a structure.1. In this example Γ could be any denumerable infinite set. Using the covers α [ γ ] yet mentioned we construct, for every finite subset F of Γ , an open cover α [ F ] whose cardinality is | F | + 1 , containing two types of open sets: • For each γ in F we have an open set A γ , as before. Observe that the union (cid:83) γ ∈ F A γ consists of those points in V with p F ( v ) > . − a . • An open set A (cid:48) F is given by those v in V such that p F ( v ) < . a , therefore A (cid:48) F is the intersection (cid:84) γ ∈ F A (cid:48) γ .23ne verifies that α [ F ] is a prismatic cover for V whenever F is a finite subset of Γ , also that α [ F (cid:48) ] is finer than α [ F ] whenever F is a subset of F (cid:48) .Let { F ( n ) } n ∈ N be an increasing sequence of subsets exhausting Γ , and considerthe sequence of complexes { K ( α [ F ( n )]) } n ∈ N . Since α [ F ( m )] (cid:31) α [ F ( n )] when-ever m > n we get a directed sequence { K ( α [ F ( n )]) , T n } n ∈ N of complexes andmaps, thus the constructions/results in Section 2.7 can be used.Being α [ F ( n )] a prismatic cover for every n , the simplex K ( α [ F ( n )]) has themaximal number of simplices allowed, say. It is easy to see that { log | F ( n ) | } n ∈ N controls the simplicial growth of { K ( α [ F ( n )]) } n ∈ N , and for every k we haveent k ( α [ F ( n )] , log | F ( n ) | ) = k + 1 , showing that the estimates in Theorem 1 are sharp.In this example {| F ( n ) |} n ∈ N controls the dimension growth of { K ( α [ F ( n )]) } n ∈ N ,and we see that Dim ( α [ F ( n )] , | F ( n ) | ) is equal to one.2. We consider a representation ρ : Γ → Aut ( V ) that leaves the apex fixed andtranslates the coordinates in the base, so that if v = v (0) e + (cid:88) γ ∈ Γ v ( γ ) e γ , then ρ ( δ ) v = v (0) e + (cid:88) γ ∈ Γ v ( δγ ) e γ . For every γ the open cover α [ γ ] is given by { A γ , A (cid:48) γ } , hence ρ ( δ ) α [ γ ] is just α [ δγ ] .Therefore for every finite subset F of Γ the cover α [ γ ] F is given, according toSection 3, by (cid:92) δ ∈ F ρ ( δ ) − α [ γ ] = (cid:92) δ ∈ F α [ δ − γ ] . The covering α [ γ ] F is not irreducible if F has at least two elements, however thecover α [ F − γ ] constructed in 1 is finer than α [ γ ] F (and prismatic). Hence forevery k whenever F is a finite subset of Γ we have the equality S k ( α [ γ ] F ) = G k ( α [ F − γ ]) . As in 1, choose an increasing family of subsets { F ( n ) } n ∈ N exhausting Γ , to inferthat the simplicial growth up to dimension k for the sequence { K ( α [ γ ] F ( n ) ) } n ∈ N is polynomial of degree ( k + 1) , this for every k , and the dimension growth islinear, and equal to one. Remark 4.1.
Being the simplicial growth of polynomial type the statements inRemark 3.8 are not relevant. Finite dimensional manifolds and property-e
As explained in Section 3.2, if V admits an expansive action of a group (semigroup) Γ we can, in a precise sense, reconstruct a complex that is homeomorphic to V if wetake as an initial condition the nerve associated to an open cover that is a generatorfor ( V, Γ , ρ ) . Moreover, if V is endowed with a Riemannian metric and its dimensionis bigger than one all the estimates in Section 3.3 for the e-constant, the (minimal)complexity of generating covers, and their relation to the (topological) entropy provideinteresting information (sometimes without much effort).If the dimension of V is either one or two the classification of closed orientable man-ifolds is complete and extremely simple. In those cases if V and W are homotopicallyequivalent finite and boundaryless simplicial complexes then they are homeomorphic,and even diffeomorphic if they are endowed with a smooth structure.In the context of algebra, the simplest groups and semigroups are Z and N respec-tively. Thus to understand expansive actions of groups and/or semigroups on closedorientable manifolds it is natural to begin with the simplest examples, i.e. with Z and/or N actions on closed (orientable) manifolds, to then consider Abelian actions ofproducts of those. The only closed one dimensional manifold up to homeomorphism is S . If Γ is equalto N , then ( S , N ) is expansive if one considers the N -action n : θ (cid:55)→ k n θ for somefixed k in Z whose absolute value is bigger than one. If no confussion arises we denotesuch a representation by f so that f n ( θ ) = kf n − ( θ ) whenever n is a natural number.Consider for simplicity the case when k is equal to two. Let α be the open coverof S given by { ] − a , π + a [ , ] π − a, a [ } for some positive a that is small enough.Then α is a generator, and it is easy to see that ent ( α, Z , f ) = ent ( S , Z , f ) = log 2 (see Figure 1). f -1 f -1 f -1 Figure 1:
A schematic evolution of the nerve of (cid:84) Tn =0 f − n α when f : S → S is given by f ( θ ) = 2 θ . Here α consists of two semicircles overlapping in a neighborhood of θ = 0 and θ = π , with T being equal to , or (from leftto right). .2 Dimension two Closed manifolds of dimension two, also known as compact Riemann surfaces, are thebasic test of (almost) every theory that wishes to be extended to higher dimensions.The classification of them up to diffeomorphism is extremely simple, and everyonecan distinguish among them by the number of holes (or the intersection form in thefirst homolgy group with Z coefficients). Within the orientable ones we will constructexpansive actions of either Z , N or N , depending on the genus. Z Expansive homeomorphisms (or expansive actions of Z ) in compact Riemann surfacesof positive genus were constructed in [Ob-Re]. We briefly explain some ideas.Consider the standard Anosov homeomorphism on the 2-torus (see Section 5.3.1for general definitions), namely the one induced by the matrix A = (cid:20) (cid:21) on R . Let h : T → T be the induced homeomorphism that turns out to be expansive,and note that if h : W → W induces an expansive action of Z then so does h k : W → W whenever k is a positive integer.Let Σ g denote the orientable Riemann surface of genus g, and consider a branched cover x : Σ g → Σ = T to construct a homeomorphism f : Σ g → Σ g by lifting h k through x for some k. If the pair ( x, f ) can be constructed, then f : Σ g → Σ g providesan expansive action of Z on Σ g (observe that this is not true in higher dimensionsbecause the branch set could have strictly positive dimension, and the dynamics of thelifted map, namely f, need not be expansive therein).Considering standard relations that the map x : Σ g → T should satisfy at the levelof the fundamental groups to achieve a branched cover, lifts of iterates of h : T → T are constructed for every g bigger than one in [Ob-Re], providing the desired expansivesystems (Σ g , Z ) whenever g is different from zero.Some years later K. Hiraide and J. Lewowicz (see [Hir] and [Lew]) found a naturalrelation between expansive actions of Z on hyperbolic Riemann surfaces and neatconstructions/results on Teichmüller theory due to W. Thurston (see [Th]). The resultin [Hir]-[Lew] can be rephrased using the language developed in [Th] as follows: Theorem 6. (Hiraide-Lewowicz) Let Σ g be a closed and orientable hyperbolic Rie-mann surface, and assume that f : Σ g → Σ g induces an expansive action of Z (thoseactions are known to exist due to the constructions in [Ob-Re]). Let T (Σ g ) denote theTeichmüller space of Σ g . Then for some f ∗ conjugated to f the induced action of f ∗ on the closure of T (Σ g ) , denoted by T ( f ∗ ) : T (Σ g ) → T (Σ g ) , has exactly two fixed points. Those points are on the boundary of T (Σ g ) and correspondto projective classes of mutually transverse measured laminations on Σ g . One of thoseprojective classes has a representative that expands under the action of f ∗ , while theother contracts (one says that f is conjugated to a pseudo-Anosov diffeomorphism).
26n [Hir] and [Lew] it is stated that S does not admit an expansive action of Z . N It is rather easy to see that V admits an expansive action of N only if there exists amap f : V → V whose degree it at least two. A necessary condition for the existenceof such a map is that the simplicial volume of V is equal to zero (see [Gro2]), and then V must be either the two sphere or the two torus. On T an expansive action of N can be easily constructed, although in S it is not possible to achieve that (see Section5.3.2 for both issues).Therefore to complete the program of reconstructing every orientable closed man-ifold whose dimension is two from a simplicial complex that has a simpler structurewe are led to consider higher rank actions. N An expansive action of N on T = S × S can be achieved using expansive actionsof N on S (see Section 5.1) if one considers the remarks in Section 5.3.3 concerningcartesian products of expansive systems.An expansive action of N on S is constructed as a particular case of Theorem 7in Section 5.3.3 (see Corollary 5.2). There exist (partial) characterizations of expansive actions on closed manifolds of thesimplest groups and semigroups, namely Z and N respectively. Z Let V be a closed manifold admitting two foliations of complementary dimension thatare transversal all over V, and let f : V → V be a diffeomorphism preserving thosefoliations. Assume furthermore that f stricly expands the current corresponding toone of those foliations and strictly contracts the other one (see [Ru-Su]). One saysthat ( V, Z , f ) is Anosov , and it is easy to see that Anosov systems provide examplesof expansive Z -actions.A good introduction to Anosov systems can be found in [Sm], and modulo examplesunknown to the author in all the systems of this type the underlying space is, up toconjugation, an infra-nilmanifold , i.e. up to a finite cover and homeomorphism, aco-compact quotient of a connected simply connected nilpotent Lie group, say G ,the quotient being induced by the action of a discrete subgroup of G , say Υ , thatis finitely generated, nilpotent, and has no elements of finite order (see [Sm] again),generalizing linear automorphisms on tori. If f : G/ Υ → G/ Υ is Anosov, then thelinear map induced at the level of Lie algebras has no eigenvalues in the unit circle,and an important part of the structure of these systems can be decoded by algebraicmeans (see [Lau-Will]). 27ne interesting feature of infra-nil-automorphisms with the Anosov property is thatthe observables involved in the estimates in Section 3.3 can be easily found. For in-stance, if λ is the largest eigenvalue of the linearization of f , then ent ( V, N , f ) is equalto log λ (see [Ru-Su]), extending the pseudo-Anosov behavior, where λ corresponds tothe expansion/contraction coefficient for the transverse measured laminations. N In [Co-Re] is shown that on closed manifolds a map f : V → V represents an expansiveaction of N if and only if such a map is expanding in the sense of [Gro1], namely iffor some metric d on V and every point v in V there exists exists a neighbourhood ofthat point such that f ∗ d > d outside the diagonal therein.The following discussion is based on [ Gro1 ] . It is proved that a necessary conditionfor the existence of a map of this type on a closed manifold is that their universalcover is homeomorphic to R n . To achieve that M. Gromov notes that the lift of thosemaps to the universal cover are globally expanding for some metric invariant underdeck transformations, a condition that is easy to verify.The simplest examples of this kind are induced by linear maps on R n whose eigen-values are greater than one and that are compatible with the free action of discretegroups on R n , say Υ : R n → R n , so that V = R n / Υ is compact. An invariant metric inthose examples is of course the very -flat canonical one (see [Be]): every flat manifoldof this type admits an expanding action of N , a result that Gromov attributes to D.Epstein and M. Shub.Assuming an upper bound on the Jacobian of the map one sees that a necessarycondition for the existence of an expanding map on a closed manifold, say V, is thatthe fundamental group must have polynomial growth. The analogous result withoutusing the assumption that the map is differentiable and obtained using techniquesfrom geometric group theory is due to J. Franks.Hence the candidates are closed aspherical manifolds that do no admit metrics ofnegative sectional curvature (see [Be] or [Gro2], for example). Needless to say, thoseare necessary conditions.Posterior work of Shub (see [Gro1]) enables to assert that an expanding system ( V, N ) is conjugated to an infra-nil-endomorphism if and only if the fundamental groupof V contains a nilpotent subgroup of finite index.Since the main result in [Gro1] claims that every finitely generated group withpolynomial growth is virtually nilpotent, one concludes that every expansive N -actionon a closed manifold is conjugated to an infra-nil-endomorphism.It is worth mentioning the result of D. Epstein and M. Shub: it provides the onlyknown examples of closed manifolds, that are not products, with special holonomy(see [Be], and the foundational [Har-Law]) and of dimension larger than two, allowingan expansive action. Indeed, all complex tori belong to this class, and the estimates ofSection 3.3 enriched with the Monge-Ampère-Aubin-Calabi-Yau developments providea play-ground. 28 .3.3 Higher rank actions Let { V ω } ω ∈ Ω be a finite collection of closed manifolds so that for each ω in Ω thespace V ω admits an expansive action of a group or semigroup (Γ ω , ρ ω ) . By means ofthe set-theoretic characterization of property-e (Section 3.2) one readily sees that theCartesian product of them, say V := (cid:81) ω ∈ Ω V ω , also admits an expansive action of (Γ , ρ ) := (cid:81) ω ∈ Ω (Γ ω , ρ ω ) .Consider now the wedge sum of the finite collection of spaces { V ω } ω ∈ Ω , denoted by ∨ ω ∈ Ω V ω , where in each of the V ω ’s a base point v ω, is understood. Inside the Cartesianproduct of the V ω ’s collapse the wedge (sum) of the spaces to a point, to get the smash of { V ω } ω ∈ Ω , usually written as ∧ ω ∈ Ω V ω .In the category of topological spaces (with base points) the smash product is acommutative and associative self-functor. Hence if (Γ ω , ρ ω ) is a group or semigroupacting on V ω having the base point v ω, as a fixed element for every ω , then there is anatural action of { (Γ ω , ρ ω ) } ω ∈ Ω on ∧ ω ∈ Ω V ω , denoted by ∧ ω ∈ Ω (Γ ω , ρ ω ) : ∧ ω ∈ Ω V ω → ∧ ω ∈ Ω V ω , that is commutative and associative with respect to the different ω coordinates (in thesame way as (cid:81) ω ∈ Ω (Γ ω , ρ ω ) : (cid:81) ω ∈ Ω V ω → (cid:81) ω ∈ Ω V ω ).The next result asserts that the property-e is preserved under the smash product. Theorem 7.
Let ( V ω , Γ ω , ρ ω ) ω ∈ Ω be a finite family of systems with property-e, each ofthem having at least one fixed point, where Γ ω is a given group or semigroup, and V ω is a closed manifold. Then the system ( ∧ ω ∈ Ω V ω , ∧ ω ∈ Ω Γ ω , ∧ ω ∈ Ω ρ ω ) is expansive as wellprovided the base points are taken as invariant ones for the representation (Γ ω , ρ ω ) ,for every ω in Ω .Proof. For simplicity consider the case when Ω has two elements, and Γ ω coincideswith N for both ω ’s. So assume that ( V, N , f ) and ( W, N , h ) correspond to expansiveactions of N on V and W repectively, with v and w being fixed points for f and h ,respectively. Then the system ( V ∧ W, N , f ∧ h ) corresponds to an action of N on V ∧ W .Let v and w denote the base points of V and W, to construct a family { g ( t ) } t ∈ ]0 , of Riemannian metrics on V × W as follows. Let κ : V → [0 , and ρ : W → [0 , befunctions different from zero outside the base points, smooth enough, but such that lim v → v κ ( v ) = 0 and lim w → w ρ ( w ) = 0 . Define for each t in ]0 , the Riemannianmetric g ( t ) on V × W by g ( t ) := ( ( t + (1 − t ) ρ ) g V ) ⊕ ( ( t + (1 − t ) κ ) g W ) , where g V and g W are metrics on V and W, both of finite diameter and of a suitableregularity.Denote by d g ( t ) the distance on V × W induced by g ( t ) , and consider the family ofmetric spaces { ( V × W, d g ( t ) ) } t ∈ ]0 , . As t goes to zero the couple ( V × W, d g ( t ) ) ceases29o be a metric space because all the elements in V ∨ W (recall that base points areunderstood) are at zero distance.After those remarks it is interesting to note: Lemma 5.1.
One has the convergence ( V × W, d g ( t ) ) (cid:32) ( V ∧ W, d g (0) ) in the Gromov-Hausdorff sense as t goes to zero (see [Gro2]). In the Gromov-Hausdorff metric space identify ( V × W, d g ( t ) ) with ( V × W ) t forevery t in [0 , , to denote by { ( ( V × W ) t , N , ( f × h ) t ) } t ∈ [0 , the collection of systems obtained, where ( ( V × W ) , N , ( f × h ) ) = ( ( V ∧ W, d g (0) ) , N , f ∧ h ) . Since by assumption both ( V, N , f ) and ( W, N , h ) are expansive systems, we con-clude that ( ( V × W ) t , N , ( f × h ) t ) is also expansive for every t different from zero;indeed, the property of being expansive is a conjugacy invariant that does not dependon the metric chosen (see Section 3.2).To conclude the proof we add further conditions to the functions κ and ρ to ensurethe expansive property on ( V ∧ W, N , f ∧ h ) thanks to the metric d g (0) .Let c V and c W be expansivity constants for ( V, d V , N , f ) and ( W, d W , N , h ) , where d V and d W are the distance functions induced by the Riemannian metrics g V and g W , respectively. If d V ( v, v ) is larger than c V we require that κ ( v ) = 1 , and if d W ( w, w ) is bigger than c W we demand that ρ ( w ) = 1 .Denote by [ v, w ] the point in V ∧ W that is the image of ( v, w ) under the map from V × W to V ∧ W . Observe that [ v, w ] = [ v , w ] = [ v , w ] for every ( v, w ) in V × W ,where [ v , w ] is a fixed point for f n ∧ h n = ( f n ∧ W ) · (1 V ∧ h n ) = (1 V ∧ h n ) · ( f n ∧ W ) whenever ( n , n ) is in N .Choose different points [ v, w ] and [ v (cid:48) , w (cid:48) ] in V ∧ W, and exhaust all the possibilitiesto infer the expansiveness of (( V ∧ W, d g (0) ) , N , f ∧ h ) with e-constant min { c V , c W } .The extension to the general case is direct.Consider the case when Ω has d elements, and for every ω in Ω choose ( V ω , Γ ω , ρ ω ) as being conjugated to ( S , N , f ) with the N -action on S given by f ( θ ) = 2 θ. Take θ = 0 as the base point in S to construct the bouquet of d circles ∨ d S , and notethat ∧ d S = S d is endowed with an action of N d induced by ∧ di =1 f n ( i ) .Theorems 3 and 7 together with Lemma 3.12 give, thanks to (the proof of) thegeneralized Poincaré conjecture : Corollary 5.2.
Let V be an homotopy S d . Then there exists an expansive action of N d on V . If α is a generator for ( V, N d , ∧ d f ) and { F ( n ) } n ∈ N is an increasing sequenceexhausting N d , then when n goes to infinity the nerve of α F ( n ) is homeomorphic to S d . Finished in the work of W. Thurston, R. Hamilton and G. Perelman in dimension 3, M. Freedmanin dimension 4, and S. Smale in higher dimensions. eferences [Ale-Pon] P. Alexandrov, V. Ponomarev: Projection-Spectra, Proceedings of the Sec-ond Prague Topological Symposium, 25-30, Czechoslovak Academy of Sciences(1967).[Be] A. Besse: Einstein Manifolds, Springer Verlag (1987).[Co-Re] E. Coven, W. Reddy: Positively Expansive Maps of Compact Manifolds,Lecture Notes in Mathematics 819, 96-110, Springer Verlag (1980).[Fa-Jo] T. Farrell, J. Jones: Anosov Diffeomorphisms constructed from π ( Diff ( S n )) ,Topology 17, 273-282 (1978).[Fed] H. Federer: Geometric Measure Theory, Springer Verlag (1969).[Gro1] M. Gromov: Groups of Polynomial Growth and Expanding Maps, Ins. HautesÉtudes Sci. Publ. Math., 53, 53-78 (1981).[Gro2] M. Gromov: Metric Structures for Riemannian and Non-Riemannian Spaces,Birkhäuser (1999).[Gro3] M. Gromov: Topological Invariants of Dynamical Systems and Spaces of Holo-morphic Maps, Math. Phys. Anal. Geom. 2, 323-415 (1999).[Har-Law] R. Harvey, B. Lawson: Calibrated Geometries, Acta Math., 148, 47-157(1982).[Hat] A. Hatcher: Algebraic Topology, Cambridge University Press (2002).[Hir] K. Hiraide: Expansive Homeomorphisms of Compact Surfaces are Pseudo-Anosov, Proc. Japan Acad. 63-A, 337-338 (1987).[Hur-Wall] W. Hurewicz, H. Wallman: Dimension Theory, Princeton University Press(1959).[Lau-Will] J. Lauret, C. Will: Nilmanifolds of Dimension ≤≤