aa r X i v : . [ m a t h . K T ] J a n A NOTE ON HOMOLOGY FOR SMALE SPACES by Valerio Proietti
Abstract . —
We collect three observations on the homology for Smale spaces definedby Putnam. The definition of such homology groups involves four complexes. It isshown here that a simple convergence theorem for spectral sequences can be usedto prove that all complexes yield the same homology. Furthermore, we introduce asimplicial framework by which the various complexes can be understood as suitable“symmetric” Moore complexes associated to the simplicial structure. The last sectiondiscusses projective resolutions in the context of dynamical systems. It is shown thatthe projective cover of a Smale space is realized by the system of shift spaces andfactor maps onto it.
Contents
1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1. Dimension groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2. Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82. C s, A ( π ) is quasi-isomorphic to C s ( π ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1. Filtrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2. Spectral sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123. Simplicial viewpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1. Symmetric simplicial groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2. Symmetric cosimplicial groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184. Projective covers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Mathematics Subject Classification . —
Key words and phrases . —
Smale spaces, Dimension groups, Homology, Spectral sequences, Pro-jective resolutions.Supported by the Danish National Research Foundation through the Centre for Symmetry andDeformation (DNRF92).
NOTE ON HOMOLOGY FOR SMALE SPACES Introduction and main results
When Steven Smale initiated his study of smooth maps on manifolds, he definedthe notion of Axiom A diffeomorphism [ ]. The main condition is that the map,restricted to its set of non-wandering points, has a hyperbolic structure. The non-wandering set of these systems can be canonically decomposed into finitely manydisjoints sets, called basic sets , each of which is irreducible in a certain sense. One ofSmale’s great insights was that, even though one began with a smooth system, thenon-wandering set itself would not usually be a submanifold, but rather an object offractal-like nature. This can be taken as a motivation for moving from the smoothcategory to the topological one.Smale spaces were introduced by Ruelle as a purely topological description of thebasic sets of Smale’s Axiom A diffeomorphisms [ ]. In this paper we consider thehomology theory for Smale spaces introduced by Putnam in [ ]. This can be viewedas a solution to Smale’s problem of classifying Axiom A systems by relatively simplecombinatorial data, in the same fashion that Morse-Smale systems could be described.Shifts of finite type are the zero dimensional examples of Smale spaces and arethe basic building blocks of the theory. Putnam’s homology can be viewed as a far-reaching generalization of Krieger’s dimension groups for shifts of finite type [ ]. Inthe preliminaries of this paper, we review the notion of Krieger’s invariant and explainits connection to K -theory by examining the stable and unstable equivalence relationswhich define the associated C ∗ -algebras (this is a well-known result, here it is simplyexpressed in a slightly unusual form, see Theorem 1.10).There are many interesting and open questions concerning Putnam’s homology forSmale spaces. In the literature, computations of the homology groups have been donemostly by resorting to the definition, e.g., [ , Chapter 7]. It is desirable to have somemachinery, as it occurs with algebraic topology, which would aid in these calculationsby appealing to techniques such as long exact sequences, excision, etc.Exact analogues are at the moment not so clear, but it is reasonable that analternative, perhaps more conceptual definition of the homology could shed somelight on these issues. Moreover, this could also lead to clarifying the relations withČech cohomology and K -theory (beyond the case of shifts of finite type). More onthese questions can be found in [ , Chapter 8]. This paper started as an effort toresearch in this direction.The technical definition of Putnam’s homology groups involves four bicomplexes[ , Chapter 5]. Only three of these are shown to be quasi-isomorphic, leaving outthe largest (but perhaps most natural) double complex, which has a clear connectionto K -theory. The first result of this paper fills this gap by showing, thanks to a simpleconvergence theorem for spectral sequences, that this double complex also yields thesame homology groups. The formal statement is given in Corollary 2.7. NOTE ON HOMOLOGY FOR SMALE SPACES Section 3 is concerned with proving a collection of results that are already proved inPutnam’s memoir, by taking a slightly different and somewhat more unified perspec-tive. The key observation stems from the simplicial nature of the homology theoryfor Smale spaces: a given Smale space is suitably “replaced” by a bisimplicial shiftof finite type, to which Krieger’s invariant is applied to get (in conjunction with theDold-Kan correspondence) a bicomplex which defines the homology groups of interest.From this viewpoint, the different variants of this bicomplex appear as the associ-ated Moore complexes (i.e., the normalized chain complexes). There is also an actionof the symmetric group which is exploited to obtain all of Putnam’s complexes as“reduced” complexes with respect to this mixed simplicial-symmetric structure. Themain result in this section, proved as application of these methods, is Theorem 3.10.The last section introduces the concept of projectivity for dynamical systems andattempts to justify the definition of the homology theory for Smale spaces by drawinga parallel with sheaf cohomology. The main result here is that the projective cover of aSmale space can be defined as a certain projective limit over the symbolic presentationsfor the given space. The rigorous statement is found in Theorem 4.3.Most of the conventions and notations in this paper are taken directly from [ ].No attempt is made to put the results in broader context or expand on detail. Forthese reasons the reader is advised to have a copy of Putnam’s A Homology Theoryfor Smale Spaces [ ] handy. Acknowledgements
My gratitude goes to Ian F. Putnam for many stimulating conversations and forthe warm hospitality I received during my stay at the Department of Mathematicsand Statistics of the University of Victoria. I would also like to thank Ryszard Nestfor proposing to look into projective resolutions.
1. Preliminaries
A Smale space (
X, φ ) is a dynamical system consisting of a homeomorphism φ on acompact metric space ( X, d ) such that the space is locally the product of a coordinatethat contracts under the action of φ and a coordinate that expands under the actionof φ . The precise definition requires the definition of a bracket map satisfying certainaxioms [
15, 17 ].The most essential feature of Smale spaces is given by the definition of two equiv-alence relations, named respectively stable and unstable , which reads as follows: – given x, y ∈ X , we say they are stably equivalent iflim n → + ∞ d ( φ n ( x ) , φ n ( y )) = 0; NOTE ON HOMOLOGY FOR SMALE SPACES – given x, y ∈ X , we say they are unstably equivalent iflim n → + ∞ d ( φ − n ( x ) , φ − n ( y )) = 0 . The orbit of x ∈ X under the stable (respectively unstable) equivalence relationis called the global stable (resp. unstable ) set and is denoted X s ( x ) (resp. X u ( x )).Given a small enough ǫ > local stable and unstable sets are also defined, andthey are denoted respectively X s ( x, ǫ ) and X u ( x, ǫ ). They provide the local productstructure in the following sense: each x ∈ X admits an open neighborhood which ishomeomorphic (via the bracket map) to the product X u ( x, ǫ ) × X s ( x, ǫ ). Local andglobal sets are related through the following identities: X s ( x ) = [ n ≥ φ − n ( X s ( φ n ( x ) , ǫ )) X u ( x ) = [ n ≥ φ n ( X s ( φ − n ( x ) , ǫ )) . Let (
X, φ ) be a Smale space. We will assume that (
X, φ ) is non-wandering , so thatthere exists an s/u -bijective pair π = ( Y, ψ, π s , Z, ζ, π u ) (see [ , Section 2.6] for thisnotion). Recall from [ , Sections 2.5 and 2.6] that we can assume Y and Z to benon-wandering, and also π s and π u to be finite-to-one.We define a subshift of finite type for each L, M ≥ L,M ( π ) = { ( y , . . . , y L , z , . . . , z M ) | y l ∈ Y, z m ∈ Z,π s ( y l ) = π u ( z m ) , ≤ l ≤ L, ≤ m ≤ M } . We have maps δ l : Σ L,M → Σ L − ,M (1) δ ,m : Σ L,M +1 → Σ L,M which delete respectively entries y l and z m . Theorem 2.6.13 in [ ] asserts that themaps δ l are s -bijective and the maps δ ,m are u -bijective (these will be defined shortly).Given a subshift of finite type Σ, we can associate to it an abelian group, denoted D s (Σ), defined in [ , Chapter 3] (see also [ ]). It will be called the (stable) dimensiongroup of Σ. This construction is covariant for s -bijective maps and contravariant for u -bijective maps [ , Sections 3.4 and 3.5]. We summarize here these definitions: Definition 1.1 . —
Let f : ( X, φ ) → ( Y, ψ ) be a map of Smale spaces. Consider foreach x ∈ X the restrictions f : X s ( x ) → Y s ( f ( x )) (2) f : X u ( x ) → Y u ( f ( x )) . (3) – If (2) is injective, we say that f is s -resolving . If it is injective and surjective,then we say f is s -bijective . NOTE ON HOMOLOGY FOR SMALE SPACES – If (3) is injective, we say that f is u -resolving . If it is injective and surjective,then we say f is u -bijective . Let us start with the definition of Krieger’s dimensiongroups.
Definition 1.2 . —
Let (Σ , σ ) be a subshift of finite type. For e ∈ Σ, consider thefamily of compact open subsets in the stable orbit Σ s ( e ) and denote it by CO s (Σ , σ, e ).Define CO s (Σ , σ ) = ∪ e ∈ Σ CO s (Σ , σ, e ). Let ∼ be the smallest equivalence relationsuch that, for E, F ∈ CO s (Σ , σ ), we have – E ∼ F if [ E, F ] = E, [ F, E ] = F , assuming both sets are defined; – E ∼ F if and only if σ ( E ) ∼ σ ( F ).We define D s (Σ , σ ) (abbreviated D s (Σ)) to be the free abelian group on the ∼ -equivalences [ E ], modulo the subgroup generated by [ E ∪ F ] − [ E ] − [ F ], where E, F belong to CO s (Σ , σ ) and E ∩ F = ∅ .There is a definition of D u (Σ , σ ), which is left to the imagination of the reader,since it won’t be used in the rest of this paper.It is easy to see that, in the construction above, it is sufficient to consider clopenslying in the local stable sets. Lemma 1.3 . —
Define a family of sets CO sǫ (Σ , σ ) , composed of clopens E ⊆ Σ s ( e, ǫ ) for some e ∈ Σ and ǫ < / . Consider the abelian group D sǫ (Σ , σ ) , defined as inDefinition 1.2, but replacing CO s (Σ , σ ) with CO sǫ (Σ , σ ) . Then we have D s (Σ , σ ) ∼ = D sǫ (Σ , σ ) .Proof . — Given E ∈ CO s (Σ , σ ) , E ⊆ Σ s ( f ), there is a well-defined function E → N ,defined assigning to e ∈ E the minimum number N ( e ) such that e n = f n whenever n ≥ N ( e ). In other words, N ( e ) is the minimum natural number such that e ∈ σ − N ( e ) (Σ s ( σ N ( e ) ( f ) , ǫ )) . By definition E ∩ σ − n (Σ s ( σ n ( f ) , ǫ )) is clopen for each n ∈ N , which implies theassignment e N ( e ) is continuous. Since E is compact, there is N ( E ) ∈ N such that E ⊆ σ − N ( E ) (Σ s ( σ N ( E ) ( f ) , ǫ )) . Therefore E can be partitioned in a finite number of disjoint clopens E i with E i ∈ CO sǫ (Σ , σ ). We conclude [ E ] ∈ D sǫ (Σ , σ ). All is left to show is the equivalence relationdefining D s (Σ , σ ) is determined within the clopens in CO sǫ (Σ , σ ). Let E ∼ F be setsin CO s (Σ , σ ) and take N to be the maximum between N ( E ) and N ( F ). By definition E ∼ F if and only if σ N ( E ) ∼ σ N ( F ), and of course σ N ( E ) and σ N ( F ) belong tolocal stable sets. This completes the proof. NOTE ON HOMOLOGY FOR SMALE SPACES Σ u ( x )Σ s ( x ) x E Fe f ∼ Figure 1.
In this figure, E and F are compact opens in CO sǫ (Σ , σ ), with E ⊆ Σ s ( e, ǫ ) and F ⊆ Σ s ( f, ǫ ). The shaded area in yellow indicates that[ E, F ] = E, [ F, E ] = F and therefore E and F are equivalent sets. A consequence of the previous lemma is that we can illustrate the definition ofdimension group by a simple figure (Figure 1).When (Σ , σ ) is non-wandering we can simplify the computation of the dimensiongroup even further, because we can decompose Σ in basic pieces as follows (see [ ,Theorem 2.1.13]). Theorem 1.4 . —
Given a non-wandering Smale space ( X, φ ) , there are closed pair-wise disjoint sets X , . . . , X n and a permutation α ∈ S n such that φ ( X i ) = X α ( i ) forall i = 1 , . . . , n . Moreover, for any i and k such that α k ( i ) = i , the system ( X i , φ k ) isa mixing Smale space. Since the stable and unstable orbits are the same for (
X, φ ) and (
X, φ k ), it is asimple matter to see that, applying the previous theorem to (Σ , σ ), we get a decom-position D s (Σ) ∼ = D s (Σ ) ⊕ · · · ⊕ D s (Σ n ) , (4)(see also [ , Section 2]). Remark 1.5 . —
In this paper we consider the dimension group merely as a group-invariant, without keeping track of the positive cone and of the induced automorphism(for more details, see [ , Chapter 7]). Since the decomposition in (4) holds at thelevel of C ∗ -algebras, the positive cones decompose along the same shape. The inducedautomorphism (which also exists at the C ∗ -level) permutes the summands accordingto α as in Theorem 1.4.In view of the preceding discussion, for the rest of this subsection we assume that(Σ , σ ) is mixing, in particular the global stable sets are dense. Lemma 1.6 . —
Let f ∈ Σ and define CO sf (Σ , σ ) = { E ∈ CO sǫ (Σ , σ ) | E ⊆ Σ s ( f ) } . Consider the abelian group D sf (Σ , σ ) , defined as in Definition 1.2, but re-placing CO s (Σ , σ ) with CO sf (Σ , σ ) . Then we have D s (Σ , σ ) ∼ = D sf (Σ , σ ) .Proof . — Given E ∈ CO sǫ (Σ , σ ), it is sufficient to prove [ E ] = [ F ] for some F ∈ CO sf (Σ , σ ). Suppose E ⊆ Σ s ( e, ǫ ) and let Σ( e, ǫ ) denote the open ball centered NOTE ON HOMOLOGY FOR SMALE SPACES at e of radius ǫ . Note that Σ s ( e, ǫ ) ⊆ Σ( e, ǫ ). Since Σ s ( f ) is dense, we can find f ′ ∈ Σ s ( f ) ∩ Σ( e, ǫ ) and define F = [ f ′ , E ]. The basic properties of the bracket imply[ E, F ] = E, [ F, E ] = F . Remark 1.7 . —
It is clear that CO sf (Σ , σ ) gives a basis for the topology of Σ s ( f ).Thus D s (Σ) is generated by equivalence classes of basic clopens in some global sta-ble set.Let R u (Σ , f ) be the set of pairs of unstably equivalent points which belong to thestable orbit through f ∈ Σ. This is an amenable, étale groupoid when endowed withthe topology as in [ , Section 1.2] (see also [ , Theorem 3.6]). Remark 1.8 . —
In [ , page 14] the question arises if stable and unstable equiv-alence relations of any mixing Smale space are locally compact amenable groupoids.The answer is positive and the proof is as follows: by [ , Corollary 3.8], in the equiv-alence class (in the sense of [ ]) of such equivalence relations we can find étaleamenable groupoids, because their corresponding C ∗ -algebras have finite nuclear di-mension (see [ , Theorem 5.6.18]). Amenability is invariant under this sort of equiv-alence by [ , Theorem 2.2.17].A subbase for the topology on R u (Σ , f ) is given by triples ( E, F, γ ) where
E, F are basic clopens of the unit space Σ s ( f ) and γ : E → F is homeomorphism such that( e, γ ( e )) ∈ R u (Σ , f ) for all e ∈ E . We consider the following “categorification” of R u (Σ , f ): define a category C (Σ , f ) whose objects are the clopens in CO sf (Σ , σ ) andmorphisms E → F are inclusions E ֒ → F and triples ( E, F, γ ) as above.Recall that the K -theory K ( C ) of an additive category ( C , ⊕ ) is the abelian groupgenerated by isomorphism classes [ E ] of objects E ∈ C subject to the relation [ E ⊕ F ] = E + F . If we interpret isomorphism classes as ( E, F, γ )-orbits in C (Σ , f ), and we take E ⊕ F to mean E ∪ F, E ∩ F = ∅ , then we obtain a well-defined abelian group K ( C (Σ , f )). Remark 1.9 . —
Note the condition E ⊕ F is completely determined by inclusions.Indeed unions and intersections are specific colimits and limits in C (Σ , f ). Theorem 1.10 . —
We have D s (Σ) ∼ = K ( C (Σ , f )) for any f ∈ Σ .Proof . — Let us take E and F such that [ E ] = [ F ]. In particular there is n ∈ N and f ∈ σ − n ( F ) such that [ f, σ − n ( E )] = σ − n ( F ). It is easy to see that the map γ ( e ) = σ n ([ f, σ − n ( e )]) (5)is a homeomorphism of E onto F , and obvioulsy σ − n ( e ) belongs to the local unstableset of σ − n ( γ ( e )), therefore ( e, γ ( e )) ∈ R u (Σ , f ).Conversely, if ( E, F, γ ) is an isomorphism in C (Σ , f ), then by [ , Lemma 4.14](and compactness), we can partitition E in a finite number of clopens E , . . . , E n , NOTE ON HOMOLOGY FOR SMALE SPACES and correspondingly F in F , . . . , F n , where E i is homeomorphic to F i through a mapin the form of (5). Therefore [ E i ] = [ F i ] and by the defining relation [ E ] = [ F ].If χ E is the indicator function of the clopen E inside the groupoid C ∗ -algebra C ∗ ( R u (Σ , f )), then a little thinking over the assignment E χ E gives the followingwell-known result (for more details see [ , Section 4.3]). Corollary 1.11 . —
There is an isomorphism K ( C ∗ ( R u (Σ , f ))) ∼ = K ( C (Σ , f )) ∼ = D s (Σ) for any f ∈ Σ . Remark 1.12 . —
As was already implicitly noted in Remark 1.8, the reason whythe choice of f ∈ Σ doesn’t affect the K -theory group is to be found in the statementthat reducing a groupoid to a transversal preserves its equivalence class, as explainedin more detail in [ , Example 2.7]. The maps in (1) will induce group morphisms denoted respec-tively δ sl , δ s ∗ ,m . For each L, M ≥
0, we consider maps ∂ sL,M : D s (Σ L,M ( π )) → D s (Σ L − ,M ( π )) (6) ∂ sL,M = X ≤ l ≤ L ( − l δ sl ∂ s ∗ L,M : D s (Σ L,M ( π )) → D s (Σ L,M +1 ( π )) (7) ∂ sL,M = X ≤ m ≤ M +1 ( − L + m δ s ∗ ,m . It is clear from the definition that ∂ sL,M +1 ◦ ∂ s ∗ L,M = ∂ s ∗ L − ,M ◦ ∂ sL,M . Furthermore, by applying [ , Theorems 2.6.11, 2.6.12, 4.1.14], we have that – for each M ≥
0, (6) is a chain complex; – for each L ≥
0, (7) is a cochain complex.Altogether, we have a double complex ( C s ( π ) • , • , ∂ s , ∂ s ∗ ), where C s ( π ) L,M = ( D s (Σ L,M ( π )) if L ≥ M ≥
00 else . The totalization of this complex is the chain complex (Tot ⊕ ( C s ( π )) • , d s ), whereTot ⊕ ( C s ( π )) N = M L − M = N C s ( π ) L,M d sL,M = ∂ sL,M + ∂ s ∗ L,M d sN = M L − M = N d sL,M : Tot ⊕ ( C s ( π )) N → Tot ⊕ ( C s ( π )) N − . NOTE ON HOMOLOGY FOR SMALE SPACES L, M + 1
L, M∂ s ∗ ∂ s L − , M Figure 2.
A representation of the complexes C s ( π ) • , • and Tot ⊕ ( C s ( π )) • .The direct sums of the groups lying on the dashed diagonals giveTot ⊕ ( C s ( π )) • . The differentials d s (e.g., the zigzag arrow in the top-leftsquare) run from south-east to north-west, decreasing degree by 1. By slightly modifying the invariant Σ D s (Σ), we can introduce a cochain com-plex which is related to (7), and will give rise to another double complex. We sum-marize the details of this construction (see [ , Definition 4.1.5]): – For any L ≥
0, the symmetric group S M +1 acts by automorphisms (in particular, s -bijective maps) on Σ L,M ( π ). Define the group D s, A (Σ L,M ( π )) = { a ∈ D s (Σ L,M ( π )) | a = sgn( β ) β ( a ) for all β ∈ S M +1 } ; – By [ , Lemma 5.1.6], we have ∂ sL,M D s, A (Σ L,M ( π )) ⊆ D s, A (Σ L − ,M ( π )) ∂ s ∗ L,M D s, A (Σ L,M ( π )) ⊆ D s, A (Σ L,M +1 ( π )); – Define a bicomplex ( C s, A ( π ) • , • , ∂ s , ∂ s ∗ ) by setting C s, A ( π ) L,M = D s, A (Σ L,M ( π )); – The inclusion map J : D s, A (Σ L,M ( π )) → D s (Σ L,M ( π )) induces chain maps( C s, A ( π ) • , • , ∂ s , ∂ s ∗ ) → ( C s ( π ) • , • , ∂ s , ∂ s ∗ )(Tot ⊕ ( C s, A ( π )) • , d s ) → (Tot ⊕ ( C s ( π )) • , d s ) , and in particular, for each L ≥
0, a cochain map( C s, A ( π ) L, • , ∂ s ∗ ) → ( C s ( π ) L, • , ∂ s ∗ ) . The advantage in using the complex just defined lies in the following propositions,proved in [ , Theorem 4.2.12, Theorem 4.3.1] Proposition 1.13 . —
There is N ≥ such that C s, A ( π ) L,M = 0 whenever M ≥ N . Proposition 1.14 . —
For each, L ≥ , the cochain map J : ( C s, A ( π ) L, • , ∂ s ∗ ) → ( C s ( π ) L, • , ∂ s ∗ ) NOTE ON HOMOLOGY FOR SMALE SPACES is a quasi-isomorphism, i.e., for each N ∈ Z there are induced isomorphisms J ∗ : H N ( C s, A ( π ) L, • , ∂ s ∗ ) ∼ = H N ( C s ( π ) L, • , ∂ s ∗ ) . Definition 1.15 . —
By considering the symmetric group action S L +1 on Σ L,M ( π ),one can introduce another invariant D s Q (Σ L,M ( π )) = D s (Σ L,M ( π )) D s B (Σ L,M ( π )) , where D s B (Σ L,M ( π )) is the subgroup of D s (Σ) generated by – all elements a satisfying α ( a ) = a for some non-trivial transposition α in thesymmetric group S L +1 ; – all elements of the form a − sgn( α ) α ( a ), where α ∈ S L +1 .Associated to this invariant is the bicomplex denoted C s Q ( π ) in [ , Chapter 5]. Thiscomplex enjoys the analogous property of Proposition 1.13, i.e., it is zero outside abounded region in the L -direction.By combining both approaches, one can also introduce a fourth bicomplex, denoted C s Q , A ( π ) and based on the following “dimension group”: D s Q , A (Σ L,M ( π )) = D s, A (Σ L,M ( π )) D s B (Σ L,M ( π )) ∩ D s, A (Σ L,M ( π )) . This complex is zero outside a bounded rectangle of the first quadrant. Further detailson these constructions are found in [ , Definition 5.1.7]. It is proved in [ , Section5.3] that there are quasi-isomorphisms C s, A ( π ) → C s Q , A ( π ) → C s Q ( π ) . These results and constructions will be obtained through different methods in thenext sections of this paper.By definition, the (stable) homology groups of ( X, φ ) are given by ( N ∈ Z ) H sN ( X, φ ) = H N (Tot ⊕ ( C s Q , A ( π )) • , d s ) . It is proved in [ , Section 5.5] that this definition does not depend on the particularchoice of s/u -bijective map π . C s, A ( π ) is quasi-isomorphic to C s ( π )We are going to prove in this section that C s, A ( π ) is quasi-isomorphic to C s ( π ).This is the missing (but conjectured) result from Putnam’s memoir [ , page 90]. Forbrevity, we write C = C s ( π ) and C A = C s ( π ) , A .There are at least two reasons why this quasi-isomorphism is important: firstly, itis clear that C is the most straight-forward among the definable complexes for thehomology of Smale spaces, and therefore it is a basic fundamental result that it’scomputing the same invariants as the other complexes. Secondly, and maybe more NOTE ON HOMOLOGY FOR SMALE SPACES ∂ s ∗ ∂ s p, Mp, M +1 Figure 3.
The vetical filtration. importantly, C is also the complex with the most evident connection to K -theory forthe associated C ∗ -algebras. Indeed, if we consider the C ∗ -morphisms induced by δ l and δ ,m as explained in [ ], their corresponding K -theory maps agree with δ sl and δ s ∗ ,m after the identification given in Corollary 1.11. We proceed by defining the vertical filtration on C • , • , i.e., thefamily of subcomplexes given by ( p ∈ Z ) F p C L,M = ( C L,M if L ≤ p , in other words everything to the right of the vertical line L = p is set to zero, seeFigure 3.The resulting family { Tot ⊕ ( F p C ) • | p ∈ Z } is a filtration of the totalization chaincomplex. Note there is a chain of inclusions · · · ⊆ Tot ⊕ ( F p C ) • ⊆ Tot ⊕ ( F p +1 C ) • ⊆ · · · . (8)In complete analogy, we get a filtration · · · ⊆ Tot ⊕ ( F p C A ) • ⊆ Tot ⊕ ( F p +1 C A ) • ⊆ · · · . (9)The following remarks will be important in the next subsection. Remark 2.1 . —
The filtration in (8) is exhaustive , i.e., the union over all p ofTot ⊕ ( F p C ) • is Tot ⊕ ( C ) • . Note this implies the induced filtration on homology is alsoexhaustive. The same holds for (9). Remark 2.2 . —
The filtration in (8) is bounded below , i.e., for each N ∈ Z thereexists s ∈ Z such that Tot ⊕ ( F s C ) N = 0. For N ≥ s = N −
1; when
N < s = −
1. Note this implies the induced filtration on homology is alsobounded below. The same holds for (9).
NOTE ON HOMOLOGY FOR SMALE SPACES A filtration of a chain complex gives rise to a spectralsequence, see [ , Theorem 5.4.1] for a proof. Proposition 2.3 . —
A filtration F of a chain complex C determines a spectral se-quence: E pq = F p C p + q /F p − C p + q E pq = H p + q ( E p • ) . In order to discuss convergence for the spectral sequence in Proposition 2.3, weintroduce a bit of terminology (we follow [ , Chapter 5]). The expert reader mayskip to Corollary 2.7.Recall that a (homology) spectral sequence is bounded below if for each n there is s = s ( n ) such that the terms E rpq with p + q = n vanish for all p < s . A spectralsequence is regular if for each p and q the differentials d rpq leaving E rpq are zero for alllarge r . Remark 2.4 . —
Bounded below spectral sequences are regular. If F is a boundedbelow filtration (see Remark 2.2), then the spectral sequence in Proposition 2.3 isbounded below, hence regular.For each n ∈ Z , the homology group H n ( C ) receives an induced filtration · · · ⊆ F p H n ( C ) ⊆ F p +1 H n ( C ) ⊆ · · · ⊆ H n ( C ) . We say the spectral sequence abuts to to H ∗ ( C ) if, for all p, q, n ∈ Z ,1. there are isomorphisms β pq : E ∞ pq ∼ = F p H p + q ( C ) / F p − H p + q ( C ); (10)2. H n ( C ) = ∪ p F p H n ( C );3. ∩F p H n ( C ) = 0.When ( F , C ) = ( F, Tot ⊕ ( C )) or ( F , C ) = ( F, Tot ⊕ ( C A )), items 2 and 3 above followfrom Remarks 2.1 and 2.2 respectively.We say the spectral sequence converges to H ∗ ( C ) if it abuts to H ∗ ( C ), it is regular,and it holds for each n ∈ Z that H n ( C ) = lim ←− p ∈ Z H n ( C ) F p H n ( C ) . Note that a bounded below (hence regular) spectral sequence always satisfies the con-dition above, therefore it converges to H ∗ ( C ) as soon as the abutment condition holds.This applies to the spectral sequences associated to ( F, Tot ⊕ ( C )) and ( F, Tot ⊕ ( C A )),because of Remarks 2.2 and 2.4.Suppose { E rpq } and { E ′ rpq } satisfy (10) with respect to H ∗ and H ′∗ respectively. Wesay that a map h : H ∗ → H ′∗ is compatible with a morphism f : E → E ′ if – h ( F p H n ) ⊆ F p H ′ n for all n ∈ Z ; NOTE ON HOMOLOGY FOR SMALE SPACES – the induced maps F p H n / F p − H n → F p H ′ n / F p − H ′ n correspond under β and β ′ to f ∞ pq : E ∞ pq → E ′∞ pq , q = n − p .We recall the following result [ , Theorem 5.5.1]. Theorem 2.5 . —
Condition (10) holds for bounded below spectral sequences.In particular, if ( F , C ) is a filtered chain complex where F is exhaustive and boundedbelow, then the associated spectral sequence is bounded below and converges to H ∗ ( C ) .Moreover, the convergence is natural: if f : C → C ′ is a map of filtered complexes,then the map f ∗ : H ∗ ( C ) → H ∗ ( C ′ ) is compatible with the corresponding morphism ofspectral sequences. Corollary 2.6 . —
With notations as above, if f r : E rpq ∼ = E ′ rpq is an isomorphismfor all p, q and some r (hence for r = ∞ , see [ , Lemma 5.2.4] ), then f ∗ : H ∗ ( C ) → H ∗ ( C ′ ) is an isomorphism. Corollary 2.7 . —
There are convergent spectral sequences E pq = H q (( C A ) p, • , ∂ s ∗ ) ⇒ H p + q (Tot ⊕ ( C A ) • , d s ) ∼ = H sp + q ( X, φ ) E ′ pq = H q ( C p, • , ∂ s ∗ ) ⇒ H p + q (Tot ⊕ ( C ) • , d s ) . Furthermore, the inclusion chain map J : (Tot ⊕ ( C A ) • , d s ) → (Tot ⊕ ( C ) • , d s ) is a quasi-isomorphism.Proof . — The spectral sequences arise by applying Proposition 2.3 with ( F , C ) =( F, Tot ⊕ ( C )) and ( F , C ) = ( F, Tot ⊕ ( C A )). Convergence follows from Theorem 2.5.The map J induces isomorphisms J : E pq = H q (( C A ) p, • , ∂ s ∗ ) ∼ = E ′ pq = H q ( C p, • , ∂ s ∗ )for all p, q by Proposition 1.14. The result follows from Corollary 2.6 above. Corollary 2.8 . —
Homology groups for the Smale space ( X, φ ) can be equally definedas ( N ∈ Z ) H sN ( X, φ ) = H N (Tot ⊕ ( C s ( π )) • , d s ) .
3. Simplicial viewpoint
The s/u -bijective pair π = ( Y, ψ, π s , Z, ζ, π u ) for ( X, φ ) gives rise to a bisimplicial
Smale space (Σ
L,M ( π )) L,M ≥ . We will drop the reference to π for brevity. NOTE ON HOMOLOGY FOR SMALE SPACES The face maps are given by (1). We stress that the δ l ’s are s -bijective and the δ ,m ’s are u -bijective. The degeneracy maps are as follows: s l : Σ L,M → Σ L +1 ,M ( y , . . . , y l , . . . , y L , z , . . . , z M ) ( y , . . . , y l , y l . . . , y L , z , . . . , z M ) s ,m : Σ L,M → Σ L,M +1 ( y , . . . , y L , z , . . . , z m , . . . , z M ) ( y , . . . , y L , z , . . . , z m , z m , . . . , z M ) , for l = 0 , · · · , L and m = 0 , . . . , M . Remark 3.1 . —
Let us point out a notational difference between the present paperand [ ]. Whenever we write δ l , Putnam appends an extra comma to the subscript,i.e., δ l, . We deviated from this convention because the distinction between δ l, and δ ,m is less important in our case, for we rarely use δ ,m . In accordance with this usage, wedenote the map s l and not s l, . Remark 3.2 . —
Note that s l (Σ L,M ) ⊆ Σ L +1 ,M is a closed shift-invariant system,clearly isomorphic to Σ L,M . The same holds for s ,m (Σ L,M ) ⊆ Σ L,M +1 . Remark 3.3 . —
It is not difficult to see that, for each l , the map s l is s -bijectivebecause its inverse is given by δ l . The situation is different for the maps s ,m : theyare only s -resolving. Proposition 3.4 . —
There are induced maps s sl : D s (Σ L,M ) → D s (Σ L +1 ,M ) s s ∗ ,m : D s (Σ L,M +1 ) → D s (Σ L,M ) . Moreover, the map s sl is split-injective.Proof . — Recall that (equivalence classes of) compact open sets inside stable or-bits provide generators for the dimension groups. Given one of such classes [ E ] ∈ D s (Σ L,M ), the assignment [ E ] [ s l ( E )] is a well-defined group morphism becausethe map s l : Σ sL,M ( e ) → Σ sL +1 ,M ( s l ( e )) is a homeomorphism. The splitting for themap s sl is given by δ sl . The definition of s s ∗ ,m is given by E s − ,m ( E ). This preim-age is compact because s ,m is proper, as s -resolving maps are proper [ , Theorem2.5.4]. Since E ∩ F = ∅ implies s ,m ( E ) ∩ s ,m ( F ) = ∅ , the map respects the groupoperation.The theorem below follows easily from the discussion so far (and some simpleverifications). See [ , Chapter 8] for the Dold-Kan correspondence. Theorem 3.5 . —
Applying the D s -functor to the bisimplicial space Σ • , • results ina simplicial cosimplicial group ( D s (Σ • , • ) , δ sl , s sl , δ s,m , s s ∗ ,m ) . Furthermore the unnor-malized double complex associated to said group via the Dold-Kan correspondenceis ( C L,M , ∂ s , ∂ s ∗ ) , as defined in Section 1. NOTE ON HOMOLOGY FOR SMALE SPACES Remark 3.6 . —
As was mentioned at the beginning of Section 2, the complex( C L,M , ∂ s , ∂ s ∗ ) is also the result of applying the K -theory functor to Σ • , • . The in-termediate step in this case is constructing the associated C ∗ -algebras, which are AF[ , Section 4.3], so the odd K -groups vanish.By considering the normalizations (sometimes called the Moore complexes ) associ-ated to D s (Σ • , • ) we obtain simplicial versions of the bicomplexes C A , C Q , C A , Q thatwere previously introduced. It is well-known that these all yield isomorphic homologygroups (see [ , Theorem 8.3.8]). However, it should be noted that these complexesare not as useful as their “symmetric” counterpart (to be introduced in the next sec-tion), because they don’t allow for computational simplifications as in Proposition1.13. Fix M ≥ • ,M , δ sl , s sl ). It carries an action of the symmetric group S L +1 . Recall thatthis group is generated by the adjacent transpositions t l = ( l l + 1), l = 0 , . . . , L − , Section 5, Theorem 3]).The functorial properties of the D s -invariant easily give the theorem below. Thenotion of symmetric simplicial group is inspired by [ ]. Theorem 3.7 . —
The simplicial group (Σ • ,M , δ sl , s sl ) is a symmetric object , i.e., itcarries an action of the transpositions t i ’s, subject to the defining relations of S L +1 and to the following mixed relations : δ sj t i = t i δ sj s sj t i = t i s sj ( i < j − δ si t i = δ si +1 s si t i = t i +1 t i s si +1 δ sj t i = t i − δ sj s sj t i = t i +1 s sj ( i > j ) t i s si = s si . For some l and j = 1 , . . . , L + 1 − l , we are going to consider the cycle σ j =( l + j l + j − · · · l + 1) in S L +1 and the compositions σ j s l . Note σ s l = s l . In otherwords σ j s l is an additional degeneracy map which repeats entry y l at coordinate l + j :( y , . . . , y l , . . . , y L ) σ j s l / / ( y , . . . , y l , . . . , y l + j − , y l , y l + j , . . . y L ) ∈ Σ L +1 ,M . As composition of s -bijective maps, the σ j s l ’s induce group morphisms D s (Σ L,M ) ( σ j s l ) s / / D s (Σ L +1 ,M ) . It is then natural to define the groups of degenerate chains,˜ DC L,M = X l,j ( σ j s l ) s ( C L − ,M ) . NOTE ON HOMOLOGY FOR SMALE SPACES The subgroup P l ( σ s l ) s ( C L − ,M ) is preserved by the differential ∂ s thanks to thesimplicial identities, but when j > ∂ s ( ˜ DC L,M ) ⊆ ˜ DC L − ,M + h σ j ( a ) − sgn( σ j )( a ) | a ∈ C L,M i , (11)because δ sl ( σ j s l ) s ( a ) = σ j ( a ) and δ sl + j ( σ j s l ) s ( a ) = a . Lemma 3.8 . —
There is an equality ( a ∈ C L,M ) h σ j ( a ) − sgn ( σ j )( a ) i = h t i ( a ) + a | i = 1 , . . . , L − i = h α ( a ) − sgn ( α )( a ) | α ∈ S L +1 i . Proof . — Since the t i ’s are generators we can write α ( a ) = t i · · · t i n ( a ). Then wehave ( t i · · · t i n ( a ) + t i · · · t i n ( a )) − ( t i · · · t i n ( a ) + t i · · · t i n ( a ))+ ( t i · · · t i n ( a ) + t i · · · t i n ( a )) − · · · ± ( t i n ( a ) + a ) = α ( a ) ± a. The sign is positive when n is odd and negative when n is even, i.e., it is in accordancewith − sgn( α ). Note that our notation for σ j does not make reference to the index l ,so that σ j ( a ) for j = 2 includes all elements of the form t i ( a ).We can now “correct” our definition of degenerate chains by setting DC L,M to bethe group generated by ˜ DC L,M and h α ( a ) − sgn( α )( a ) | a ∈ D s (Σ L,M ) , α ∈ S L +1 i . Lemma 3.9 . — ( DC • ,M , ∂ s ) is a well-defined subcomplex of ( C • ,M , ∂ s ) .Proof . — In view of the remark in (11), we only need to check what happens to ∂ s ( t i ( a ) + a ). By looking at the identities in Theorem 3.7, we see that it suffices tocheck the expression δ si ( t i ( a ) + a ) − δ si +1 ( t i ( a ) + a ). It is easy to see that δ si +1 t i = δ si so we get δ si ( t i ( a ) + a ) − δ si +1 ( t i ( a ) + a ) = δ si +1 ( a ) + δ si ( a ) − δ si ( a ) − δ si +1 ( a ) = 0 . Therefore ∂ s preserves the subgroup h α ( a ) − sgn( α )( a ) | a ∈ D s (Σ L,M ) , α ∈ S L +1 i . Theorem 3.10 . —
Consider the short exact sequence / / DC • ,M / / C • ,M / / C • ,M DC • ,M / / . The complex DC • ,M is acyclic, hence the projection map is a quasi-isomorphism. NOTE ON HOMOLOGY FOR SMALE SPACES Proof . — Set DC L = DC L,M for brevity. We filter DC • ,M by setting F DC L = 0and F p DC L = k X l =0 L − l X j =1 ( σ j s l ) s ( C L − ,M )+ n X j =1 ( σ j s k +1 ) s ( C L − ,M ) + h α ( a ) − sgn( α )( a ) i when p = L +( L − · · · +( L − k )+ n and 0 ≤ n ≤ L − k −
1. When p ≥ L ( L +1) / F p DC L = DC L . The simplicial (and mixed) identities show that each F p DC • isa subcomplex. This filtration F is bounded, so there is a convergent spectral sequence(see [ , Theorem 5.5.1]) E pq = H p + q ( F p DC • /F p − DC • ) ⇒ H p + q ( DC • ) . We have reduced ourselves to showing that each F p DC/F p − is acyclic. We take x ∈ DC L − and compute in F p DC/F p − : ∂ s ( σ n s k +1 ) s ( x ) = L X i = k + n +2 ( − i ( σ n s k +1 ) s ( δ si − )( x ) ∂ s ( σ n s k +1 ) s ( σ n s k +1 ) s ( x ) + ( σ n s k +1 ) s ∂ s ( σ n s k +1 ) s ( x )= L +1 X i = k + n +2 ( − i ( σ n s k +1 ) s ( δ si − )( σ n s k +1 ) s ( x ) − L X i = k + n +2 ( − i ( σ n s k +1 ) s ( σ n s k +1 ) s ( δ si − )( x )= ( − p ( σ n s k +1 ) s ( x ) . Hence ψ L = ( − p ( σ n s k +1 ) s is a chain contraction of the identity map which implies F p DC/F p − is acyclic. Corollary 3.11 . —
There is an isomorphism of double complexes: (( C Q ) • , • ) , ∂ s , ∂ s ∗ ) ∼ = C • ,M DC • ,M , ∂ s , ∂ s ∗ ! . In particular, for each M ≥ there is a quasi-isomorphism of chain complexes (( C • ,M ) , ∂ s ) → (( C Q ) • ,M , ∂ s ) . (12) Proof . — All we need to do is identifying D s B (Σ L,M ) with DC L,M . Obviously h α ( a ) − sgn( α )( a ) i = h a − sgn( α ) α ( a ) i , and elements in the image of the degeneracy maps are clearly left invariant by somenon-trivial transposition. Given [ E ] ∈ D s B (Σ L,M ) such that [ E ] = [ α ( E )] for some NOTE ON HOMOLOGY FOR SMALE SPACES transposition, we need to show [ E ] = [ F ] for some clopen F in the image of a degen-eracy map. Now suppose α = ( i i + k ) and define F to be ( σ k s i ) δ i + k ( E ). Then thecondition [ E, F ] = E, [ F, E ] = F trivially holds separately on each coordinate y l with l = i + k , and when l = i + k we can check the condition replacing F by α ( E ), becausethe coordinate of index i + k is pointwise the same in F and α ( E ).Note that (12) is a chain version of Proposition 1.14 and is proved in [ , Theorem4.3.1]. We have used Proposition 1.14 in order to establish the quasi-isomorphism(Tot ⊕ ( C A ) • , d s ) → (Tot ⊕ ( C ) • , d s ) given by inclusion. Dually, it is natural to seek aquasi-isomorphism (Tot ⊕ ( C ) • , d s ) → (Tot ⊕ ( C Q ) • , d s ) induced by projection, whichmakes use of (12). Of course the strategy is completely similar to Corollary 2.7, butconsidering the horizontal filtration instead of the vertical one. We skip the details. Corollary 3.12 . —
The projection map in Theorem 3.10 induces a quasi-isomorphism (Tot ⊕ ( C ) • , d s ) → (Tot ⊕ ( C Q ) • , d s ) . To complete the picture, we also give the dual version of Proposition 1.13.
Proposition 3.13 . —
There is N ≥ such that C L,M = DC L,M whenever L ≥ N .Therefore ( C Q ) L,M = 0 whenever L ≥ N .Proof . — Recall that we can choose the s/u -bijective pair π = ( Y, ψ, π s , Z, ζ, π u ) sothat π s is finite-to-one. Let N − π s and L ≥ N . A generic generator for D s (Σ L,M ) is a compact open in some stable orbit E ⊆ Σ sL,M ( e ). By the choice of L , there are i and k such that e = ( y , y , . . . , y L , . . . )with y i = y i + k . Since δ i + k : Σ sL,M ( e ) → Σ sL − ,M ( δ i + k ( e )) is a homeomorphism, we seethat E = ( σ k s i )( E ). As was hinted at the end of the previoussection, the methods so far can be promptly dualized by considering the symmetriccosimplicial group (Σ L, • , δ s ∗ ,m , ( σ i s ,m ) s ∗ ) for fixed L ≥ h a − sgn( α ) α ( a ) i ,we are led to consider the dual notion of “invariants”. This brings to defining CC L,M = { a ∈ C L,M | ( σ j s ,m ) s ∗ ( a ) = 0 , a − sgn( α ) α ( a ) = 0for all m = 0 , . . . , M, j = 1 , . . . , M + 1 − m, α ∈ S M } , that is the invariant elements lying in intersection of kernels for all degeneracy maps. NOTE ON HOMOLOGY FOR SMALE SPACES By the (dual) argument of 3.10 one can proceed to show that the quotient complex C L, • /CC L, • is acyclic, so the inclusion CC L, • → C L, • is a quasi-isomorphism. Finally, we mention that the projection map in Theorem 3.10clearly induces a chain map CC • ,M → CC • ,M DC • ,M ∩ CC • ,M , which is an isomorphism on homology. An explicit inverse for the map is constructedin [ , page 98].
4. Projective covers
Given a sheaf S over a paracompact Hausdorff space X , sheaf cohomology H ∗ ( X, S )is computed from the complex Hom X ( Z , I • ), where I • is an injective resolution of S .It is true, but perhaps less well-known, that the same calculation can be performedby means of the complex Hom X ( E • X, S ), where E • X is a semi-simplicial resolutionof X arising from a projective cover E of X (see [ ]).This alternative path to computing sheaf cohomology calls for an analogy with thehomology theory for Smale spaces. Indeed, we have seen how the defining complexarises by applying Krieger’s invariant to the bisimplicial space induced by a chosen s/u -bijective pair. So the role of the global section functor is played, in our context,by the dimension group construction for subshifts.The analogy is stronger when we start with a Smale space with totally disconnectedstable sets. In this case, the homology is computed by the complex ( C Q ( π ) • , , ∂ s ) andthe s/u -bijective pair is reduced to a simple s -bijective map π : Σ → X, where Σ is a subshift of finite type (see [ , Section 7.2]). Thus in this case the anal-ogy calls for considering Σ as a “projective” cover of X , together with its associatedsimplicial resolution Σ • obtained by taking iterated fibered products over π .It should be noted that, while the usage of the term “resolution” is somewhatjustified (since by definition Σ • computes the “right” homology groups), the attribute projective requires further reasons. This section contains a simple theorem in thisdirection.In the category of compact Hausdorff spaces and continuous maps, a projectiveobject is a space E such that, whenever we are given f : E → A and g : B ։ A (onto),there is h : E → B with f = g ◦ h .A projective cover of X is a pair ( E, e ) with E projective and e : E ։ X irreducible ,i.e., mapping proper closed sets onto proper subsets. NOTE ON HOMOLOGY FOR SMALE SPACES Gleason [ ] has proved that projective covers exist and are unique (up to a homeo-morphism making the obvious diagram commute). Moreover he showed that a spaceis projective if and only if it is extremally disconnected, i.e., the closure of each of itsopen sets is open. Recall that Σ is a compact, Hausdorff, totally disconnected space.In general extremally disconnected Hausdorff spaces are totally disconnected, but theconverse does not hold.Let ( X, φ ) be a non-wandering Smale space and (
E, e ) its projective cover. Notethat φ induces a self-homeomorphism ˜ φ of E and e intertwines ˜ φ, φ . Consider thetotally disconnected space Σ , ( π ) associated to a choice of s/u -bijective pair π (thisis the correct analogue of Σ when X is not totally disconnected along the stabledirection). The difference between E and Σ , ( π ) can be recast in terms of thedependence of the latter space on π . This suggests that in order to make sense ofprojectivity in the context of Smale spaces we ought to consider all s/u -bijective pairsat the same time.The discussion on projectivity will inevitably bring us outside the category ofSmale spaces (e.g., extremally disconnected spaces are not metrizable, unless theyare discrete), therefore the following setup is in the context of (invertible) dynamicalsystems. See also Remark 4.2 below. An open set in a space X is called regular if itis the interior of its closure. A regular partition P of X is a finite collection of disjointregular opens in X whose union is dense.Let ( X, φ ) be an invertible dynamical system and P a regular partition of X . View P as an alphabet and let a a · · · a n be a word. We say this word is allowed if ∩ ni =1 φ − i ( a i ) = ∅ and let L P be the family of allowed words. It can be checked [ ,Section 6.5] that L P is the language of a shift space that we denote Σ P . Note thatfor each x ∈ Σ P and n ∈ N , the set D n ( x ) = n \ i = − n φ − i ( x i ) ⊆ X is nonempty. Definition 4.1 . —
We say that P is a symbolic presentation of ( X, φ ) if for every x ∈ Σ P the set ∩ ∞ n =0 D n ( x ) consists of exactly one point. We call P a Markov partition if Σ P is a subshift of finite type.Other definitions of Markov partitions are common in the literature, e.g., [ ]. No-tice that the set of regular partitions is directed: we write P ≤ P if P is a refinementof P , i.e., each member of P is contained in a member of P . Given partitions P , P we can define an upper bound P ∩ P , obtained by taking pairwise intersections ofelements from each partition.If ( X, φ ) admits a symbolic presentation P , then given any regular partition P wehave that P ∩ P is again a symbolic presentation. In other words, once a symbolic NOTE ON HOMOLOGY FOR SMALE SPACES presentation exists, we can guarantee that the family of symbolic presentations iscofinal among all regular partitions.Associated to P we get a factor map (i.e., an equivariant surjection) π P : Σ P → X (see [ , Proposition 6.5.8]). If P is a refinement of P , then π P : Σ P → X isa factor map which factors through Σ P . Indeed if we view P and P as alphabets,there is a code µ P , P which assigns to each letter a ∈ P the unique letter b ∈ P such that a ⊆ b , and π P = π P ◦ µ P , P .As a result, if I denotes the family of symbolic presentations of ( X, φ ) (assumingit is nonempty), then (Σ i , µ ij , π i ) i ≤ j ∈ I defines a projective system in the category ofdynamical systems over X . Let E be the inverse limit of (Σ i , µ ij , π i ) i ≤ j ∈ I . Since E ⊆ Q i Σ i , the shift map σ applied componentwise turns E into a dynamical system.Given P ∈ I , denote by p P the canonical projection E → Σ P .The case of Smale spaces is as follows. A non-wandering Smale space ( X, φ ) alwaysadmits a Markov partition [ , Section 7]. If we denote such partition by M , then Σ M is a subshift of finite type endowed with an almost one-to-one factor map π M : Σ M → X (i.e., an equivariant surjection that is finite-to-one, and the set of points in X withsingle preimage is a dense G δ ). If P is a refinement of M , then π P : Σ P → X is analmost one-to-one factor map which factors through Σ M . As a result, in this case wecan take I to be the family of refinements of M . Remark 4.2 . —
It is worth noting that { Σ i } i ∈ I is a collection of shift spaces thatare not necessarily of finite type (in particular, they are not Smale spaces). That isbecause the refinement of a Markov partition is not a Markov partition (in general).It is unclear to the author if there are conditions under which a Smale space admitsa cofinal collection of Markov partitions. Theorem 4.3 . —
Let ( X, φ ) be a dynamical system which admits a symbolic presen-tation P . Suppose (Σ i , µ ij , π i ) i ≤ j ∈ I is the projective system associated to the collectionof symbolic presentations of ( X, φ ) and denote by ( E, σ ) the associated inverse limit.Then ( E, σ ) is a projective cover of ( X, φ ) and the map e : E → X is given by thecomposition E p P / / Σ P π P / / X .
Proof . — Let J be the family of regular partitions of X . Given P ∈ J , denoteby X ( P ) the topological space given by the disjoint union ∪ Y ∈P Y . Then by [ ,Proposition 17] we have that E ′ = lim ←− j ∈ J ( X ( j ) , f jk )is a projective cover of X (here f jk : X ( k ) ։ X ( j ) when j ≤ k is the obvious surjectioninduced by the refinement). First of all we notice that I is cofinal in J so that thelimit can be taken over the index set I . Secondly, notice that for each i ∈ I there isa natural surjection p i : X ( i ) → X . We claim that π i : Σ i → X factors through p i . NOTE ON HOMOLOGY FOR SMALE SPACES Indeed, note that if x ∈ Σ i , then π i ( x ) belongs to x ∈ i and π i ( x ) admits a uniquelift ˜ x ∈ x ⊆ X ( i ). Define ˜ π i ( x ) = ˜ x and by construction π i = p i ◦ ˜ π i .It is easy to check that ( i ≤ j ) Σ j µ ij / / ˜ π j (cid:15) (cid:15) Σ i ˜ π i (cid:15) (cid:15) X ( j ) f ij / / X ( i )is a commuting diagram so that { ˜ π i } i ∈ I induces a (continuous) map of spaces˜ π : E → E ′ . Since ˜ π is a map of compact Hausdorff spaces, we only need to showit is bijective in order to get the required homeomorphism E ∼ = E ′ . In fact, it issufficient to show that it is one-to-one, because ˜ π ( E ) ⊆ E ′ is a closed set mappingonto X , thus by irreducibility ˜ π ( E ) = E ′ .Suppose x, y ∈ E, x = y , so there is i ∈ I with x i = y i . Recall that x i and y i arebi-infinite sequences in Σ i , let us denote their components by ( x ki ) k ∈ Z , ( y ki ) k ∈ Z .There is m ∈ Z with x mi = y mi . Note that φ − m ( i ) is also a symbolic presentation,and if we set α = i ∩ φ − m ( i ) we have i ≤ α , thus there are elements x α , y α ∈ Σ α , appearing at the α -th component of respectively x, y , and satisfying µ iα ( x α ) = x i , µ iα ( y α ) = y i .We claim ˜ π α ( x α ) = ˜ π α ( y α ). In fact, there are A x , B x , A y , B y ∈ i with x α = x i ∩ φ − m ( A x ) x mα = x mi ∩ φ − m ( B x ) y α = x i ∩ φ − m ( A y ) x mα = x mi ∩ φ − m ( B y ) x i ∩ φ − m ( A x ) ∩ φ − m ( x mi ) ∩ φ − m ( B x ) = ∅ y i ∩ φ − m ( A y ) ∩ φ − m ( y mi ) ∩ φ − m ( B y ) = ∅ . From the above we derive A x = x mi , A y = y mi and in particular A x = A y . But bydefinition ˜ π α ( x α ) ∈ x i ∩ φ − m ( A x ) ⊆ X ( α )˜ π α ( y α ) ∈ y i ∩ φ − m ( A y ) ⊆ X ( α )so ˜ π α ( x α ) cannot be equal to ˜ π α ( y α ). This proves injectivity of ˜ π and concludes theproof. Remark 4.4 . —
At first sight, it it reasonable to view E as the “universal” versionof the spaces of the form Σ , ( π ). In the same spirit, one could think of defining a“universal s/u -bijective pair” π = ( E s , ˜ ψ, e s , E u , ˜ ζ, e u ), where E s and E u would beprojective with respect to s -bijective and u -bijective maps.The first step towards this program would be applying Putnam’s lifting theorem[ ] to the projective system { Σ i } i ∈ I of Theorem 4.3 (assuming the system, or acofinal replacement, consists entirely of shifts of finite type). Unfortunately, in orderto lift the entire (infinite) system, limits of spaces are necessary, thus we run once NOTE ON HOMOLOGY FOR SMALE SPACES again into the problem that these limits are not Smale spaces, and the notions of s - and u -bijective maps don’t work well in this context. This suggests that, if onedesires importing the machinery of homological algebra in the setting of Smale spaces,the ambient category should be chosen with care. A good candidate for this categorymight be the equivariant (with respect to the stable or unstable equivalence relation)KK-category, but this idea will not be pursued in the present paper. References [1]
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