Definable Eilenberg--Mac Lane Universal Coefficient Theorems
aa r X i v : . [ m a t h . A T ] O c t DEFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS
MARTINO LUPINI
Abstract.
We prove definable versions of the Universal Coefficient Theorems of Eilenberg–Mac Lane expressingthe (Steenrod) homology groups of a compact metrizable space in terms of its integral cohomology groups, and the(ˇCech) cohomology groups of a polyhedron in terms of its integral homology groups. Precisely, we show that, givena compact metrizable space X , a (not necessarily compact) polyhedron Y , and an abelian Polish group G with thedivision closure property, there are natural definable exact sequences0 → Ext (cid:0) H n +1 ( X ) , G (cid:1) → H n ( X ; G ) → Hom ( H n ( X ) , G ) → → Ext ( H n − ( Y ) , G ) → H n ( Y ; G ) → Hom ( H n ( Y ) , G ) → H n ( X ; G ) is the n -dimensional definable homology group of X with coefficients in G and H n ( Y ; G ) is the n -dimensional definable cohomology group of Y with coefficients in G .Both of these results are obtained as corollaries of a general algebraic Universal Coefficient Theorem relating thecohomology of a cochain complex of countable free abelian groups to the definable homology of its G -dual chaincomplex of Polish groups. Introduction
The integral (Steenrod) homology H • ( X ) of a compact metrizable space X was initially defined by Steenrod inthe seminal paper [40]. Motivated by the problem of computing H • ( X ) when X is the diadic solenoid (a spacehomeomorphic to the Pontryagin dual group of Z [1 / H • ( X ). More generally,Eilenberg and Mac Lane considered the homology H • ( X ; G ) of a compact metrizable space X with coefficients inan abelian group G , and expressed it purely in terms of the cohomology of X via the so-called Universal CoefficientTheorem; see also [25, Section 21.3].The possibility of enriching the homology of a compact metrizable space X was considered in [3]. It is showntherein that H • ( X ) can be considered as a definable graded group (see Section 2.2). Furthermore, H • ( − ) as adefinable graded group is a complete invariant for torus-free solenoids up to homeomorphism, while the same doesnot hold when H • ( − ) is regarded as a purely algebraic graded group.In this paper we consider the definable homology of a compact metrizable space with coefficients in an abelianPolish group G with the division closure property (see Definition 2.1). We prove in this context the natural definableversion of the Universal Coefficient Theorem (UCT) of Eilenberg and Mac Lane for homology, including the caseof pairs of spaces; see Theorem 7.7 and Theorem 7.10. In particular, this shows that the definable graded group H • ( X ; G ) only depends on the integral cohomology of the compact metrizable space X .In [11], Eilenberg and Mac Lane also proved a Universal Coefficient Theorem expressing the cohomology of a(not necessarily compact) polyhedron in terms of its homology. In this paper, we consider definable cohomologyof polyhedra with coefficients in a Polish group with the division closure property. In this context, we prove adefinable version of the UCT of Eilenberg and Mac Lane for cohomology of (pairs of) polyhedra; see Theorem 8.6and Theorem 8.9.In fact, we obtain both the UCT for homology and the UCT for cohomology as particular instances of a generalalgebraic UCT, relating the cohomology of a cochain complex of countable groups to the homology of its G -dual Date : October 13, 2020.2000
Mathematics Subject Classification.
Primary 54H05, 55N07; Secondary 55N10, 18G10, 18G35.
Key words and phrases.
Polish group, definable group, homology, cohomology, Universal Coefficient Theorem, group extension.The author was partially supported by a Marsden Fund Fast-Start Grant VUW1816 from the Royal Society of New Zealand. chain complex of
Polish groups. In particular, our approach can be seen as providing a new proof of the UniversalCoefficient Theorems from [11], although inspired by the original proof of Eilenberg and Mac Lane.The rest of this paper is divided as follows. In Section 2 we recall some fundamental notions concerning Polishgroups and definable groups as introduced in [4, 23]. In Section 3 we present some basic facts about Polish chaincomplexes, and a purely algebraic UCT. The coherent category of towers of Polish chain complexes is introduced inSection 4, and the coherent category of inductive sequences of countable chain complexes is introduced in Section 5.Section 6 recalls some classical notions concerning simplicial complexes and their homological invariants. The UCTfor (pairs of) compact metrizable spaces is inferred from the algebraic version in Section 7. Finally, the UCT for(pairs of) polyhedra is proved in Section 8. In what follows, we let ω denote the set of positive integers includingzero. 2. Polish and definable groups
In this section, we recall the definition and fundamental properties of Polish spaces and definable sets, as well asPolish groups and definable groups, as can be found in [2, 14, 17] and [4, 23].2.1.
Polish spaces and Polish groups. A Polish space is a topological space whose topology is second-countableand induced by some complete metric. Let X be a Polish space. The σ -algebra of Borel sets is the smallest σ -algebraof subsets of X that contains all open sets. A subset of X is Borel if it belongs to the Borel σ -algebra. A closedsubset of a Polish space is a Polish space when endowed with the subspace topology. We regard every countableset as a Polish space endowed with the discrete topology. The product X × Y of two Polish spaces is a Polish spacewhen endowed with the product topology. Similarly, the product Q n ∈ ω X n of a sequence ( X n ) n ∈ ω of Polish spacesis Polish when endowed with the product topology. The category of Polish spaces has Polish spaces as objects andcontinuous functions as morphisms. The Borel category of Polish spaces has Polish spaces as objects and Borelfunctions as morphisms. Let X, Y be a Polish spaces, and f : X → Y be a Borel function. In general the image ofa Borel subset of X under f need not be a Borel subset of Y . However, if f is an injective Borel function, then f maps Borel subsets of X to Borel subsets of Y [17, Theorem 15.1].A Polish group is, simply, a group object in the category of Polish groups in the sense of [24, Section III.6].In other words, a Polish group is a Polish space G endowed with a continuous group operation G × G → G , andsuch that the function G → G mapping each element to its inverse is also continuous. A subgroup H of a Polishgroup G is Polishable if it is Borel and can be endowed with a (necessarily unique) Polish topology that induces theBorel structure on H and turns H into a Polish group. This is equivalent to the assertion that H is the image ofa continuous group homomorphism ϕ : ˆ G → G for some Polish group ˆ G . If G is a Polish group, and H is a closedsubgroup, then H is a Polish group with the subspace topology. If H is closed and normal in G , then G/H is aPolish group with the quotient topology.A continuous action of a Polish group G on a Polish space X is an action G y X that is continuous as a function G × X → X . A Polish G -space is a Polish space endowed with a continuous action of the Polish group G . FollowingSteenrod [39] ad Lefshetz [19], we consider the following notion. Definition 2.1.
An additively-denoted abelian Polish group G has the division closure property if, for every k ∈ N , kG = { kg : g ∈ G } is a closed subgroup of G .Clearly, every countable discrete group has the division closure property.2.2. Definable sets and definable groups.
Suppose that X is a Polish space. We regard an equivalence relation E on X as a subset of the product space X × X . Consistently, we say that E is Borel if it is Borel as a subset of X × X . We now recall the notion of idealistic equivalence relation as formulated in [23, Definition 1.6], which isslightly more generous than the original definition from [16]; see also [14, Definition 5.4.9] and [18]. Definition 2.2.
Let C be a set. A nonempty collection F of nonempty subsets of C is a σ - filter if it satisfies thefollowing: • S n ∈ F for every n ∈ ω implies T n ∈ ω S n ∈ F ; • if S ∈ F and S ⊆ T ⊆ C , then T ∈ F . Definition 2.3.
Let E be an equivalence relation on a Polish space X . Then E is idealistic if there exist a Borelfunction s : X → X and a function C
7→ F C assigning to each E -class C a σ -filter F C of subsets of C such that: EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 3 (1) xEs ( x ) for every x ∈ X ;(2) for every Borel subset A of X × X , the set (cid:8) x ∈ X : { y ∈ [ x ] E : ( s ( x ) , y ) } ∈ F [ x ] E (cid:9) is Borel, where [ x ] E = { y ∈ X : ( x, y ) ∈ E } for x ∈ X .One can equivalently reformulate the notion of idealistic equivalence relation in terms of σ -ideals, due to theduality between σ -ideals and σ -filters. As in [23], we let a definable set X be a pair ( ˆ X, E ) where ˆ X is a Polish spaceand E is a Borel and idealistic equivalence relation on E . We consider the pair ( ˆ X, E ) as an explicit presentationof X as the quotient of a Polish space ˆ X by a “well-behaved” equivalence relation E . Consistently, we denote sucha definable set also by X = ˆ X/E . Every Polish space ˆ X is, in particular, a definable set X = ˆ X/E where X = ˆ X and E is the identity relation on ˆ X . Suppose that X = ˆ X/E and Y = ˆ Y /F are definable sets. A Borel subset Z of X is a subset of the form ˆ Z/E for some E -invariant Borel subset ˆ Z of ˆ X . (Notice that Z is itself a definable set.)Similarly, one defines a closed subset Z of X to be a subset of the form ˆ Z/E for some E -invariant closed subset ˆ Z of X ˆ X . A function ˆ f : ˆ X → ˆ Y is a lift of (or induces ) the function f : ˆ X/E → ˆ Y /F if f ([ x ] E ) = [ ˆ f ( x )] F for every x ∈ ˆ X . Definition 2.4.
Suppose that X = ˆ X/E and Y = ˆ Y /F are definable sets. A function f : X → Y is Borel-definable (or, briefly, definable ) if it admits a Borel lift ˆ f : ˆ X → ˆ Y , and continuously-definable if it admits a continuous liftˆ f : ˆ X → ˆ Y .The category of definable sets has definable sets as objects and definable functions as morphisms. The continuouscategory of definable sets has definable sets as objects and continuously-definable functions as morphisms. Noticethat the category of Polish spaces is a full subcategory of the continuous category of definable sets, and the Borelcategory of Polish spaces is a full subcategory of the category of definable sets.The category of definable sets has nice properties, which generalize analogous properties of the Borel categoryof Polish spaces; see [23, Proposition 1.10]. For instance, if f : X → Y is an injective definable function betweendefinable sets, then its image is a Borel subset of Y . Furthermore, if f : X → Y is a bijective definable functionbetween definable sets, then its inverse f − : X → Y is also definable. The (continuous) category of definable setshas products, where the product of X = ˆ X/E and Y = ˆ Y /F is the definable set X × Y = ( ˆ X × ˆ Y ) / ( E × F ), wherethe equivalence relation E × F on ˆ X × ˆ Y is defined by setting ( x, y ) ( E × F ) ( x ′ , y ′ ) if and only if xEx ′ and yF y ′ .(It is easily seen that E × F is Borel and idealistic whenever both E and F are Borel and idealistic.)A definable group is, simply, a group object in the category of definable sets. Thus, G is a definable group if itis a definable set that is also a group, and such that the group operation G × G → G and the function G → G thatmaps each element to its inverse are definable.If G is a Polish group and ˆ X is a Polish G -space, the corresponding orbit equivalence relation E ˆ XG on ˆ X is definedby setting xE ˆ XG y if and only if there exists g ∈ G such that g · x = y . Then one has that E ˆ XG is idealistic. If E ˆ XG is furthermore Borel, then X := ˆ X/E ˆ XG is a definable set. In particular, if G is a Polish group and H is a normalPolishable subgroup of G , then the coset equivalence relation E GH of H in G is Borel and idealistic, and hence thequotient G/H = G/E GH is a definable group, called a group with a Polish cover in [4]. Definition 2.5.
Given a group with a Polish cover
G/H , its corresponding weak
Polish group is the quotient
G/H of G by the closure of H in G , and its corresponding asymptotic group with a Polish cover is H/H . Notice that theassignment
G/H → G/H is a functor from the continuous category of groups with a Polish cover to the categoryof Polish groups. Similarly, the assignment
G/H → H/H is a functor from the continuous category of groups witha Polish cover to itself.Recall that a graded group G • is a sequence ( G n ) n ∈ Z of groups indexed by Z , where a (degree 0) homomorphism f from G • to H • is a sequence ( f n ) n ∈ Z of group homomorphisms f n : G n → H n . More generally, for k ∈ Z , a degree MARTINO LUPINI k homomorphism f from G • to H • is a sequence ( f n ) n ∈ Z of group homomorphisms f n : G n → H n + k . The notionsof graded definable group and (continuously) definable degree k homomorphism between graded definable groupsare obtained by replacing groups with definable groups and group homomorphisms with (continuously) definablegroup homomorphisms.2.3. Group extensions.
In what follows, we will assume all the groups to be abelian and additively denoted . Wewill also assume that G is a Polish group with the division closure property . Let also A be a countable group. A(normalized 2-) cocycle on A with values in G is a function c : A × A → G such that, for every x, y, z ∈ A : • c ( x,
0) = 0; • c ( x, y ) = c ( y, x ); • c ( x, y ) + c ( x + y, z ) = c ( x, z ) + c ( x + z, y ).We regard the set Z ( A, G ) of cocycles on A with values in G as a closed subgroup of the Polish group G A × A . A coboundary is a cocycle of the form c h ( x, y ) = h ( x ) + h ( y ) − h ( x + y )for some function h : A → G such that h (0) = 0. The set B ( A, G ) of coboundaries is a
Polishable subgroupof Z(
A, G ), being the image of the Polish group G = (cid:8) h ∈ G A : h (0) = 0 (cid:9) under the continuous homomorphism G → Z ( A, G ), h c h . Two cocycles are cohomologous if they belong to the same B ( A, G )-coset. The definable group Ext(
A, G ) is the group with a Polish cover Z ( A, G ) / B ( A, G ); see [4, Section 7]. We let [ c ] be the element ofExt( A, G ) determined by the cocycle c ∈ Z ( A, G ). Remark 2.6.
Given a cocycle c on A with values in G one can then define by recursion on n ≥ c : A n → G by setting c ( x , . . . , x n +1 ) = c ( x , . . . , x n ) + c ( x + · · · + x n , x n +1 ) .Then one can prove by induction that this is a permutation-invariant function satisfying c ( x , . . . , x n , y , . . . , y m ) = c ( x , . . . , x n ) + c ( y , . . . , y n ) + c ( x + · · · + x n , y + · · · + y m )for n, m ≥
1, where c ( x ) = 0 for x ∈ A .The following lemma is also easily proved by induction. Lemma 2.7.
Suppose that F is a countable free group, G is a Polish group, and c is a cocycle on F with values in G . Consider for every n ≥ c : F n → G as in Remark 2.6. For x ∈ F and k ≥ x ( k ) to be the k -tuple ( x, x, . . . , x ). Suppose that B ⊆ F is a free Z -basis of F . Define the function ρ : F → G bysetting, for x = ( k b + · · · + k ℓ b ℓ ) − ( m d + · · · + m s d s ) ∈ F ,where k , . . . , k ℓ , m , . . . , m s are strictly positive integers and b , . . . , b ℓ , d , . . . , d s are distinct elements of B , ρ ( x ) := c ( b ( k )0 , . . . , b ( k ℓ ) ℓ , − d ( m )0 , . . . , − d ( m s ) s ) − s X i =0 m i c ( d i , − d i ) .Then ρ satisfies, for every x, x ′ ∈ F and e ∈ B :(1) ρ ( x ) + c ( x, e ) = ρ ( x + e ) and ρ ( x ) + c ( x, − e ) − c ( e, − e ) = ρ ( x − e );(2) ρ ( x + x ′ ) = c ( x, x ′ ) + ρ ( x ) + ρ ( x ′ ).A group extension of A by G is an exact sequence of Polish groups0 → G i → X π → A → c ] ∈ Ext(
A, G ) defined by c ( x, y ) = i − ( t ( x ) + t ( y ) − t ( x + y )))for x, y, z ∈ A , where t : A → X is a right inverse for the function π : X → A such that t (0) = 0. Notice that [ c ]does not depend on the choice of t . Conversely, given a definable cocycle c on A with values in G one can considerthe corresponding extension X c defined by 0 → G i → X c π → A → EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 5 where X c = A × G is endowed with the product topology and the operation defined by( x, y ) + ( x ′ , y ′ ) = ( x + x ′ , c ( x, x ′ ) + y + y ′ ) .Two cocycles c, c ′ are cohomologous if and only if the corresponding extensions X c and X c ′ are equivalent [35,Section 7.2], namely there exists a (topological) isomorphism ψ : X c → X ′ c that makes the following diagramcommute. G X c AG X ′ c A ψ The assignments described above establish mutually inverse bijections between Ext(
A, G ) and the set of equivalenceclasses of extensions of A by G .We let B w ( A, G ) be the closure of B ( A, G ) inside Z ( A, G ). This is the closed subgroup of Z ( A, G ) consistingof cocycles c that are weak coboundaries , namely c | S × S is a coboundary for every finite (or equivalently, for everyfinitely-generated) subgroup S of A . Define then the pure Ext group (also known as phantom
Ext group) PExt(
A, G )to be the group with a Polish cover B w ( A, G ) / B ( A, G ) [7, 36]. Thus, PExt(
A, G ) is the asymptotic group with aPolish cover associated with Ext(
A, G ); see Definition 2.5. Its name is due to the fact that the cocycles in PExt(
A, G )corresponds to extensions 0 → G i → X p → A → A by G that are pure , in the sense that i ( G ) is a pure subgroup of X ; see [13, Chapter V].Following [11, Section 5], one can give an alternative description of Ext( A, G ) and PExt(
A, G ), as follows.Suppose that F is a free countable abelian group, and R is a subgroup. Define: • Hom(
R, G ) to be the Polish group of homomorphisms R → G , regarded as a closed subgroup of G R , • Hom( F | R, G ) to be the Polishable subgroup of Hom(
R, G ) consisting of group homomorphisms R → G thatextend to a group homomorphisms F → G , and • Hom f ( F | R, G ) to be the subgroup of Hom(
R, G ) consisting of all group homomorphisms R → G that extendto a group homomorphism F → R for every subgroup F of F containing R as a finite index subgroup.By [11, Lemma 5.1 and Lemma 5.2], we have that, for ϕ ∈ Hom(
R, G ), the following assertions are equivalent: • ϕ ∈ Hom f ( F | R, G ); • for every subgroup F of F containing R and such that F /R is finitely-generated, ϕ extends to a grouphomomorphism F → G ; • whenever t ∈ F and m ∈ Z satisfy mt ∈ R , one has that ϕ ( mt ) ∈ mG .This shows that, as G is assumed to have the division closure property , Hom f ( F | R, G ) is a closed subgroup ofHom(
R, G ), and in fact it is equal to the closure of the Polishable subgroup Hom( F | R, G ) of Hom(
R, G ); see [11,Lemma 5.3 and Lemma 5.5].Consider now the group with a Polish cover Hom(
R, G ) / Hom( F | R, G ). The following proposition is the definableversion of [11, Theorem 10.1], and it can be proved in a similar fashion using Lemma 2.7.
Proposition 2.8.
Suppose that G is a Polish group with the division closure property, F is a countable free group,and R ⊆ F is a subgroup. Fix a right inverse t : F/R → F for the quotient map F → F/R in the category of setssuch that t (0) = 0 . Define then the cocycle ζ on F/R with values in R by setting ζ ( x, y ) = t ( x ) + t ( y ) − t ( x + y ) for x, y ∈ F/R . The function
Hom (
R, G ) → Z (
F/R, G ) , θ θ ◦ ζ induces a natural isomorphism Hom(
R, G )Hom( F | R, G ) ∼ −→ Ext(
F/R, G ) in the continuous category of definable groups, which restricts to a natural isomorphism Hom f ( R, G )Hom( F | R, G ) ∼ −→ PExt(
F/R, G ) .The continuously-definable inverse Ext(
F/R, G ) ∼ −→ Hom(
R, G )Hom( F | R, G ) MARTINO LUPINI is induced by the function Z ( F/R, G ) → Hom (
R, G ) , σ ρ σ | R , where ρ σ : F → G satisfies ρ σ ( x ) + ρ σ ( y ) − ρ σ ( x + y ) = σ ( x + F, y + F ) , and can be obtained from σ as in Lemma 2.7. Chain complexes
In this section, we continue to assume all the groups to be abelian and additively denoted. Recall that we denoteby G a Polish group with the division closure property. We consider a duality between chain complexes and cochaincomplexes, and prove a general Universal Coefficient Theorem relating the cohomology of a cochain complex of freecountable groups to the homology of its G -dual chain complex of Polish groups; see Theorem 3.11.3.1. Polish chain complexes. A Polish chain complex is simply a chain complex in the additive category ofPolish groups; see [35, Section 5.5]. Thus, a Polish chain complex is a sequence A = ( A n , ∂ n ) n ∈ Z such that, forevery n ∈ Z , A n is a Polish group, and the differential ∂ n : A n → A n − is a continuous group homomorphismsatisfying ∂ n − ◦ ∂ n = 0. In the following we will omit the homomorphisms ∂ n from the notation for a Polish chaincomplex. We will also omit the subscripts in the homomorphisms ∂ n . The elements of A n are called n -chains of A .Let A and B be Polish chain complexes. A continuous chain map f : A → B with source s ( f ) = A and target t ( f ) = B is a sequence ( f n ) n ∈ Z of continuous group homomorphisms f n : A n → B n satisfying ∂f n − f n − ∂ = 0for every n ∈ Z . The composition gf of f : A → B and g : B → C is defined by setting ( gf ) n = g n f n for n ∈ Z .The identity chain map 1 A of A is defined by (1 A ) n = 1 A n for every n ∈ Z . We say that a Polish chain complex iscountable (free countable, free finitely-generated, respectively) if for every n ∈ Z , A n is a countable (free countable,free finitely-generated) group. One can regard Polish chain complexes as objects of a category, where morphismsare continuous chain maps. An isomorphism in this category is called a continuous chain isomorphism. The productof a sequence ( A ( m ) ) m ∈ ω of chain complexes is the chain complex B = Q m A ( m ) defined by setting B n = Q m A ( m ) n ,where the map ∂ : B n → B n − is induced by the maps ∂ : A ( m ) n → A ( m ) n − for m ∈ ω by the universal property ofthe product of Polish groups.Suppose that A, B are Polish chain complexes, and f, g : A → B are continuous chain maps. A continuous chainhomotopy L : f ⇒ g from f to g is a sequence ( L n ) n ∈ Z of continuous group homomorphisms L n : A n → B n +1 suchthat ∂L n + L n − ∂ = g n − f n for every n ∈ Z . We say that f is the source s ( L ) of L and g is the target t ( L ) of L .If f, g, h : A → B are continuous chain maps, and L : f ⇒ g and L ′ : g ⇒ h are continuous chain homotopies, thentheir composition (over 1-cells) L ′ ◦ L : f ⇒ h is defined by setting ( L ′ ◦ L ) n = L ′ n + L n for n ∈ Z . If f, g : A → B , h : A ′ → A , and k : B → B ′ are continuous chain maps, and L : f ⇒ g is a continuous chain homotopy, then we let Lh : f h ⇒ gh and kL : kf ⇒ kg be the continuous chain homotopies defined by ( Lh ) n = L n h n and ( kL ) n = k n +1 L n for n ∈ Z . The identity chain homotopy 1 f of f : A → B is defined by setting (1 f ) n = 0 : A n → B n +1 for n ∈ Z .It is observed in [22, Section 2] that Polish chain complexes form a strict ω -category, with continuous chainmaps as 1-cells, and continuous chain homotopies as 2-cells; see [20, Chapter 1]. In this paper, we will only referto the notion of k -cells for k ≤
3. If
A, B are Polish chain complexes, f, f ′ : A → B are continuous chain maps,and L, L ′ : f ⇒ f ′ are continuous chain homotopies, then a 3-cell H : L ⇛ L ′ is a sequence of continuous grouphomomorphisms H n : A n → B n +2 for n ∈ Z satisfying ∂H n − H n − ∂ = L ′ n − L n for n ∈ Z .We say that two continuous chain maps f, g : A → B are continuously chain homotopic if there is a continuouschain homotopy h : f ⇒ g . We let [ f ] be the continuous chain homotopy class of the continuous chain map f : A → B . We call [ f ] a continuous chain h-map from A to B . We define the homotopy category of Polishchain complexes to be the category that has Polish chain complexes as objects, and continuous chain h-maps asmorphisms. Isomorphisms in this category are called continuous chain h-isomorphisms .Let A be a Polish chain complex. Then one defines: • the (closed) subgroup of n -cycles Z n ( A ) = ker( ∂ n ) ⊆ A n ; • the (Polishable) subgroup of n -boundaries B n ( A ) = ran( ∂ n +1 ) ⊆ A n ; • the (closed) subgroup of weak n -boundaries B n ( A ) ⊆ Z n ( A ) equal to the closure of B n ( A ) inside A n . Definition 3.1.
The n -th definable homology group of A is the definable group H n ( A ) = Z n ( A ) / B n ( A ). Thehomology H • ( A ) of A is the graded definable group ( H n ( A )) n ∈ Z . Given a ∈ Z n ( A ), we let [ a ] be the correspondingelement of H n ( A ). EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 7
We call H w n ( A ) = Z n ( A ) / B n ( A ) the n -th weak homology group of A , and H ∞ n ( A ) = B n ( A ) / B n ( A ) the n -thasymptotic homology group of A . Since B n ( A ) is a Borel subgroup of Z n ( A ), and since a countable index Borelsubgroup of a Polish group is clopen, H ∞ n ( A ) is either trivial or uncountable. Definition 3.2.
A Polish chain complex A is proper if, for every n ∈ Z , B n ( A ) is a closed subgroup of Z n ( A ).Notice that, if A is proper, then H • ( A ) = H w • ( A ) is a graded Polish group.If f is a continuous chain map from A to B then, f induces a continuously-definable homomorphism H • ( f ) from H • ( A ) to H • ( B ). Furthermore, H • ( f ) only depends on the continuous chain homotopy class of f . Definition 3.3.
The definable homology functor for Polish chain complex is the functor A H • ( A ), [ f ] H • ( f )from the homotopy category of Polish chain complexes to the continuous category of graded definable groups.3.2. Polish cochain complexes.
As in the case of chain complexes, a Polish cochain complex is simply a cochaincomplex in the additive category of Polish groups, namely a sequence A = ( A n , δ n ) n ∈ Z where A n is a Polish group,and the codifferential δ n : A n → A n +1 is a continuous homomorphism. We will often omit the superscripts from themaps δ n : A n → A n +1 . We also denote the Polish cochain complex A as above simply by ( A n ) n ∈ Z . A Polish cochaincomplex A is countable, free countable, free finitely-generated, respectively, if for every n ∈ Z , A n is countable,free countable, free finitely-generated, respectively. A continuous cochain map f : A → B is a sequence ( f n ) n ∈ Z of continuous group homomorphisms f n : A n → B n such that δf n − f n +1 δ = 0 for every n ∈ Z . A continuouscochain homotopy from a continuous cochain map f : A → B to g : A → B is a sequence ( L n ) n ∈ Z of continuousgroup homomorphisms L n : A n → B n − such that δL n + L n +1 δ = g n − f n for every n ∈ Z . Two continuouscochain maps are continuously cochain homotopic if there is a continuous cochain homotopy from one to the other.A continuous cochain h-map is a continuous cochain homotopy class of continuous cochain maps. The homotopycategory of Polish cochain complexes is the category that has Polish cochain complexes as objects and continuouscochain h-maps as morphisms. As in the case of Polish chain complexes, one can also define 3-cells (or, more general, k -cells) in this context. If f, f ′ : A → B are continuous cochain maps, and L, L ′ : f ⇒ f ′ are continuous cochainhomotopies, then a 3-cell H : L ⇛ L ′ is a sequence ( H n ) n ∈ Z of continuous group homomorphisms H n : A n → B n − such that δH n − H n +1 δ = L ′ n − L n for every n ∈ Z .Clearly, there is a correspondence between Polish cochain complexes and Polish chain complexes. Indeed, if( A n , ∂ n ) n ∈ Z is a Polish chain complex, then by setting A n := A − n and δ n := ∂ − n one defines a Polish cochaincomplex. It is easily seen that this establishes an isomorphism from the strict ω -category of Polish chain complexesto the strict ω -category of Polish cochain complexes. This perspective allows one to define the analogue for cochaincomplexes of the homology groups and any other notion about chain complexes. For instance, one defines: • the (closed) subgroup of n -cocycles Z n ( A ) = ker ( δ n ) ⊆ A n ; • the (Polishable) subgroup of n -coboundaries B n ( A ) = ran( δ n − ) ⊆ A n ; • the (closed) subgroup of weak n -coboundaries defined as the closure B n ( A ) of B n ( A ) inside A n . Definition 3.4.
The n -th definable cohomology group of A is the definable group H n ( A ) = Z n ( A ) / B n ( A ). Thehomology H • ( A ) of A is the graded definable group ( H n ( A )) n ∈ Z . Given a ∈ Z n ( A ), we let [ a ] be the correspondingelement of H n ( A ).By definition, the weak group H n w ( A ) of H n ( A ) is the Polish group Z n ( A ) / B n ( A ), which we call the n -th weakcohomology group of A . The asymptotic group of H n ( A ) is the group with a Polish cover H n ∞ ( A ) = B n ( A ) / B n ( A ),which we call the n -th asymptotic cohomology group of A . Definition 3.5.
A Polish cochain complex A is proper if, for every n ∈ Z , B n ( A ) is a closed subgroup of Z n ( A ). Definition 3.6.
The definable cohomology functor for Polish cochain complexes is the functor A H • ( A ),[ f ] H • ( f ) from the homotopy category of Polish cochain complexes to the continuous category of graded definablegroups.3.3. Duality. If A is a countable group, define the G -dual A ∗ of A to be the Polish group Hom( A, G ) of homo-morphisms A → G , regarded as a closed subgroup of G A . We can then consider a G -valued pairing A ∗ × A → G , h ϕ, a i 7→ ϕ ( a ). MARTINO LUPINI
The assignment A A ∗ and f f ∗ , where f ∗ ( ϕ ) = ϕ ◦ f , defines a contravariant functor from the category ofcountable groups to the category of Polish groups. Definition 3.7.
Let A be a countable cochain complex. The G -dual complex A ∗ = Hom( A, G ) is the Polish chaincomplex defined by setting A ∗ n := ( A n ) ∗ and ∂ n := ( δ n − ) ∗ : A ∗ n → A ∗ n − for n ∈ Z . Remark 3.8.
Notice that, in Definition 3.7, if A is a free finitely-generated cochain complex, then A ∗ is a proper Polish chain complex since G is assumed to satisfy the division closure property. This easily follows from thestructure of group homomorphisms between free finitely-generated groups given by the Smith Normal Form forinteger matrices [33].Suppose that A and B are countable chain complexes. A cochain map f : A → B induces a continuous chainmap f ∗ : B ∗ → A ∗ obtained by setting f ∗ n = ( f n ) ∗ : B ∗ n → A ∗ n for n ∈ Z . Similarly, a cochain homotopy L : f ⇒ g induces a continuous chain homotopy L ∗ : f ∗ ⇒ g ∗ by setting L ∗ n = (cid:0) L n +1 (cid:1) ∗ : B ∗ n → A ∗ n +1 .A 3-cell H : L ⇛ L ′ similarly induces a 3-cell H ∗ : L ′∗ ⇛ L ∗ by setting H ∗ n = (cid:0) H n +2 (cid:1) ∗ : B ∗ n → A ∗ n +2 . Definition 3.9. If A is a free countable chain complex, define A ⊗ G to be the Polish chain complex Hom(Hom( A, Z ) , G ).Notice that A ⊗ G is proper if A is a free finitely-generated chain complex by Remark 3.8.3.4. An algebraic Universal Coefficient Theorem.
Suppose that A is a free countable cochain complex. Welet A ∗ be the G -dual Polish chain complex of A . Consider for n ∈ Z the closed subgroupsAnn ( Z n ( A )) = { z ∈ A ∗ n : ∀ a ∈ Z n ( A ) , h z, a i = 0 } ⊆ Z n ( A ∗ )and ◦ H n ( A ∗ ) := Ann ( Z n ( A )) / B n ( A ) ⊆ H n ( A ∗ ) .The pairing between A ∗ and A induces the index homomorphism Index : H • ( A ∗ ) → Hom ( H • ( A ) , G ). This isthe natural continuously-definable homomorphism obtained by setting Index ([ z ]) ([ a ]) = h z, a i for [ z ] ∈ H n ( A ∗ )and [ a ] ∈ H n ( A ). We also define a co-index homomorphismcoIndex : Hom (cid:0) B • +1 ( A ) , G (cid:1) Hom ( Z • +1 ( A ) | B • +1 ( A ) , G ) → H • ( A ∗ ) .This is the natural continuously-definable homomorphism obtained by settingcoIndex ([ f ]) = [( − n ( f ◦ δ n )]for n ∈ Z and f ∈ Hom (cid:0) B n +1 ( A ) , G (cid:1) . Lemma 3.10.
Suppose that A is a free countable cochain complex, and let A ∗ be its G -dual Polish chain complex.(1) The index homomorphism Index : H • ( A ∗ ) → Hom ( H • ( A ) , G )is a natural continuously-definable homomorphism with kernel equal to ◦ H • ( A ∗ ), and it is a split epimor-phism in the continuous category of graded definable groups.(2) The co-index homomorphismcoIndex : Hom (cid:0) B • +1 ( A ) , G (cid:1) Hom ( Z • +1 ( A ) | B • +1 ( A ) , G ) → H • ( A ∗ )is a natural continuously-definable group homomorphism with image equal to ◦ H • ( A ∗ ), and it is a splitmonomorphism in the continuous category of graded definable groups. EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 9
Proof. (1) It is easy to see that Index is a natural continuously-definable homomorphism with kernel equal to ◦ H • ( A ∗ ). We now prove that Index is a split epimorphism in the continuous category of graded definable groups.Recall that A is a free countable cochain complex. Since subgroups of free abelian groups are free abelian, and freeabelian groups are projective, we can fix a left inverse i n † : A n → Z n ( A ) for the inclusion map Z n ( A ) → A n in thecategory of groups. Let also p n : Z n ( A ) → H n ( A ) be the quotient map. We can then consider the continuously-definable homomorphism Hom ( H • ( A ) , G ) → H • ( A ∗ ) , ϕ (cid:2) ϕ ◦ p n ◦ i n † (cid:3) for ϕ ∈ Hom ( H n ( A ) , G ). We claim that this is a well-defined right inverse for Index. Indeed, if n ∈ Z , ϕ ∈ Hom ( H n ( A ) , G ), and z ϕ := ϕ ◦ p n ◦ i n † , then we have that, for every a ∈ A n − , h ∂z ϕ , a i = h z ϕ , δa i = ( ϕ ◦ p n ) ( δa ) = 0.Hence, ∂z ϕ = 0 and z ϕ ∈ Z n ( A ∗ ). Furthermore, we have that for a ∈ Z n ( A ),Index ([ z ϕ ]) ([ a ]) = h z ϕ , a i = (cid:0) ϕ ◦ p n ◦ i n † (cid:1) ( a ) = ϕ ([ a ]).This shows that Index ([ z ϕ ]) = ϕ , concluding the proof.(2) It is easy to see that coIndex is a natural continuously-definable homomorphism. We now prove that coIndexis a split monomorphism in the continuous category of graded definable groups with image equal to ◦ H • ( A ∗ ). For n ∈ Z , consider a right inverse δ n † : B n +1 ( A ) → A n for the split epimorphism δ n : A n → B n +1 ( A ) in the category ofgroups. Then we can consider the continuously-definable homomorphism H • ( A ∗ ) → Hom (cid:0) B • +1 ( A ) , G (cid:1) Hom ( Z • +1 ( A ) | B • +1 ( A ) , G ) , [ z ] (cid:2) ( − n (cid:0) z ◦ δ n † (cid:1)(cid:3) for z ∈ H n ( A ∗ ). It is immediate to verify that this is a right inverse for coIndex. It remains to show that the imageof coIndex is equal to ◦ H • ( A ∗ ). If f ∈ Hom (cid:0) B n +1 ( A ) , G (cid:1) then we have that ( − n f ◦ δ n ∈ Ann ( Z n ( A )), and hencethe image of coIndex is contained in ◦ H • ( A ∗ ). Conversely, if z ∈ Ann ( Z n ( A )), then [ z ] is equal to the image of[( − n ( z ◦ δ n † )] under coIndex. Indeed, for every a ∈ A n , a − ( δ n † ◦ δ n )( a ) ∈ Z n ( A )and hence, since z ∈ Ann ( Z n ( A )), z ( a ) = ( z ◦ δ n † ◦ δ n ) ( a ) = coIndex (cid:0) [( − n (cid:0) z ◦ δ n † (cid:1) ] (cid:1) ([ a ]) . This concludes the proof. (cid:3)
Considering the natural definable isomorphismHom (cid:0) B • +1 ( A ) , G (cid:1) Hom ( Z • +1 ( A ) | B • +1 ( A ) , G ) ∼ = Ext (cid:0) H • +1 ( A ) , G (cid:1) in the continuous category of definable groups as in Proposition 2.8, we can consider the co-index homomorphismas a natural continuously-definable isomorphismcoIndex : Ext (cid:0) H • +1 ( A ) , G (cid:1) → ◦ H • ( A ∗ ) .From Lemma 3.10 we immediately obtain the following. Theorem 3.11.
Suppose that G is a Polish group with the division closure property. Let A be a free countablecochain complex, and let A ∗ be its G -dual Polish chain complex. Then we have a natural continuously-definableexact sequence of definable graded groups (cid:0) H • +1 ( A ) , G (cid:1) H • ( A ∗ ) Hom ( H • ( A ) , G ) 0 coIndex Index which continuously-definably splits, called the UCT exact sequence for A . Exact sequences of chain complexes.
A (short) locally split exact sequence of Polish chain complexes → A → B → C → A, B, C and continuous chain maps i : A → B and π : B → C such that, for every n ∈ Z , the sequence0 → A n i n → B n π n → C n → → A → B → C → Proposition 3.12.
Let → A → B → C → be a locally split exact sequence of Polish chain complexes. Forevery n ∈ Z , let π † n : C n → B n be a right inverse for π n : B n → C n in the category of Polish groups, and let i † n : B n → A n be right inverses for i n : A n → B n in the category of Polish groups, such that i † n π † n = 0 . Define ˆ d n := ( i † n − ◦ ∂ ◦ π † n ) : C n → A n − . Then one has that ˆ d n for n ∈ Z induce a natural degree − continuously-definablehomomorphism d • : H • ( C ) → H •− ( A ) ,called connecting homomorphism , which does not depend on the choice of π † n and i † n . Furthermore, d n fits into theexact sequence · · · → H n ( A ) H n ( i ) → H n ( B ) H n ( π ) → H n ( C ) d n → H n − ( A ) H n − ( i ) → H n − ( B ) → · · · Notice that one can equivalently state Proposition 3.12 in terms of cochain complexes.If 0 → A → B → C → G -dual0 → C ∗ → B ∗ → A ∗ → d • : H • ( C ) → H • +1 ( A ) and a connecting homomorphism d • : H • ( A ∗ ) → H •− ( C ∗ ). Inturn, d • induces natural continuously-definable homomorphisms Ext ( d • , G ) : Ext (cid:0) H • +1 ( A ) , G (cid:1) → Ext ( H • ( C ) , G )and Hom (cid:0) d •− , G (cid:1) : Hom ( H • ( A ) , G ) → Hom (cid:0) H •− ( C ) , G (cid:1) . Lemma 3.13.
Suppose that 0 → A → B → C → → C ∗ → B ∗ → A ∗ → G -dual locally split exact sequence of Polish chain complexes. Thediagram Ext (cid:0) H • +1 ( A ) , G (cid:1) H • ( A ∗ ) Hom ( H • ( A ) , G )Ext ( H • ( C ) , G ) H •− ( C ∗ ) Hom (cid:0) H •− ( C ) , G (cid:1) coIndex A Ext( d • ,G ) d • Index A Hom( d •− ,G )coIndex C Index C commutes. Proof.
Fix, for n ∈ Z , a right inverse π n † : C n → B n for the map π n : B n → C n and a left inverse i n † : B n → A n forthe map i n : A n → B n in the category of groups such that i n † π n † = 0. Define then i † n := ( π n † ) ∗ : B ∗ n → C ∗ n and π † n :=( i n † ) ∗ : A ∗ n → B ∗ n , and observe that they are left inverses in the category of Polish groups for i n = ( π n ) ∗ : C ∗ n → B ∗ n and π n = ( i n ) ∗ : B ∗ n → A ∗ n , respectively, such that i † n π † n = 0. Define thenˆ d n := i n +1 † δ n π n † : C n → A n +1 and ˆ d n := i † n − ∂ n π † n : A ∗ n → C ∗ n − .By definition, we have that ˆ d n = ( ˆ d n − ) ∗ , which easily gives communitativity of the right square.Via the natural isomorphisms as in Proposition 2.8 for H n +1 ( A ) = Z • +1 ( A ) / B • +1 ( A ) and H • ( C ) = Z • ( C ) / B • +1 ( C ),the map Ext ( d • , G ) corresponds to the mapHom (cid:0) B • +1 ( A ) , G (cid:1) Hom ( Z • +1 ( A ) | B • +1 ( A ) , G ) → Hom ( B • ( C ) , G )Hom ( Z • ( C ) | B • +1 ( C ) , G ) EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 11 given by [ θ ] [ θ ◦ ˆ d n | B n ( C ) ]for θ ∈ Hom (cid:0) B n +1 ( A ) , G (cid:1) . Thus, to obtain the commutativity of the left square, it suffices to see that the identity( − n δ n ˆ d n − = ( − n − ˆ d n δ n − holds. Indeed, π n ( δ n − π n − † − π n † δ n − ) = π n δ n − π n − † − π n π n † δ n − = δ n − π n − π n − † − π n π n † δ n − = 0.Therefore i n i n † δ n − π n − † = i n i n † δ n − π n − † − i n i n † π n † δ n − = i n i n † ( δ n − π n − † − π n † δ n − ) = ( δ n − π n − † − π n † δ n − ).Hence, − δ n π n † δ n − = δ n ( δ n − π n − † − π n † δ n − ) = δ n i n i n † δ n − π n − † = i n +1 δ n i n † δ n − π n − † and − ˆ d n δ n − = − i n +1 † δ n π n † δ n − = i n +1 † i n +1 δ n i n † δ n − π n − † = δ n i n † δ n − π n − † = δ n ˆ d n − .This concludes the proof. (cid:3) As an immediate consequence of Lemma 3.13, we obtain the following.
Theorem 3.14.
Let G be a Polish group with the division closure property. Suppose that → A → B → C → isa locally split exact sequence of free countable cochain complexes, and → C ∗ → B ∗ → A ∗ → is its G -dual locallysplit exact sequence of Polish chain complexes. Then the connecting homomorphisms for → A → B → C → and → C ∗ → B ∗ → A ∗ → induce a natural degree − homomorphism from the UCT exact sequence for A tothe UCT exact sequence for C . Towers of Polish chain complexes
In this section, we present the coherent category of towers of Polish chain complexes from [22, Section 3], and thenotion of homotopy limit of a tower. This category can be seen as the algebraic counterpart of the coherent categoryof towers of topological spaces from strong shape theory, which underpins the approach to Steenrod homology inthat context; see [25]. Recall that we let G denote a Polish group with the division closure property.4.1. Morphisms.
A tower of Polish chain complexes is an inverse sequence A = (cid:0) A ( m ) , p ( m,m +1) (cid:1) of Polish chaincomplexes and continuous chain maps p ( m,m +1) : A ( m +1) → A ( m ) . We then define p ( m ,m ) to be the continuouschain map p ( m ,m +1) ◦ · · · ◦ p ( m − ,m ) from A ( m ) to A ( m ) for m < m , and p ( m ,m ) = 1 A ( m . We now recallthe notion of coherent morphisms between towers of Polish chain complexes. Definition 4.1.
Let A = (cid:0) A ( m ) , p ( m,m +1) (cid:1) and B = (cid:0) B ( k ) , p ( k,k +1) (cid:1) be towers of Polish chain complexes. A 1-cellfrom A to B is given by a sequence f = ( m k , f ( k ) , f ( k,k +1) ) such that: • ( m k ) is an increasing sequence in ω , • f ( k ) : A ( m k ) → B ( k ) is a continuous chain map from A ( m k ) to B ( k ) , and • f ( k,k +1) : p ( k,k +1) f ( k +1) ⇒ f ( k ) p ( m k ,m k +1 ) is a continuous chain homotopy.One then defines for k ≤ k the continuous chain homotopy f ( k ,k ) : p ( k ,k ) f ( k ) ⇒ f ( k ) p ( m k ,m k ) by setting f ( k ,k ) = 1 f ( k and, for k < k , f ( k ,k ) = k − X k = k p ( k ,k ) f ( k,k +1) p ( m k +1 ,m k ) One can also define f ( k ,k ) recursively by setting f ( k ,k +1) = f ( k ,k ) p ( m k ,m k ) ◦ p ( k ,k ) f ( k ,k +1) .The identity A of A is the 1-cell (cid:0) m k , f ( k ) , f ( k,k +1) (cid:1) where m k = k , f ( k ) = 1 A ( k ) , and f ( k,k +1) = 1 p ( k,k +1) . We now define composition of 1-cells.
Definition 4.2.
Suppose that A , B , and C are towers of Polish chain complexes, and f = ( m k , f ( k ) , f ( k,k +1) ) and g = (cid:0) k t , g ( t ) , g ( t,t +1) (cid:1) are 1-cells from A to B and from B to C , respectively. We define gf = g ◦ f = (cid:16) m k t , g ( t ) f ( k t ) , g ( t ) f ( k t ,k t +1 ) ◦ g ( t,t +1) f ( k t +1 ) (cid:17) . Remark 4.3.
Composition of 1-cells is not associative. However, it induces a well-defined associative operationon the set of equivalence classes of 1-cells, where we identify 1-cells that are equal up to
Definition 4.4.
Let A = (cid:0) A ( m ) , p ( m ) (cid:1) and B = (cid:0) B ( k ) , p ( k ) (cid:1) be towers of Polish chain complexes. Let f =( m k , f ( k ) , f ( k,k +1) ) and f = (cid:0) m ′ k , f ′ ( k ) , f ′ ( k,k +1) (cid:1) be 1-cells from A to B . A 2-cell from f to f ′ is a sequence L = (cid:0) ˜ m k , L ( k ) , L ( k,k +1) (cid:1) such that: • ( ˜ m k ) is an increasing sequence in ω such that ˜ m k ≥ max { m k , m ′ k } , • L ( k ) : f ( k ) p ( m k , ˜ m k ) ⇒ f ′ ( k ) p ( m ′ k , ˜ m k ) is a continuous chain homotopy, • for every k ∈ ω , L ( k,k +1) : f ′ ( k,k +1) p ( m ′ k +1 , ˜ m k +1 ) ◦ p ( k,k +1) L ( k +1) ⇛ L ( k ) p ( ˜ m k , ˜ m k +1 ) ◦ f ( k,k +1) p ( m k +1 , ˜ m k +1 ) is a 3-cell.Let A and B be two towers of Polish chain complexes. We say that two 1-cells f, f ′ : A → B represent thesame coherent morphism if there exists a 2-cell f ⇒ f ′ . This defines an equivalence relation on the set of 1-cellsfrom A to B . Given a 1-cell f we let [ f ] be its equivalence class, and we call [ f ] the coherent morphism from A to B represented by f . Setting [ g ] ◦ [ f ] = [ gf ] gives a well-defined associative operation between coherent morphisms.The identity morphism of A is [1 A ] where 1 A is the identity 1-cell of A . This yields a category which we call the coherent category of towers of Polish chain complexes; see [22, Theorem 3.6].4.2. The homotopy limit of a tower.
We associate as in [25, Section 17] with a tower of Polish chain complexes A a Polish chain complex holim A , called its homotopy limit ; see also [22, Section 4] and [8]. For n ∈ Z , define C n ( A ) = (holim A ) n to be the closed subgroup of Y m ∈ ω A ( m ) n ⊕ Y m ≤ m A ( m ) n +1 consisting of those elements z = ( z m , z m ,m ) such that z m m + p ( m ,m ) n ( z m ,m ) = z m m for every m ≤ m ≤ m . The differential d n : C n +1 ( A ) → C n ( A )is defined by setting, for z ∈ C n ( A ), ( d n z ) m = ∂ n z m and ( d n z ) m ,m = ∂ n +1 z m ,m + ( − n ( p ( m ,m ) n ( z m ) − z m )for every m ∈ ω and m ≤ m . This defines a Polish chain complex · · · → C n +1 ( A ) d n +1 → C n ( A ) d n → C n − ( A ) → · · · which we denote by holim A . Remark 4.5. If z ∈ C n ( A ), then one has that z m ,m = 0and z m ,m = m − X m = m p ( m ,m ) n ( z m,m +1 ) EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 13 for every m < m . Thus the values z m,m +1 for m ∈ ω determine the values z m ,m for every m ≤ m .Suppose that A and B are towers of Polish chain complexes, and f = ( m k , f ( k ) , f ( k,k +1) ) is a 1-cell from A to B . We define a continuous chain map f ( ∞ ) = holim f from holim A to holim B , called the (homotopy) limit of f ,as follows. For n ∈ Z , z ∈ C n ( A ), we define f ( ∞ ) n ( z ) ∈ C n ( B ), by setting f ( ∞ ) n ( z ) k = f ( k ) n ( z m k )and f ( ∞ ) n ( z ) k,k +1 = f ( k ) n +1 ( z m k ,m k +1 ) + ( − n f ( k,k +1) n ( z m k +1 ).for every k ∈ ω .One can show that the continuous chain homotopy class of holim f only depends on the coherent morphismrepresented by f . Furthermore, the assignment A holim A , [ f ] holim [ f ] := [holim f ] yields a functor from thecoherent category of towers of Polish chain complexes to the homotopy category of Polish chain complexes; see [22,Theorem 4.2]. Thus, the assignment A H • (holim A ) is a functor from the coherent category of towers of Polishchain complexes to the continuous category of graded definable groups.Recall that a Polish chain complex A is proper if, for every n ∈ Z , B n ( A ) is a closed subset of A n . This impliesthat H • ( A ) is a graded Polish group. Suppose that A = ( A ( k ) ) is a tower of proper Polish chain complexes. Forevery k ∈ ω , we have a continuous cochain mapholim A → A ( k ) , z z k .This induces a continuously-definable homomorphism H • (holim A ) → H • ( A ( k ) ).These homomorphisms for k ∈ ω induce a natural continuously-definable homomorphismLocal : H • (holim A ) → Y k ∈ ω H • ( A ( k ) ). Lemma 4.6.
Let A = ( A ( k ) ) be a tower of proper Polish chain complexes. Then the image of the naturalcontinuously-definable homomorphism Local : H • (holim A ) → Y k ∈ ω H • ( A ( k ) )is equal to the (inverse) limit lim k H • (cid:0) A ( k ) (cid:1) , while the kernel is equal to H ∞• (holim A ). Proof.
The first assertion is easy to prove, so we just prove the second assertion. We have that the kernel of Local isa closed subgroup of H • (holim A ). Thus, it suffices to prove that the kernel of Local is contained in H ∞• (holim A ).Suppose that z ∈ Z n (holim A ) is such that [ z ] belongs to the kernel of Local. Then we have that z m ∈ B n ( A ( m ) )for m ∈ ω .Fix m ∈ ω . Set w k = 0 for k > m and w k,k +1 = 0 for every k ∈ ω . Since z m ∈ B n ( A ( m ) ) we can choose w m ∈ A ( m ) n +1 such that ∂w m = z m . We then define recursively w k for 0 ≤ k < m by setting w k := ( − n z k,k +1 + p ( k,k +1) n +1 ( w k +1 ) ∈ A ( k ) n +1 .This defines an element w ∈ C n +1 ( A ). We show by induction that, for 0 ≤ k ≤ m , ( d n +1 w ) k = ∂w k = z k . For k = m this holds for the choice of w m . Suppose that 0 ≤ k < m is such that the conclusion holds for k + 1.Then we have that ( d n +1 w ) k = ∂w k = ∂ (( − n z k,k +1 + p ( k,k +1) n +1 ( w k +1 ))= ( z k − p ( k,k +1) n ( z k +1 )) + p ( k,k +1) n ( z k +1 ) = z k .Fix now 0 ≤ k < m . Then we have that( d n +1 w ) k,k +1 = ∂w k,k +1 + ( − n +1 ( p ( k,k +1) n +1 ( w k +1 ) − w k )= ( − n +1 ( p ( k,k +1) n +1 ( w k +1 ) − (( − n z k,k +1 + p ( k,k +1) n +1 ( w k +1 )))= z k,k +1 . Since m ∈ ω is arbitrary, this concludes the proof that z ∈ B n (holim A ) and [ z ] ∈ H ∞ n (holim A ). (cid:3) Remark 4.7.
Although we will not need this fact, one can similarly show that, if A = ( A ( k ) ) is a tower of properPolish chain complexes, then H ∞• (holim A ) is naturally isomorphic in the category of definable graded groups tolim k H • +1 ( A ( k ) ). Here, the lim of a tower of Polish groups is regarded as a group with a Polish cover as in [4,Section 5]. 5. Inductive sequences of countable cochain complexes
In this section we introduce the coherent category of inductive sequences of cochain complexes, and the homotopycolimit functor. Via the obvious duality between inductive sequences of cochain complexes and towers of chaincomplexes, homotopy colimits correspond to homotopy limits. This allows one to obtain from Theorem 3.11 adefinable Universal Coefficient Theorem relating the homology of the homotopy colimit of an inductive sequence tothe cohomology of the homotopy limit of its dual tower; see Theorem 5.5.5.1.
Morphisms. An inductive sequence of countable cochain complexes is a sequence A = (cid:0) A ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) ofcountable cochain complexes, and cochain maps η ( ℓ +1 ,ℓ ) : A ( ℓ ) → A ( ℓ +1) . Then we set η ( ℓ ,ℓ ) = 1 A ( ℓ and η ( ℓ ,ℓ ) = η ( ℓ ,ℓ − ◦ · · · ◦ η ( ℓ +1 ,ℓ ) : A ( ℓ ) → A ( ℓ ) for ℓ < ℓ . The notion of 1-cell between inductive sequences isthe natural dual version of the notion of 1-cell between towers of chain complexes; see Section 4.1. Definition 5.1.
Let A = (cid:0) A ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) and B = (cid:0) B ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) be inductive sequences of countable cochaincomplexes. A 1-cell from A to B is given by a sequence f = ( t ℓ , f ( ℓ ) , f ( ℓ +1 ,ℓ ) ) such that: • ( t ℓ ) is an increasing sequence in ω , • f ( ℓ ) : A ( ℓ ) → B ( t ℓ ) is a cochain map from A ( ℓ ) to B ( t ℓ ) for every ℓ ∈ ω , and • f ( ℓ +1 ,ℓ ) : f ( ℓ +1) η ( ℓ +1 ,ℓ ) ⇒ η ( t ℓ +1 ,t ℓ ) f ( ℓ ) is a cochain homotopy.One then defines f ( ℓ ,ℓ ) for ℓ ≤ ℓ recursively by setting f ( ℓ,ℓ ) = 1 f ( ℓ ) for ℓ ∈ ω and f ( ℓ +1 ,ℓ ) = f ( ℓ +1 ,ℓ ) η ( ℓ ,ℓ ) ◦ η ( t ℓ ,t ℓ ) f ( ℓ ,ℓ ) .The identity A is the 1-cell (cid:0) t ℓ , f ( ℓ ) , f ( ℓ +1 ,ℓ ) (cid:1) where t ℓ = ℓ , f ( ℓ ) = 1 A ( ℓ ) , and f ( ℓ +1 ,ℓ ) = 1 η ( ℓ +1 ,ℓ ) .In a similar fashion, one can define the notion of 2-cell between 1-cells. Definition 5.2.
Let A = (cid:0) A ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) and B = (cid:0) B ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) be inductive sequences of countable cochaincomplexes. Let f = ( t ℓ , f ( ℓ ) , f ( ℓ +1 ,ℓ ) ) and f ′ = ( t ′ ℓ , f ′ ( ℓ ) , f ′ ( ℓ +1 ,ℓ ) ) be 1-cells from A to B . A 2-cell from f to f ′ is asequence L = (cid:0) ˜ t ℓ , L ( ℓ ) , L ( ℓ +1 ,ℓ ) (cid:1) such that: • (cid:0) ˜ t ℓ (cid:1) is an increasing sequence in ω such that ˜ t ℓ ≥ max { t ℓ , t ′ ℓ } for ℓ ∈ ω , • L ( ℓ ) : η ( ˜ t ℓ ,t ℓ ) f ( ℓ ) ⇒ η ( ˜ t ℓ ,t ′ ℓ ) f ′ ( ℓ ) is a cochain homotopy, and • for every ℓ ∈ ω , L ( ℓ +1 ,ℓ ) : η ( ˜ t ℓ +1 , ˜ t ℓ ) L ( ℓ ) ◦ η ( ˜ t ℓ +1 ,t ℓ +1 ) f ( ℓ +1 ,ℓ ) ⇛ η ( ˜ t ℓ +1 ,t ′ ℓ +1 ) f ′ ( ℓ +1 ,ℓ ) ◦ L ( ℓ +1) η ( ℓ +1 ,ℓ ) is a 3-cell.We define composition of 1-cells as follows. Definition 5.3.
Let A , B , C be inductive sequences of countable cochain complexes. Let f = ( t ℓ , f ( ℓ ) , f ( ℓ +1 ,ℓ ) ) : A → B and g = (cid:0) r t , g ( t ) , g ( t +1 ,t ) (cid:1) : B → C be 1-cells. We define their composition gf = g ◦ f to be the 1-cell (cid:0) r t ℓ , g ( t ℓ ) f ( ℓ ) , g ( t ℓ +1 ,t ℓ ) f ( ℓ ) ◦ g ( t ℓ +1 ) f ( ℓ +1 ,ℓ ) (cid:1) .Letting two 1-cells f, g : A → B between inductive sequences of countable cochain complexes be equivalent ifthere exists a 2-cell L : f ⇒ g defines an equivalence relation. Denoting by [ f ] the equivalence class of f , anddefining [ g ] ◦ [ f ] = [ gf ], one obtains a category with inductive sequences of countable cochain complexes as objectsand equivalence classes of 1-cells as arrows. We refer to this category as the coherent category of inductive sequencesof countable cochain complexes, and to its arrows as coherent morphisms .If A = (cid:0) A ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) is an inductive sequence of free finitely-generated cochain complexes, we let A ∗ be the towerof proper Polish chain complexes ( A ∗ ( ℓ ) , η ∗ ( ℓ +1 ,ℓ ) ), where A ∗ ( ℓ ) is the G -dual Polish chain complex of A ( ℓ ) . We call A ∗ EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 15 the G -dual tower of A . The assignment A A ∗ defines a functor from the coherent category of inductive sequencesof free finitely-generated chain complexes to the coherent category of towers of proper Polish chain complexes.5.2. Homotopy colimits.
We associate with an inductive sequence A of countable cochain complexes a countablecochain complex hocolim A , called the homotopy colimit of A . For n ∈ Z , define C n ( A ) = (hocolim A ) n to be thecountable group M m ∈ ω A n +1( m ) ⊕ M k ∈ ω A n ( k ) .For ℓ ∈ ω and z ∈ A n ( ℓ ) , we let ze ℓ ∈ C n ( A ) be the element where all the coordinates are 0 apart from the oneindexed by k = ℓ . If ℓ ∈ ω and z ∈ A n +1( ℓ ) , then we let ze ℓ,ℓ +1 ∈ C n ( A ) be the element where all the coordinates are0 apart from the one indexed by m = ℓ . We also set e ℓ,ℓ = 0 and, for ℓ < ℓ , and z ∈ A n +1( ℓ ) , ze ℓ ,ℓ := ℓ − X ℓ = ℓ η ( ℓ,ℓ ) ( z ) e ℓ,ℓ .We define the codifferential d n : C n ( A ) → C n +1 ( A ) to be the group homomorphism such that, for ℓ ∈ ω , z ∈ A n ( ℓ ) ,and w ∈ A n +1( ℓ ) , d n ( ze ℓ ) = ( δ n z ) e ℓ and d n ( we ℓ,ℓ +1 ) = (cid:0) δ n +1 w (cid:1) e ℓ,ℓ +1 + ( − n ( we ℓ − η ( ℓ +1 ,ℓ ) ( w ) e ℓ +1 ) . Notice that this definition implies that, for ℓ < ℓ , and w ∈ A n +1( ℓ ) , d n ( we ℓ ,ℓ ) = (cid:0) δ n +1 w (cid:1) e ℓ ,ℓ + ( − n (cid:0) we ℓ − η ( ℓ ,ℓ ) ( w ) e ℓ (cid:1) .We define hocolim A to be the countable cochain complex · · · → C n − ( A ) d n − → C n ( A ) d n → C n +1 ( A ) → · · · Suppose now that A and B are inductive sequences of countable cochain complexes, and f = ( t ℓ , f ( ℓ ) , f ( ℓ +1 ,ℓ ) )is a 1-cell from A to B . We define a cochain map f ( ∞ ) := hocolim( f ) from hocolim A to hocolim B , as follows. For n ∈ Z , ℓ ∈ ω , z ∈ A n ( ℓ ) , and w ∈ A n +1( ℓ ) , we define f n ( ∞ ) ( ze ℓ ) = f n ( ℓ ) ( z ) e t ℓ and f n ( ∞ ) ( we ℓ,ℓ +1 ) = f n +1( ℓ ) ( w ) e t ℓ ,t ℓ +1 + ( − n f n +1( ℓ +1 ,ℓ ) ( w ) e t ℓ +1 .This defines a cochain map f ( ∞ ) = hocolim f : hocolim A ⇒ hocolim B . One can show that the cochain homotopyclass hocolim [ f ] := [hocolim f ] of hocolim f only depends on the coherent morphism [ f ] represented by f . Further-more, the assignment A hocolim A , [ f ] [hocolim f ] defines a functor from the coherent category of inductivelimits of countable cochain complexes to the homotopy category of countable cochain complexes.Suppose that A is an inductive sequence of countable cochain complexes. Then for every ℓ ∈ ω we have a cochainmap A ( ℓ ) → hocolim A , z ze ℓ .This induces a homomorphism H • ( A ( ℓ ) ) → H • (hocolim A ) .Such homomorphisms for ℓ ∈ ω induce a natural homomorphismcolim ℓ H • ( A ( ℓ ) ) → H • (hocolim A ) ,which can be seen to be an isomorphism. We thus have the following. Proposition 5.4.
Suppose that A is an inductive sequence of countable cochain complexes. Then H • (hocolim A ) and colim ℓ H • ( A ( ℓ ) ) are naturally isomorphic. Duality.
We now observe that the construction of the homotopy colimit is dual to the construction of thehomotopy limit. Suppose that A = (cid:0) A ( ℓ ) , η ( ℓ +1 ,ℓ ) (cid:1) is an inductive sequence of free finitely-generated cochaincomplexes. Recall that we let A ∗ = ( A ∗ ( m ) , η ∗ ( m +1 ,m ) ) be its G -dual tower of proper Polish chain complexes. Forevery n ∈ Z , we have a pairing C n ( A ∗ ) × C n ( A ) → G defined by h z, ae ℓ i = h z ℓ , a ih z, be ℓ,ℓ +1 i = h z ℓ,ℓ +1 , b i for z ∈ C n ( A ∗ ), ℓ ∈ ω , a ∈ A n ( ℓ ) , and b ∈ A n +1( ℓ ) . This pairing establishes a natural isomorphism C n ( A ∗ ) ∼ = C n ( A ) ∗ , z
7→ h z, ·i .It is easy to verify that, via these isomorphisms, the differential d n − : C n − ( A ∗ ) → C n ( A ∗ ) corresponds to thedual ( d n ) ∗ of the codifferential d n : C n ( A ) → C n − ( A ). Thus, such isomorphisms for n ∈ Z induces a naturalisomorphism of Polish chain complexes holim( A ∗ ) ∼ = (hocolim A ) ∗ .We can therefore infer from Theorem 3.11 the following result. Recall the definition of the index and co-indexhomomorphisms as in Section 3.4, and the homomorphism Local as in Section 4.2. Theorem 5.5.
Let G be a Polish group with the division closure property. Suppose that A is an inductive sequenceof free finitely-generated cochain complexes, and let A ∗ be its G -dual tower of proper Polish chain complexes. Define A := hocolim A . Then there is a natural continuously definable exact sequence (cid:0) H • +1 ( A ) , G (cid:1) H • (holim( A ∗ ); G ) Hom ( H • ( A ) , G ) 0 coIndex Index which continuously-definably splits, and a natural continuously definable exact sequence (cid:0) H • +1 ( A ) , G (cid:1) H • (holim( A ∗ )) lim ℓ H • ( A ∗ ( ℓ ) ) 0 coIndex ∞ Local where coIndex ∞ : PExt (cid:0) H • +1 ( A ) , G (cid:1) ∼ −→ H ∞• (holim( A ∗ )) is the restriction of coIndex .Proof. By the above remarks, we can identify holim( A ∗ ) with A ∗ . The first exact sequence is simply the UCT exactsequence for A as in Theorem 3.11. It remains to prove the second assertion. Since coIndex is continuously-definable,it restricts to a homomorphism coIndex ∞ : PExt (cid:0) H • +1 ( A ) , G (cid:1) → H ∞• ( A ∗ ). By Lemma 4.6,Local : H • ( A ∗ ) → lim ℓ H • ( A ∗ ( ℓ ) )is surjective with kernel H ∞• ( A ∗ ). It thus remains to prove that the image of coIndex ∞ contains H ∞• ( A ∗ ).For ℓ ∈ ω , the canonical cochain map A ( ℓ ) → A induces a continuous chain map A ∗ → A ∗ ( ℓ ) , and a continuously-definable homomorphism Ext ( H • ( A ) , G ) → Ext (cid:0) H • ( A ( ℓ ) ) , G (cid:1) .Consider the co-index isomorphism coIndex ( ℓ ) : Ext (cid:0) H • ( A ( ℓ ) ) , G (cid:1) → ◦ H • ( A ∗ ( ℓ ) )associated with A ( ℓ ) . By naturality, the following diagram commutes0 Ext ( H • ( A ) , G ) H • ( A ∗ )0 Ext (cid:0) H • ( A ( ℓ ) ) , G (cid:1) H • ( A ∗ ( ℓ ) ) coIndexcoIndex ( ℓ ) We have that H ∞• ( A ∗ ) = Ker( H • ( A ) Local → lim ℓ H • ( A ∗ ( ℓ ) )) = \ ℓ ∈ ω Ker( H • ( A ∗ ) → H • ( A ∗ ( ℓ ) )). EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 17
Furthermore, since H • ( A ( ℓ ) ) is finitely-generated for every ℓ ∈ ω , we have by definition thatPExt ( H • ( A ) , G ) = \ ℓ ∈ ω Ker (cid:0)
Ext ( H • ( A ) , G ) → Ext (cid:0) H • ( A ( ℓ ) ) , G (cid:1)(cid:1) .This shows that the image of coIndex ∞ contains to H ∞• ( A ∗ ), concluding the proof. (cid:3) Remark 5.6.
In view of Remark 4.7, the second continuously-definable exact sequence in Theorem 5.5 can be seenas the
Milnor exact sequence → lim ℓ H • +1 ( A ∗ ( ℓ ) ) → H • (holim A ∗ ) → lim ℓ H • ( A ∗ ( ℓ ) ) → ℓ H • +1 ( A ∗ ( ℓ ) ) is naturally definably isomorphic to lim ℓ Hom (cid:0) H • +1 ( A ( ℓ ) ) , G (cid:1) .This follows from the (definable) six-term exact sequence relating lim and lim [36, Proposition 3.3(5)] applied tothe UCT exact sequences for A ∗ ( ℓ ) for ℓ ∈ ω , considering that lim ℓ Ext (cid:0) H • (cid:0) A ( ℓ ) (cid:1) , G (cid:1) = 0 by Roos’ theorem [36,Theorem 6.2], 6. Simplicial complexes
In this section we recall the combinatorial notion of simplicial complex, simplicial map, and simplicial carrier ascan be found in [12]. We then describe the classical homological invariants of simplicial complexes, including theirchain complexes and homology groups, and a correspondence between simplicial carriers and chain maps.6.1.
Simplicial complexes and carriers. A simplicial complex K is a family of nonempty finite sets that isclosed downwards, i.e., σ ⊆ τ ∈ K ⇒ σ ∈ K . A simplex of K is any element σ ∈ K . If σ, σ ′ are simplices of K such that σ ⊆ σ ′ , then σ is a face of σ ′ . A vertex of K is any element v of dom( K ) := S K . A subcomplex K ′ of a simplicial complex K is a subfamily of K that is itself a simplicial complex. A subcomplex K ′ of K is full ifits simplices are precisely the simplices σ of K such that σ ⊆ dom( K ′ ). We identify a simplex σ of K with thefull subcomplex of K whose simplices are the faces of σ . Let v be a vertex of K . The (closed) star St K ( v ) of K isthe subcomplex of K consisting of the simplices σ of K such that σ ∪ { v } is a simplex of K . Notice that, if τ is asimplex of K and v is a vertex of τ , then τ = St τ ( v ).Let K, L be two simplicial complexes. A simplicial map f : K → L is any function f : dom( K ) → dom( L ) so that { f ( v ) , . . . , f ( v n ) } a simplex of L for every simplex { v , . . . , v n } of K . A carrier κ : K → L is a function κ fromsimplices of K to subcomplexes of L with the property that, whenever σ, σ ′ are simplices of K such that σ ⊆ σ ′ ,one has that κ ( σ ) ⊆ κ ( σ ′ ) [12, Chapter VI, Definition 5.5]. A star carrier is a carrier κ : K → L such that forevery simplex σ of K there is a vertex w ( σ ) of κ ( σ ) such that St κ ( σ ) ( w ( σ )) = κ ( σ ). In this case, we call σ w ( σ )a choice function for κ . A simplicial carrier is a carrier κ such that, for every simplex σ of K , κ ( σ ) is a simplex of L . Every simplicial carrier is, in particular, a star carrier. We identify a simplicial map f with the simplicial carrier { v , . . . , v n } 7→ { f ( v ) , . . . , f ( v n ) } . We define an order relation among carriers by setting κ ≤ κ ′ if and only if κ ( σ ) ⊆ κ ′ ( σ ) for every simplex σ . We let the category of simplicial complexes have simplicial complexes as objectsand simplicial carriers as arrows, with composition defined as follows. Definition 6.1.
Suppose that K and L are simplicial complexes. The identity simplicial carrier 1 K : K → K isthe identity function on the set of simplices of K . If κ : K → L and τ : L → T are simplicial carriers, then theircomposition τ κ = τ ◦ κ is defined by ( τ κ ) ( σ ) = τ ( κ ( σ ))for every simplex σ of K .A simplicial complex is finite if it has finitely many vertices, countable if it has countably many vertices, and locally finite if every vertex belongs to finitely many simplices. In this paper, we will assume all the simplicialcomplexes to be countable and locally finite .One can associate with a (countable, locally finite) simplicial complex K a locally compact Polish space, calledits geometric realization , as follows. Let Ξ be the separable Hilbert space with orthonormal basis ( e v ) v ∈ dom( K ) indexed by the set of vertices of K . For each simplex σ = { v , . . . , v n } of K define | σ | = { t e v + · · · + t n e v n : t , . . . , t n ∈ [0 , , t + · · · + t n = 1 } ⊆ Ξ. Then | K | is defined to be the closed subspace of Ξ obtained as the union of | σ | where σ ranges among the simplicesof K . A polyhedron is a locally compact Polish space that is obtained in this fashion from some simplicial complex[32, Chapter 1]; see also [21, Section II.3, Proposition 3.6]. A polyhedron is compact if and only if it is the geometricrealization of a finite simplicial complex.Suppose that κ : K → L is a simplicial carrier. A continuous map α : | K | → | L | has support κ if α ( | σ | ) ⊆ | κ ( σ ) | for every simplex of σ . Any two maps | K | → | L | with support κ are homotopic. Assigning to a simplicial carrier κ : K → L the homotopy class of continuous maps | K | → | L | with support κ , one obtains a functor from thecategory of simplicial carriers to the homotopy category Ho( P ) of polyhedra. Such a category has polyhedra asobjects and homotopy classes of continuous maps as arrows.Suppose that K is a simplicial complex. The barycentric subdivision Sd( K ) of K is the simplicial complex withdom (Sd( K )) equal to the set of simplices of K . A simplex in Sd( K ) is a set { σ , . . . , σ n } of simplices of K thatis linearly ordered by inclusion. Notice that a simplicial carrier K → L induces a simplicial map Sd( K ) → Sd( L ).Thus, K Sd( K ) is a functor from the category of simplicial complexes to itself. Remark 6.2.
There is a natural simplicial carrier π K : Sd( K ) → K , { σ , . . . , σ ℓ } 7→ σ ℓ , where σ ⊆ · · · ⊆ σ ℓ .We also have star carrier Sd K : K → Sd( K ), σ Sd( σ ) = St Sd( K ) ( σ ). Notice that π Sd( K ) ◦ Sd K is the identitysimplicial carrier of K , while Sd K ◦ π K : Sd( K ) → Sd( K ) is a star carrier such that 1 Sd( K ) ≤ Sd K ◦ π K .6.2. The chain complex of a simplicial complex.
We now recall the definition of the classical homologyinvariants of a simplicial complex. Recall that we assume all simplicial complexes to be countable and locally finite.We adopt the notation and terminology from [12, Chapter VI]. For n ≥
0, an elementary n - chain is any tuple( v , . . . , v n ) ∈ dom( K ) n +1 with { v , . . . , v n } a simplex of K . Let C n ( K ) be the free abelian group generated by theset of elementary n -chains for n ≥
0, and C n = { } for n < C n ( K ) are called( ordered ) n - chains of K . The differential ∂ : C n ( K ) → C n − ( K ) is defined by ∂ ( v , . . . , v n ) = n X i =0 ( − i ( v , . . . , ˆ v i , . . . , v n ),where ˆ v i denotes the omission of v i . This gives rise to a free countable chain complex C • ( K ) which we call thechain complex of K . The chain complex of K with coefficients in G is C • ( K ; G ) := C • ( K ) ⊗ G ; see Definition 3.9.The cochain complex C • ( K ) of K is the Z -dual cochain complex of C • ( K ). More generally, the cochain complex C • ( K ; G ) of K with coefficients in G is the G -dual cochain complex of C • ( K ); see Definition 3.7. Notice that C • ( K ; G ) and C • ( K ; G ) are proper whenever K is finite .The index homomorphism is the group homomorphism Ind : C ( K ) → Z defined by Ind ( v ) = 1 for every vertex v of K . If K, L are simplicial complexes, then we say that a chain map f : C • ( K ) → C • ( L ) is augmentation-preserving if Ind ◦ f = Ind. Definition 6.3.
Suppose that K and L are simplicial complexes, and that κ : K → L is a carrier function. • For n, m ∈ Z , a homomorphism ϕ : C n ( K ) → C m ( L ) has support κ if ϕ ( C n ( σ )) ⊆ C m ( κ ( σ )) for everysimplex σ of K . • An augmentation-preserving chain map f : C • ( K ) → C • ( L ) has support κ if f n has support κ for every n ∈ Z . • Suppose that f, f ′ : C • ( K ) → C • ( L ) are augmentation-preserving chain maps with support κ , and F : f ⇒ f ′ is a chain homotopy. Then F has support κ if F n has support κ for every n ∈ Z .Suppose that K is a simplicial complex, and v is a vertex of K . We define the group homomorphism C n (St K ( v )) → C n +1 ( K ), x v a x by setting v a ( v , . . . , v n ) = ( v, v , . . . , v n )for every elementary n -chain ( v , . . . , v n ). Then one has that, for n ∈ ω and x ∈ C n (St K ( v )), ∂ ( v a x ) = (cid:26) x − Ind( x ) v if n = 0, x − ( v a ∂x ) if n > n ≥ Lemma 6.4.
Suppose that:
EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 19 • K and L are simplicial complexes; • κ : K → L is a star carrier and w is a choice function for κ ; • f, f ′ : C • ( K ) → C • ( L ) are augmentation-preserving chain maps with support κ ; • F, F ′ : f ⇒ f ′ are chain homotopies with support κ .(1) Define by recursion on n ∈ ω group homomorphisms f ( κ,w ) n : C n ( K ) → C n +1 ( L ) by setting f ( κ,w ) n = 0 for n < f ( κ,w ) n ( v , . . . , v n ) = w ( { v , . . . , v n } ) a ( f ( κ,w ) n − ∂ ) ( v , . . . , v n )for n ≥ n -chain ( v , . . . , v n ) ∈ C n ( K ). Then f ( κ,w ) : C • ( K ) → C • ( L ) is an augmentation-preserving chain map with support κ .(2) Define by recursion on n ∈ ω group homomorphisms F ( κ,w ) : C n ( K ) → C n +1 ( L )with support κ by setting F ( κ,w ) n = 0 for n < F ( κ,w ) n ( v , . . . , v n ) = w ( { v , . . . , v n } ) a (( f ′ n − f n ) − F ( κ,w ) n − ∂ ) ( v , . . . , v n )for n ≥ n -chain ( v , . . . , v n ) ∈ C n ( K ). Then F ( κ,w ) : f ⇒ f ′ is a chain homotopy withsupport κ .(3) Define by recursion on n ∈ ω group homomorphisms E ( κ,w ) n : C n ( K ) → C n +2 ( L )with support κ by setting E ( κ,w ) n = 0 for n < E ( κ,w ) n ( v , . . . , v n ) = w ( { v , . . . , v n } ) a (( F ′ n − F n ) + E ( κ,w ) n − ∂ ) ( v , . . . , v n )for n ≥ n -chain ( v , . . . , v n ) ∈ C n ( K ). Then E ( κ,w ) : F ⇛ F ′ is a 3-cell.If K, L are simplicial complexes, and f : K → L is a simplicial map, then f induces in the obvious wayan augmentation-preserving chain map C • ( K ) → C • ( L ) with support f , obtained by setting f n ( v , . . . , v n ) =( f ( v ) , . . . , f ( v n )) for n ∈ ω . More generally, we have the following result, which is an immediate consequence ofLemma 6.4. Lemma 6.5.
Suppose that κ : K → L is a star carrier. Then(1) There exists an augmentation-preserving chain map f : C • ( K ) → C • ( L ) with support κ ;(2) If f, f ′ : C • ( K ) → C • ( L ) are augmentation-preserving chain maps with support κ , then there exists a chainhomotopy F : f ⇒ f ′ with support κ .(3) If f, f ′ : C • ( K ) → C • ( L ) are augmentation-preserving chain maps with support κ , and F, F ′ : f ⇒ f ′ arechain homotopies with support κ , then there exists a 3-cell E : F ⇛ F ′ .By Lemma 6.5 the assignment K C • ( K ), κ [ f κ ] where f κ is a chain map with support κ , is a functorfrom the category of simplicial complexes to the homotopy category of free countable chain complexes. Thus, theassignment K C • ( K ; G ) defines a functor from the category of simplicial complexes to the homotopy categoryof Polish chain complexes, and the assignment K C • ( K ; G ) defines a contravariant functor from category ofsimplicial complexes to the homotopy category of Polish cochain complexes. Definition 6.6.
Let K be a simplicial complex. The definable homology H • ( K ; G ) of K with coefficients in G isthe definable homology of the Polish chain complex C • ( K ; G ), and the definable cohomology H • ( K ; G ) of G is thedefinable cohomology of the Polish cochain complex C • ( K ; G ). Remark 6.7.
Notice that if K is a simplicial complex, and f K : C • (Sd( K )) → C • ( K ) is a chain map with supportthe canonical simplicial carrier π K : Sd( K ) → K , then f K is a natural chain h-isomorphism by Lemma 6.5 andRemark 6.2; see also [12, Section VI.7]. Definition 6.8.
A tower of finite simplicial complexes is a sequence K = ( K ( m ) , p ( m,m +1) ) where K ( m ) is a finitesimplicial complex and p ( m,m +1) : K ( m +1) → K ( m ) is a simplicial carrier. We assign to K the tower of freefinitely-generated chain complexes C • ( K ) = (cid:0) C • (cid:0) K ( m ) (cid:1)(cid:1) m ∈ ω , where the bonding map C • (cid:0) K ( m +1) (cid:1) → C • (cid:0) K ( m ) (cid:1) is an augmentation-preserving chain map with support p ( m,m +1) . We also assign to K the tower of polyhedra (cid:0)(cid:12)(cid:12) K ( m ) (cid:12)(cid:12)(cid:1) m ∈ ω , where the bonding map (cid:12)(cid:12) K ( m +1) (cid:12)(cid:12) → (cid:12)(cid:12) K ( m ) (cid:12)(cid:12) is a continuous map with support p ( m,m +1) .6.3. Pairs of simplicial complexes.
A pair of simplicial complexes is a pair (
K, K ′ ) such that K is a simplicialcomplex and K ′ is a subcomplex of K . One can regard pairs of simplicial complexes as objects of a category,where a morphism from ( K, K ′ ) to ( L, L ′ ) is a simplicial carrier K → L mapping simplices of K ′ to simplices of L ′ . Suppose that ( K, K ′ ) is a pair of simplicial complexes. For every n ∈ Z , an elementary n -chain for K ′ is alsoan elementary n -chain for K . Thus, there is a canonical chain map C • ( K ′ ) → C • ( K ) such that, for every n ∈ Z , C n ( K ′ ) → C n ( K ) is a split monomorphism in the category of groups. Whence, one can define a free countablechain complex C • ( K, K ′ ) such that 0 → C • ( K ′ ) → C • ( K ) → C • ( K, K ′ ) → C • ( K, K ′ ; G ) := C • ( K, K ′ ) ⊗ G and C • ( K, K ′ ; G ) := Hom ( C • ( K, K ′ ) ; G ), we have a locally split short exact sequence of Polishchain complexes 0 → C • ( K ′ ; G ) → C • ( K ; G ) → C • ( K, K ′ ; G ) → → C • ( K, K ′ ; G ) → C • ( K ; G ) → C • ( K ′ ; G ) → C • ( K, K ′ ; G ) and C • ( K, K ′ ; G ) are proper when K is finite .7. Homology of compact metrizable spaces
In this chapter, we assume all the spaces to be compact and metrizable .7.1.
Covers.
Let X be a space. A cover of X is a family U = (cid:0) U U i (cid:1) i ∈ ω of open subsets of X such that X is theunion of { U i : i ∈ ω } . The nerve N ( U ) of a cover U of X is the simplicial complex withdom( N ( U )) = supp( U ) := (cid:8) i ∈ ω : U U i = ∅ (cid:9) and σ = { v , . . . , v n } a simplex of N ( U ) if and only if U U σ := U U v ∩ · · · ∩ U U v n = ∅ .We say that U is finite or locally finite if N ( U ) is finite or locally finite, respectively. A cover U of X is a refinement of a cover V of X if for every i ∈ supp( U ) there exists j ∈ supp( V ) such that U U i ⊆ U V j . In this case we set V ≤ U ,and define the refinement carrier κ ( U , V ) X : N ( U ) → N ( V ) by setting κ ( U , V ) X ( σ ) = (cid:8) j ∈ ω : U U σ ⊆ U V j (cid:9) for every simplex σ of N ( U ). Notice that this is a simplicial carrier. The relation ≤ renders the set cov ( X ) of finite covers of X an upward directed ordering of countable cofinality. Throughout this section, we assume all thecovers to be finite . Definition 7.1.
Let X be a compact metrizable space. A cofinal sequence (of finite open covers) for X is a cofinalincreasing sequence U = ( U m ) m ∈ ω in cov ( X ). A covered space is a pair ( X, U ) where X is a space and U is a cofinalsequence for X . A continuous map f : ( X, U ) → ( Y, W ) between covered compact metrizable spaces is a continuousmap f : X → Y . The category of covered spaces has covered spaces as objects and continuous maps as morphisms.Suppose that ( X, U ) , ( Y, V ) are covered spaces, and α : X → Y is a continuous map. Let us say that α is ( U , V )-continuous if for every i ∈ ω there exists j ∈ ω such that α (cid:0) U U i (cid:1) ⊆ U V j . We can then define the simplicial carrier κ ( U , V ) α : N ( U ) → N ( V ) associated with α by setting κ ( U , V ) α ( σ ) := (cid:8) j ∈ ω : α (cid:0) U U σ (cid:1) ⊆ U V j (cid:9) for every simplex σ of N ( U ). This subsumes the notion of refinement carrier, which corresponds to the case when X = Y and α is the identity map of X . EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 21
Homology of spaces.
Suppose now that ( X, U ) is a covered space. We let K ( X, U ) be the tower of finite sim-plicial complexes ( N ( U m )) m ∈ ω , where the bonding map N ( U m +1 ) → N ( U m ) is the refinement carrier κ ( U m +1 , U m ) X .Define then C • ( X, U ) to be the tower of free finitely-generated chain complexes associated with K ( X, U ) as inDefinition 6.8. Definition 7.2.
Suppose that ( X, U ) and ( Y, V ) are covered spaces, and α : X → Y is a continuous map. A 1-cell (cid:0) m k , f ( k ) , f ( k,k +1) (cid:1) : C • ( X, U ) → C • ( X, U ) has support α if, for every k ∈ ω , α is ( U m k , V k )-continuous, f ( k ) hassupport κ ( U mk , V k ) α , and f ( k,k +1) has support κ ( U mk +1 , V k ) α .The following lemma is an immediate consequence of Lemma 6.5. Lemma 7.3.
Suppose that ( X, U ), ( Y, V ), ( Z, W ) are covered spaces, and α : X → Y and β : Y → Z are continuousmaps.(1) There exists a 1-cell C • ( X, U ) → C • ( Y, V ) with support α ;(2) If f, f ′ : C • ( X, U ) → C • ( Y, V ) are 1-cells with support α , then there is a 2-cell L : f ⇒ f ′ ;(3) If f : C • ( X, U ) → C • ( Y, V ) and g : C • ( Y, V ) → C • ( Z, W ) are 1-cells with support α and β , respectively,then g ◦ f has support β ◦ α .It follows from Lemma 7.3 that choosing, for each continuous map α : ( X, U ) → ( Y, V ) a 1-cell f α : C • ( X, U ) → C • ( Y, V ) with support α defines a functor ( X, U ) C • ( X, U ), α [ f α ] from the category of covered spaces tothe coherent category of towers of free finitely-generated complexes.Suppose that X is a compact space. Then one can associate with X a tower (cid:0) X ( m ) (cid:1) of compact polyhedra, called polyhedral resolution of X , such that X is homeomorphic to the inverse limit lim m X ( m ) ; see [27, Theorem 7, page61]. The assignment X (cid:0) X ( m ) (cid:1) is a functor from the category of spaces to the category of towers tow(Ho( P ))associated with the homotopy category of polyhedra Ho( P ) [27, Corollary 6, page 67]. (See [10, Section 2.1] for thenotion of category of towers tow ( C ) associated with a category C .) One can find a cofinal sequence U X = (cid:0) U Xm (cid:1) m ∈ ω for X such that, setting K ( m ) := N (cid:0) U Xm (cid:1) for m ∈ ω , the tower (cid:0) | K ( m ) | (cid:1) m ∈ ω as in Definition 6.8 is isomorphic to (cid:0) X ( m ) (cid:1) m ∈ ω in tow(Ho( P )); see [12, Section IX.9]. Define then K ( X ) := K (cid:0) X, U X (cid:1) .Define the tower of free finitely-generated chain complex C • ( X ) to be C • (cid:0) X, U X (cid:1) . Consider then the cor-responding tower of proper Polish chain complexes C • ( X ; G ) := C • ( X ) ⊗ G (Definition 3.9), and the Z -dualinductive sequence C • ( X ) := Hom ( C • ( X ) , Z ) of free finitely-generated cochain complexes (Definition 3.7). Thisdefines a functor X C • ( X ; G ) from the category of spaces to the coherent category of towers of proper Polishchain complexes, which can be seen to be homotopy invariant, and a functor X C • ( X ) from the categoryof spaces to the coherent category of inductive sequences of free finitely-generated cochain complexes. Set also C • ( X ; G ) := holim( C • ( X ; G )), and C • ( X ) := hocolim( C • ( X )). Notice that, by definition, C • ( X ; G ) is the G -dual tower of C • ( X ). Definition 7.4.
Let X be a compact metrizable space, and G be a Polish group with the division closure property.The definable homology of X with coefficients in G is the definable homology of the Polish chain complex C • ( X ; G ).The integral cohomology of X is the cohomology of the free countable cochain complex C • ( X ).One defines the asymptotic homology H ∞• ( X ; G ) to be the asymptotic group with a Polish cover associated with H • ( X ; G ), and the weak homology group H w • ( X ; G ) to be the weak Polish group associated with H • ( X ; G ); seeDefinition 2.5. Remark 7.5.
The strong homology ¯ H • ( X ; G ) is defined in [25, Section 21] as the homology of the chain complex¯ C • ( X ; G ) = holim( ¯ C • ( X ; G )). Here, ¯ C • ( X ; G ) is the tower of chain complexes ( ¯ C • ( K ( m ) ; G )) m ∈ ω where (cid:0) K ( m ) (cid:1) isthe tower of finite simplicial complexes K ( X ) defined as above, and ¯ C • ( K ( m ) ; G ) is the singular chain complex ofthe geometric realization of K ( m ) . The canonical inclusions C • (cid:0) K ( m ) ; G (cid:1) → ¯ C • (cid:0) K ( m ) ; G (cid:1) are quasi-isomorphisms(i.e., they induce an isomorphism at the level of homology) by [15, Theorem 2.27] and the K¨unneth formula [41,Theorem 3.6.1], and hence they induce a natural quasi-isomorphism C • ( X ; G ) → ¯ C • ( X ; G ). This shows that H • ( X ; G ) and ¯ H • ( X ; G ) are naturally isomorphic graded groups.A similar argument shows that H • ( X ; G ) is naturally isomorphic to the Steenrod homology of X as in [10,Chapter 8], which is defined as above by replacing the ˇCech nerve with the Vietoris nerve of covers. To see this, it suffices to notice that the geometric realization of the Vietoris nerve of a cover is naturally homotopy equivalent tothe geometric realization of the ˇCech nerve [10, Theorem 8.2.10],Since ¯ H • ( − ; G ) is a homotopy-invariant functor [25, Corollary 19.2], one obtains from these observation analternative proof, not using the results about the coherent category of towers of chain complexes from [22], that H • ( − ; G ) is a homotopy-invariant functor. Lemma 7.6.
Suppose that K is a finite simplicial complex. Then H • ( K ; G ) and H • ( | K | ; G ) are naturally iso-morphic graded Polish groups. Proof.
Since H • ( K ) is finitely-generated, we have that PExt ( H • +1 ( K ) , G ) = { } . One can choose a cofinal sequence U | K | for | K | such that K (cid:0) | K | , U | K | (cid:1) is the tower of simplicial complexes (cid:0) K ( m ) (cid:1) m ∈ ω , where K (0) = K , and for m ≥ K ( m +1) = Sd (cid:0) K ( m ) (cid:1) is the ( m + 1)-st barycentric subdivision of K [12, Section II.6], and K ( m +1) → K ( m ) isthe canonical simplicial carrier π Sd ( K ( m ) ) : Sd (cid:0) K ( m ) (cid:1) → K ( m ) . Thus, by Theorem 5.5, we have that H • ( | K | ; G ) isnaturally isomorphic to lim m H • (cid:0) K ( m ) ; G (cid:1) . The bonding maps H • (cid:0) K ( m +1) ; G (cid:1) → H • (cid:0) K ( m ) ; G (cid:1) are isomorphismsby Remark 6.7. Therefore, we have that lim m H • (cid:0) K ( m ) ; G (cid:1) is naturally isomorphic to H • ( K ; G ). This concludesthe proof. (cid:3) In view of Lemma 7.6, as a particular instance of Theorem 5.5 we obtain the following.
Theorem 7.7.
Let X be a compact metrizable space with polyhedral resolution (cid:0) X ( m ) (cid:1) m ∈ ω , and let G be a Polishgroup with the division closure property. Then there exist a natural continuously-definable exact sequence (cid:0) H • +1 ( X ) , G (cid:1) H • ( X ; G ) Hom ( H • ( X ) , G ) 0 coIndex Index which continuously-definably splits, called the UCT exact sequence for X , and a natural continuously definable exactsequence (cid:0) H • +1 ( X ) , G (cid:1) H • ( X ; G ) lim m H • ( X ( m ) ; G ) 0 coIndex ∞ Local where coIndex ∞ : PExt (cid:0) H • +1 ( X ) , G (cid:1) ∼ −→ H ∞• ( X ; G ) is the restriction of coIndex . Remark 7.8.
Considering Remark 5.6, the second exact sequence in Theorem 7.7 can be seen as the Milnor exactsequence for homology 0 → lim m H • +1 ( X ( m ) ; G ) → H • ( X ; G ) → lim m H • ( X ( m ) ; G ) → m H • +1 ( X ( m ) ; G ) is definably isomorphic to lim m Hom (cid:0) H • +1 (cid:0) X ( m ) (cid:1) , G (cid:1) .7.3. Pairs of spaces.
A pair of spaces is a pair (
X, X ′ ) such that X is a compact metrizable space, and X ′ isa closed subspace. We identify a single space X with the pair ( X, ∅ ). Pairs of spaces form a category, where amorphism from ( X, X ′ ) to ( Y, Y ′ ) is a continuous function X → Y mapping X ′ to Y ′ . If U is a cover of X , define U| X ′ to be the cover (cid:0) U U i ∩ X ′ (cid:1) i ∈ ω of X ′ . One can associate with a pair of spaces ( X, X ′ ) a tower (cid:0) X ( m ) , X ′ ( m ) (cid:1) of pairs of compact polyhedra, called polyhedral resolution of ( X, X ′ ), such that ( X, X ′ ) is isomorphic to the (inverse)limit lim m (cid:0) X ( m ) , X ′ ( m ) (cid:1) in the category of pairs of spaces [27, Section I.3]. As in the previous section, it is easy tosee that there is a cofinal sequence U ( X,X ′ ) = ( U ( X,X ′ ) m ) m ∈ ω for X such that: • ( U ( X,X ′ ) m | X ′ ) m ∈ ω is a cofinal sequence for X ′ , and • setting K ( m ) = N ( U ( X,X ′ ) m ) and K ′ ( m ) = N ( U ( X,X ′ ) m | X ′ ), one has that ( | K ( m ) | , | K ′ ( m ) | ) m ∈ ω is isomorphicto (cid:0) X ( m ) , X ′ ( m ) (cid:1) m ∈ ω in the category of towers associated with the homotopy category of pairs of compactpolyhedra.One then defines C • ( X, X ′ ; G ) to be the homotopy limit of the tower of proper Polish chain complexes( C • ( K ( m ) , K ′ ( m ) ; G )) m ∈ ω ,and C • ( X, X ′ ) to be the homotopy colimit of the tower of free finitely-generated cochain complexes( C • ( K ( m ) , K ′ ( m ) )) m ∈ ω . EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 23
Definition 7.9.
Let (
X, X ′ ) be a pair of compact metrizable spaces, and let G be a Polish group with the divisionclosure property. The definable homology of ( X, X ′ ) with coefficients in G is the definable homology of the Polishchain complex C • ( X, X ′ ; G ). The integral cohomology H • ( X, X ′ ) of ( X, X ′ ) is the cohomology of the countablecochain complex C • ( X, X ′ ).As a consequence of Theorem 5.5, Proposition 3.12, and Theorem 3.14, one obtains the following. Theorem 7.10.
Let ( X, X ′ ) be a pair of compact metrizable spaces with polyhedral resolution (cid:0) X ( m ) , X ′ ( m ) (cid:1) m ∈ ω ,and let G be a Polish group with the division closure property. Then there is a continuously-definable exact sequence (cid:0) H • +1 ( X, X ′ ) , G (cid:1) H • ( X, X ′ ; G ) Hom ( H • ( X, X ′ ) , G ) 0 coIndex Index which continuously-definably splits, called the UCT exact sequence for ( X, X ′ ) , and a natural continuously definableexact sequence (cid:0) H • +1 ( X, X ′ ) , G (cid:1) H • ( X, X ′ ; G ) lim m H • ( X m , X ′ m ; G ) 0 coIndex ∞ Local where coIndex ∞ : PExt (cid:0) H • +1 ( X, X ′ ) , G (cid:1) ∼ −→ H ∞• ( X ; G ) is the restriction of coIndex .Furthermore, there are a natural continuously-definable degree − homomorphism d • : H • ( X, X ′ ; G ) → H •− ( X ′ ; G ) and a natural continuously-definable degree homomorphism d • : H • ( X ′ ) → H • +1 ( X, X ′ ) , called connecting ho-momorphisms , that fit into the exact sequences · · · → H n ( X ′ ; G ) H n ( i ) → H n ( X ; G ) H n ( j ) → H n ( X, X ′ ; G ) d n → H n − ( X ′ ; G ) H n − ( i ) → H n − ( X ; G ) → · · · and · · · → H n ( X, X ′ ) H n ( j ) → H n ( X ) H n ( i ) → H n ( X ′ ) d n → H n +1 ( X, X ′ ) H n +1 ( j ) → H n +1 ( X ) → · · · where i : X ′ → X and j : X → ( X, X ′ ) are the inclusion maps. The connecting homomorphisms d • and d • inducea natural homomorphism from the UCT exact sequence for ( X, X ′ ) to the UCT exact sequence for X ′ . Cohomology of polyhedra
In this section, we discuss limits of towers of Polish cochain complexes, and colimits of inductive sequences ofcountable chain complexes. We then prove a definable version of the Eilenberg–Mac Lane Universal CoefficientTheorem for (pairs of) polyhedra; see Theorem 8.6 and Theorem 8.9.8.1.
Towers of Polish cochain complexes.
Suppose that A = ( A ( k ) ) k ∈ ω is a tower of Polish cochain complexes.Then one can define its limit lim A to be the Polish cochain complex obtained by setting (lim A ) n = lim k ( (cid:0) A ( k ) (cid:1) n )with codifferentials induced by the codifferentials of A ( k ) for k ∈ ω . This defines a functor A lim A from the strict category of towers of Polish cochain complexes to the category of Polish cochain complexes. An arrow in thestrict category of Polish cochain complexes ( strict morphism ) is a coherent morphism that is represented by a 1-cell (cid:0) m k , f ( k ) , f ( k,k +1) (cid:1) such that p ( k,k +1) f ( k +1) = f ( k ) p ( m k ,m k +1 ) for every k ∈ ω .As in the case of homotopy limits, if A is a tower of Polish cochain complexes, then we have a natural continuously-definable homomorphism Local : H • (lim A ) → Y k ∈ ω H • ( A ( k ) ).We say that a tower of proper Polish cochain complexes A = (cid:0) A ( k ) (cid:1) with bonding maps p ( k,k +1) : A ( k +1) → A ( k ) is split epimorphic if, for every k ∈ ω and n ∈ Z , ( p ( k,k +1) ) n : ( A ( k +1) ) n → ( A ( k ) ) n is a split epimorphism in thecategory of Polish groups. Notice that we do not require p ( k,k +1) to be a split epimorphism in the category of Polishchain complexes. A similar proof as for Lemma 4.6 shows the following. Lemma 8.1.
Let A be a split epimorphic tower of proper Polish cochain complexes. Then the image of the naturalcontinuously-definable homomorphism Local : H • (lim A ) → Y k ∈ ω H • ( A ( k ) )is equal to the (inverse) limit lim k H • ( A ( k ) ), while the kernel is equal to H •∞ (lim A ). Remark 8.2.
Although we will not use this fact, one can also show that, if A is a split epimorphic tower of properPolish cochain complexes, then H •∞ (lim A ) is naturally isomorphic to lim k H •− (cid:0) A ( k ) (cid:1) , where the lim of a towerof Polish groups is regarded as a group with a Polish cover as in [4, Section 5].8.2. Inductive sequences of countable chain complexes.
Suppose that A = (cid:0) A ( ℓ ) (cid:1) ℓ ∈ ω is an inductive sequenceof countable chain complexes. Then one can define its colimit colim A by setting (colim A ) n = colim ℓ (cid:0)(cid:0) A ( ℓ ) (cid:1) n (cid:1) for n ∈ Z , with differential maps induced by the differential maps of A ( ℓ ) for ℓ ∈ ω . This defines a functor A colim A from the strict category of inductive sequences of countable chain complexes to the category of countable chaincomplexes. An arrow in the strict category ( strict morphism ) is a coherent morphism that is represented by a1-cell (cid:0) t ℓ , f ( ℓ ) , f ( ℓ +1 ,ℓ ) (cid:1) such that η ( t ℓ +1 ,t ℓ ) f ( ℓ ) = f ( ℓ +1) η ( ℓ +1 ,ℓ ) for every ℓ ∈ ω . For ℓ ∈ ω , we have a chain map A ( ℓ ) → colim A , which induces a homomorphism H • (cid:0) A ( ℓ ) (cid:1) → H • (colim A ). In turn, these homomorphisms inducea natural homomorphism colim ℓ H • (cid:0) A ( ℓ ) (cid:1) → H • (colim A ), which can be seen to be an isomorphism.We say that an inductive sequence A = (cid:0) A ( ℓ ) (cid:1) of countable chain complexes with bonding maps η ( ℓ +1 ,ℓ ) : A ( ℓ ) → A ( ℓ +1) is split monomorphic if, for every ℓ ∈ ω and n ∈ Z , ( η ( ℓ +1 ,ℓ ) ) n : ( A ( ℓ ) ) n → ( A ( ℓ +1) ) n is a split monomorphismin the category of countable groups. We do not require η ( ℓ +1 ,ℓ ) to be a split monomorphism in the category ofcountable chain complexes. Notice that, if A is a split monomorphic inductive sequence of free finitely-generatedchain complexes, then its G -dual tower A ∗ is a split epimorphic tower of proper Polish cochain complexes. Observefurthermore that lim( A ∗ ) is naturally isomorphic to (colim A ) ∗ . Thus, the same proof as in Theorem 5.5 allows oneto infer from Theorem 3.11 (where one reverses the roles of chain complexes and cochain complexes) and Lemma8.1 the following. Theorem 8.3.
Let G be a Polish group with the division closure property. Suppose that A is a split monomorphicinductive sequence of free finitely-generated chain complexes, and let A ∗ be its G -dual split epimorphic tower ofproper Polish cochain complexes. Define A := colim A . Then there is a natural continuously definable exact sequence H •− ( A ) , G ) H • (lim( A ∗ ) , G ) Hom ( H • ( A ); G ) 0 coIndex Index which continuously-definably splits, and a natural continuously definable exact sequence H •− ( A ) , G ) H • (lim( A ∗ )) lim ℓ H • ( A ∗ ( ℓ ) ) 0 coIndex ∞ Local where coIndex ∞ : PExt ( H •− ( A ) , G ) ∼ −→ H •∞ (lim( A ∗ )) is the restriction of coIndex . Remark 8.4.
In view of Remark 5.6 and Remark 8.2, the second continuously-definable exact sequence in Theorem8.3 can be seen as the Milnor exact sequence0 → lim ℓ H •− ( A ∗ ( ℓ ) ) → H • (lim( A ∗ )) → lim ℓ H • ( A ∗ ( ℓ ) ) → ℓ H •− ( A ∗ ( ℓ ) ) is definably isomorphic to lim ℓ Hom (cid:0) H •− (cid:0) A ( ℓ ) (cid:1) , G (cid:1) ; see [41, Theorem 3.5.8].8.3. Cohomology of polyhedra.
In this section we assume all the simplicial complexes to be countable and locally finite , but not necessarily finite. Recall that a polyhedron is a topological space X that is the geometricrealization | K | of a simplicial complex K . If | K | is a polyhedron, then one defines its corresponding chain complex C • ( | K | ) to be the free countable chain complex C • ( K ) associated with K as in Section 6. If α : | K | → | L | is acontinuous function between polyhedra, then one defines a corresponding chain h-map [ α ] : C • ( K ) → C • ( L ), asfollows. (Recall that a chain h-map is a morphism in the homotopy category of chain complexes.) One fixes asubdivision K ′ of K , a simplicial map f : K ′ → L that is a simplicial approximation of α , and a simplicial map g : K ′ → K that is a simplicial approximation of the identity map | K ′ | → | K | [32, Theorem 16.5]. (Here, thegeometric realization of the subdivision K ′ of K is identified with the geometric realization of K as in [32, Section15].) Let κ : K → K ′ be the carrier defined by letting κ ( σ ) be the full subcomplex of K ′ consisting of the simplicesof K ′ whose vertices belong to | σ | . One then lets f : C • ( K ′ ) → C • ( L ) be an augmentation-preserving chain mapwith support f , and g : C • ( K ′ ) → C • ( K ) be an augmentation preserving chain map with support g . One has that g is a chain h-isomorphism [32, Theorem 17.2], and the chain homotopy class [ α ] := [ f ] ◦ [ g ] − only depends on α [32, Section 18]. Furthermore, the assignment | K | 7→ C • ( K ), α [ α ] defines a homotopy-invariant functor fromthe category of polyhedra to the homotopy category of free countable chain complexes [32, Theorem 18.1, Theorem19.2]. EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 25 If | K | is a polyhedron, then one defines the corresponding cochain complex C • ( | K | ; G ) to be the G -dual of C • ( | K | ); see [32, Section 42]. The assignment | K | 7→ C • ( | K | ; G ) defines a homotopy-invariant functor from thecategory of polyhedra and continuous maps to the homotopy category of Polish cochain complexes. Definition 8.5.
Suppose that X is a polyhedron, and G is a Polish group with the division closure property. The(simplicial) definable cohomology H • ( X ; G ) of X with coefficients in G is the definable cohomology of the Polishcochain complex C • ( X ; G ). The integral homology H • ( X ) of X is the homology of the countable chain complex C • ( X ); see [32, Chapter 5].One similarly defines the asymptotic definable cohomology H •∞ ( X ; G ) and the weak definable cohomology H • w ( X ; G ) of a polyhedron X . Notice that Lemma 7.6 asserts that Definition 7.4 and Definition 8.5 agree inthe case where they overlap, namely for compact polyhedra.8.4. The Universal Coefficient Theorem for polyhedra.
Suppose that X = | K | is a polyhedron. A polyhedralcofiltration of X is an increasing sequence ( | K m | ) m ∈ ω of compact subspaces of X such that ( K m ) m ∈ ω is an increasingsubsequence of finite subcomplexes of K such that K is the union of { K m : m ∈ ω } . One can then consider thecorresponding split monomorphic inductive sequence of free finitely-generated chain complexes ( C • ( | K m | )) m ∈ ω .Notice that C • ( | K | ) is naturally chain h-isomorphic to colim m C ( | K m | ). Thus, H • ( | K | ) is naturally isomorphicto colim m H • ( | K m | ). As a particular instance of Theorem 8.3, we obtain the Universal Coefficient Theorem forcohomology of polyhedra. Theorem 8.6.
Let X be a polyhedron with polyhedral cofiltration ( X m ) m ∈ ω , and let G be a Polish group with thedivision closure property. Then there exist a natural continuously-definable exact sequence H •− ( X ) , G ) H • ( X ; G ) Hom ( H • ( X ) , G ) 0 coIndex Index which continuously-definably splits, called the UCT exact sequence for X , and a natural continuously definable exactsequence H •− ( X ) , G ) H • ( X ; G ) lim m H • ( X m ; G ) 0 coIndex ∞ Local where coIndex ∞ : PExt ( H •− ( X ); G ) ∼ −→ H •∞ ( X ; G ) is the restriction of coIndex . Remark 8.7.
In view of Remark 8.4, the second exact sequence from Theorem 8.6 can be seen as the Milnor exactsequence for cohomology 0 → lim m H •− ( X m ; G ) → H • ( X ; G ) → lim m H • ( X m ; G ) → m H •− ( X m ; G ) is definably isomorphic to lim m Hom ( H •− ( X m ) , G ).8.5. Cohomology of pairs of polyhedra.
A pair of polyhedra is a pair ( | K | , | K ′ | ) such that ( K, K ′ ) is a pairof simplicial complexes. Morphisms between pairs of polyhedra are defined as in the case of pairs of compactmetrizable spaces. One defines C • ( | K | , | K ′ | ; G ) := C • ( K, K ′ ; G ) and C • ( K, K ′ ) := C • ( K, K ′ ); see Section 6.3. Definition 8.8.
Let (
X, X ′ ) be a pair of polyhedra, and G be a Polish group with the divisor closure property.The definable cohomology of ( X, X ′ ) with coefficients in G is the definable homology of the Polish cochain complex C • ( X, X ′ ; G ). The homology of ( X, X ′ ) is the homology of the countable chain complex C • ( X, X ′ ).Similarly, one defines the weak and asymptotic homology of a pair of polyhedra ( | K | , | K ′ | ). A polyhedralcofiltration for ( | K | , | K ′ | ) is a sequence ( | K m | , | K ′ m | ) m ∈ ω such that ( | K m | ) m ∈ ω is a polyhedral cofiltration of | K | ,( | K ′ m | ) m ∈ ω is a polyhedral cofiltration of | K ′ | , and K ′ m is a subcomplex of K m for m ∈ ω . As a consequence ofTheorem 5.5, Proposition 3.12, and Theorem 3.14, one obtains the following. Theorem 8.9.
Let ( X, X ′ ) be a pair of polyhedra, with polyhedral cofiltration ( X m , X ′ m ) m ∈ ω , and G be a Polishgroup with the division closure property. Then there exist a natural continuously-definable exact sequence H •− ( X, X ′ ) , G ) H • ( X, X ′ ; G ) Hom ( H • ( X, X ′ ) , G ) 0 coIndex Index which continuously-definably splits, called the UCT exact sequence for ( X, X ′ ) , and a natural continuously definableexact sequence H •− ( X, X ′ ) , G ) H • ( X, X ′ ; G ) lim m H • ( X ( m ) , X ′ ( m ) ; G ) 0 coIndex ∞ Local6 MARTINO LUPINI where coIndex ∞ : PExt ( H •− ( X, X ′ ) , G ) ∼ −→ H •∞ ( X, X ′ ; G ) is the restriction of coIndex .Furthermore, there are a natural continuously-definable degree homomorphism d • : H • ( X ′ ; G ) → H • +1 ( X, X ′ ; G ) and a natural continuously-definable degree − homomorphism d • : H • ( X, X ′ ) → H •− ( X ′ ) , called connecting ho-momorphisms , that fit into the exact sequences · · · → H n ( X, X ′ ; G ) H n ( j ) → H n ( X ; G ) H n ( i ) → H n ( X ′ ; G ) d n → H n +1 ( X, X ′ ; G ) H n +1 ( j ) → H n +1 ( X ; G ) → · · · and · · · → H n ( X ′ ) H n ( i ) → H n ( X ) H n ( j ) → H n ( X, X ′ ) d n → H n − ( X ′ ) H n − ( i ) → H n − ( X ) → · · · where i : X ′ → X and j : X → ( X, X ′ ) are the inclusion maps. The connecting homomorphisms d • and d • inducea natural homomorphism from the UCT exact sequence for ( X, X ′ ) to the UCT exact sequence for X ′ . Cohomology of homotopy polyhedra.
By homotopy invariance, one can extend Definition 8.5 to the classof homotopy polyhedra , namely spaces that are homotopy equivalent to a polyhedron. Indeed, if X is homotopyequivalent to a polyhedron | K X | , as witnessed by continuous maps s X : X → | K X | and t X : | K X | → X , thenone can set H • ( X ; G ) := H • ( | K X | ; G ) and H • ( X ) := H • ( | K X | ). If α : X → Y is a continuous map, then thecorresponding continuously-definable homomorphisms H • ( Y ; G ) → H • ( X ; G ) and H • ( X ) → H • ( Y ) are the onesassociated with the continuous map s Y ◦ α ◦ t X : | K X | → | K Y | .The class of homotopy polyhedra contains all countable CW complexes [1, Section 1.5]. A CW complex iscountable if it is obtained by attaching countably many cells. Every polyhedron is a countable simplicial complex.Conversely, every countable CW complex is a homotopy polyhedron by [29, Theorem 1], [21, Section IV.6, Theorem6.1], [42, Section I.9, Theorem 13]. Similar considerations apply to absolute neighborhood retract (ANR) [27,Section I.3]: every polyhedron is an ANR and, conversely, every ANR is a homotopy polyhedron [27, Section I.4,Theorem 5].We also have that a second countable paracompact space with a good cover is a homotopy polyhedron. Recallthat a second countable topological spaces is paracompact if every cover has a countable, locally finite refinement.A good cover of a paracompact space X is a countable, locally finite cover U of X such that, for every σ ∈ N ( U ), U U σ is contractible. In this case, one has that X is homotopy equivalent to the polyhedron | N ( U ) | [15, Corollary4G.3].The class of second countable paracompact spaces with a good cover includes all locally compact Polish spacesadmitting a basis that (1) is closed under intersections, and (2) consists of precompact contractible open sets. Inparticular, all second countable Riemannian manifolds have this property; see [38, Chapter 11], [6, Theorem 5.1],[34, proof of Theorem 89], [9, Section 3.4], [28, Remark after Lemma 10.3].Analogous considerations apply to the case of pairs of spaces.8.7. Definable cohomology and homotopical definable cohomology.
Suppose that G is a countable abeliangroup. A homotopical approach to definable cohomology with coefficients in G is considered in [5]. Fix n ≥
1. Let( P, ∗ ) be a pointed polyhedron that is an Eilenberg–MacLane space of type ( G, n ) [37, Section 8.1], and let (
X, X ′ )be a pair of polyhedra. It is shown in [5] that the set [( X, X ′ ) , ( P, ∗ )] of homotopy classes of continuous maps( X, X ′ ) → ( P, ∗ ) is a definable set, regaded as the quotient of the Polish space of continuous maps ( X, X ′ ) → ( P, ∗ )endowed with the compact-open topology. Furthermore, the H -space structure on ( P, ∗ ) [37, Section 1.5] inducesa definable group structure on [( X, X ′ ) , ( P, ∗ )], which we call the n -th homotopical definable cohomology group H nh ( X, X ′ ; G ) of ( X, X ′ ) with coefficients in G .We have that H n ( P, ∗ ) ∼ = π n ( P, ∗ ) ∼ = G by the Hurewicz isomorphism theorem [37, Theorem 7.5.4]. Therefore,by the Universal Coefficient Theorem for cohomology of pairs of polyhedra, there exists υ ∈ H n ( P, ∗ ; G ) such thatIndex ( υ ) ∈ Hom ( H n ( P, ∗ ) , G ) is an isomorphism. Such an element is called n -characteristic for ( P, ∗ ); see [37,Section 8.1].Fix an n -characteristic element υ ∈ H n ( P, ∗ ; G ). Given a pair of polyhedra ( X, X ′ ), one can consider a naturaldefinable group homomorphism ϕ X : H nh ( X, X ′ ; G ) → H n ( X, X ′ ; G ), [ f ] [ H n ( f ) ( υ )] where f : ( X, X ′ ) → ( P, ∗ )is a continuous map and H n ( f ) : H n ( P, ∗ ; G ) → H n ( X, X ′ ; G ) is the homomorphism induced by f . By [37,Theorem 8.1.8], ϕ X is a group isomorphism. Thus, by [23, Proposition 1.10(3)], ϕ X is an isomorphism in thecategory of definable groups. It follows that, for a pair of polyhedra ( X, X ′ ), the homotopical definable cohomology EFINABLE EILENBERG–MAC LANE UNIVERSAL COEFFICIENT THEOREMS 27 group H nh ( X, X ′ ; G ) and the definable cohomogy group H n ( X, X ′ ; G ) as in Definition 8.8 are naturally definablyisomorphic. References
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