On Axiomatic Characterization of Alexander-Spanier Normal Homology Theory of General Topological Spaces
aa r X i v : . [ m a t h . A T ] F e b On Axiomatic Characterization of Alexander-Spanier Normal Homology Theoryof General Topological Spaces
Vladimer Baladze, Anzor Beridze, Leonard Mdzinarishvili
Department of Mathematics, Faculty of exact sciences and education, Batumi Shota Rustaveli State University, 35, Ninoshvili St., Batumi,Georgia; e-mail: [email protected] of Mathematics, Kutaisi International University, Youth Avenue, 5th Lane, Kutaisi, 4600 Georgia; e-mail: [email protected] of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University, 77, Kostava St., Tbilisi, Georgia;e-mail: [email protected]
Abstract
The Alexandro ff - ˇCech normal cohomology theory [Mor ], [Bar], [Ba ],[Ba ] is the unique continuous extension [Wat]of the additive cohomology theory [Mil], [Ber-Mdz ] from the category of polyhedral pairs K Pol to the category ofclosed normally embedded, the so called, P -pairs of general topological spaces K Top . In this paper we define theAlexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to theAlexandro ff - ˇCech normal cohomology. Using this fact and methods developed in [Ber-Mdz ] we construct an exact,the so called, Alexander-Spanier normal homology theory on the category K Top , which is isomorphic to the Steenrodhomology theory on the subcategory of compact pairs K C . Moreover, we give an axiomatic characterization of theconstructed homology theory.
Keywords:
Universal Coe ffi cient Formula; Continuity of exact homology; Steenrod homology; Alexander-Spaniernormal homology. Introduction
On the category K CM of pairs of compact metric spaces the exact homology theory was defined by N. Steenrod[St], which is known as the classical Steenrod homology theory. J. Milnor [Mil] constructed the exact homology theoryon the category K C of pairs of compact Hausdor ff spaces, which is isomorphic to the Steenrod homology theory onthe subcategory K CM and which satisfies the so called ”modified continuity” property: if X ← X ← X ← . . . is aninverse sequence of compact metric spaces with inverse limit X , then for each integer n there is an exact sequence:0 → lim ←−− H n + ( X i ) β −−→ H n ( X ) γ −→ lim ←−− H n ( X i ) → , (1)where H ∗ is the Steenrod (Milnor) homology theory [Mil]. There are exact homology theories defined by other authors[Kol], [Cho] [Sit], [Bor-Moo], [In ], [Ed-Ha ],[Ed-Ha ], [Mas], [Mdz ], [Skl] which are isomorphic to the Steenrodhomology theory on the category K CM and so, satisfy the modified continuity axiom.On the category K C the axiomatic characterization is obtained by N. Berikashvili [Ber], L. Mdzinarishvili andKh. Inasaridze [In-Mdz], L. Mdzinarishvili [Mdz ], Kh. Inasaridze [In ]. Consequently, in addition to the Eilenberg-Steenrod axioms one of the following axioms is required: Universal Coe ffi cient Formula : For each ( X , A ) ∈ K C and an abelian group G , there exists a functorial exactsequence 0 −→ Ext( ˇ H n + ( X , A ); G ) −→ H n ( X , A ; G ) −→ Hom( ˇ H n ( X , A ); G ) −→ , (2)where ˇ H n + ( − , − ; G ) is the Alexandro ff - ˇCech cohomology [Ber]. Partial continuity:
Let ( X , A ) be an inverse limit of an inverse system { ( X α , A α ) , p α,α ′ } of compact polyhedra, thenfor each integer n there is a functional exact sequence:0 → lim ←−− H n + ( X α , A α ) β −−→ H n ( X , A ) γ −→ lim ←−− H n ( X α , A α ) → . (3) Preprint submitted to Elsevier February 10, 2021
In-Mdz], [Mdz ]. Continuity for an Injective Group:
Let ( X , A ) be the inverse limit of an inverse system { ( X α , A α ) , p α,α ′ } of com-pact polyhedra and G be an injective abelian group, then for each integer n there is an isomorphism: H n ( X , A ; G ) ≈ lim ←−− H n ( X α , A β ; G ) (4)[In ].In the paper we will define the exact homology theory ¯ H N ∗ ( − , − ; G ) using the method developed in [Ber-Mdz ] andthe Alexander-Spanier cochains based on the normal coverings. There is defined the continuity of an exact homologytheory, which is the generalization of partial continuity. It is shown that any exact continuous theory is continuous foran injective abelian coe ffi cient group and there is the Universal Coe ffi cient Formula. Using the obtained properties weproved the uniqueness theorem.
1. Alexander-Spanier Normal Cohomology Theory
Let X be a general topological space and A be its subspace. Let us recall that a normal covering is defined as anopen covering α = { U α } , which admits a partition of unity n ϕ U α | U α ∈ α o subordinated to α , i.e. ϕ U α ≥ P ϕ U α = U α ∈ α the support supp ϕ U α = n x ∈ X | ϕ U α ( x ) > o is contained in U α [Mar-Seg]. A is said to be P -embeddedor normally embedded in X , if for each normal covering β = { V β } of A there is a normal covering α = { U α } of X suchthat α | A = { U α ∩ A | U α ∈ α } is a refinement of the covering β . A pair ( X , A ) of topological spaces is said to be closed P -pair if A is closed and P -embedded in X . Let K Top be the category of closed P -pairs [Mar-Seg], [Wat]. Therefore,for paracompact spaces, any closed pair is closed P -pair [Mar-Seg]. In this section, using the normal coveringswe construct the Alexander-Spanier type cohomology theory, the so called, Alexander-Spanier normal cohomologytheory . The methods of construction is the standard that is considered in [Sp], but we will recall and review some ofthem which are important for our purpose - to construct and axiomatically characterize an exact homology theory onthe category K Top . Note that, it is possible to define the Alexander-Spanier normal cohomology theory for each closedpair ( X , A ), but it has ”nice” properties for closed P -pairs.Let X be a topological space, G ∈ A b be an abelian group. Consider the set C n ( X ; G ) of all functions ϕ : X n + → G ,where X n + is ( n + X . If ϕ , ϕ ∈ C n + ( X ; G ) and ( x , x , . . . , x n ) ∈ X n + , thenthe addition in C n ( X ; G ) can be defined by the formula:( ϕ + ϕ )( x , x , . . . , x n ) = ϕ ( x , x , . . . , x n ) + ϕ ( x , x , . . . , x n ) . (5)It is clear that C n ( X ; G ) is an abelian group by the given operation. The coboundary homomorphism δ : C n ( X ; G ) → C n + ( X ; G ) is defined by the formula: δ ( ϕ )( x , x , . . . , x n + ) = n + X i = ( − i ϕ ( x , . . . , ˆ x i , . . . , x n + ) , (6)where the coordinate with the hat ˆ x i is omitted. In this case, it is known that δ ◦ δ = C ∗ ( X ; G ) = (cid:8) C n ( X ; G ) , δ (cid:9) is a cochain complex [Sp].Let Cov N ( X ) be the system af all normal coverings α = { U α } of X . An element ϕ ∈ C n ( X ; G ) is said to belocally zero with respect to a normal covering (in short, N -locally zero), if there is a normal covering α = { U α } such that ϕ vanishes on any ( n + X which lies in some elements U α of α . Thus, if we define α n + = S α ∈A U n + α ⊂ X n + , then ϕ vanishes on α n + . The subset of C n ( X ; G ) consisting of all N -locally zero functions is asubgroup, denoted by C nN ( X ; G ). It is easy to check that, if ϕ vanishes on α n + , then δ ( ϕ ) vanishes on α n + ⊂ X n + and so, C ∗ N ( X ; G ) = n C nN ( X ; G ) , δ o is a cochain subcomplex of C ∗ ( X ; G ). Let ¯ C ∗ N ( X ; G ) ≃ C ∗ ( X ; G ) / C ∗ N ( X ; G ) be thequotient cochain complex of C ∗ ( X ; G ) by C ∗ N ( X ; G ). Let the n -dimensional cohomology group of the obtained cochaincomplex ¯ C ∗ N ( X ; G ) denote by ¯ H ∗ N ( X ; G ) and call the Alexander-Spanier normal cohomology of the topological space X with coe ffi cients in the group G .Let f : X → Y be a continuous map and f : C ∗ ( Y ; G ) → C ∗ ( X ; G ) be the cochain map defined by the formula: f ( ϕ )( x , x , . . . , x n ) = ϕ ( f ( x ) , f ( x ) , . . . , f ( x n )) , ϕ ∈ C n ( Y ; G ) , x , x , . . . , x n ∈ X . (7)2ote that if β = n V β o is a normal covering of Y , then α = f − ( β ) = n f − ( V β ) o is a normal covering of X . If n ψ V β | V β ∈ β o is a partition of a unity subordinated to β, then n ϕ U α = ψ V β ◦ f | U α = f − ( V β ) ∈ α o is a partition of the unity subordinatedto α [Mar-Seg]. On the other hand, if ϕ ∈ C ∗ ( Y ; G ) vanishes on β n + , then f ( ϕ ) vanishes on α n + and so, f maps C nN ( Y ; G ) into C nN ( X ; G ). Therefore, if f : X → Y is continuous, then it induces a cochain map f : ¯ C ∗ N ( Y ; G ) → ¯ C ∗ N ( X ; G ) (8)and so, a homomorphism between the cohomology groups f ∗ : ¯ H ∗ N ( Y ; G ) → ¯ H ∗ N ( X ; G ) . (9)Note that for each pair ( X , A ) ∈ K Top the inclusion i : A → X induces an epimorphism i : C ∗ ( X ; G ) → C ∗ ( A ; G ) , which maps C ∗ N ( X ; G ) into C ∗ N ( A ; G ). Consequently, it induces an epimorphism i : ¯ C ∗ N ( X ; G ) → ¯ C ∗ N ( A ; G ) . (10)Let ¯ C ∗ N ( X , A ; G ) be the kernel of i , then the cohomology groups of it is denoted by ¯ H ∗ N ( X , A ; G ) and will be said to bethe Alexander-Spanier normal cohomology of pair ( X , A ) with coe ffi cients in group G .For each closed P -pair ( X , A ) ∈ K Top , denote the subcomplex of the cochain complex C ∗ ( X ) consisting withfunctions ϕ ∈ C ∗ ( X ), such that the restrictions on A ϕ | A are N -locally zero by C ∗ N ( X , A ) . In this case, C ∗ N ( X ; G ) ⊂ C ∗ N ( X , A ; G ) and ¯ C ∗ N ( X , A ; G ) ≃ C ∗ N ( X , A ; G ) / C ∗ N ( X ; G ). Note that this result is not true for any closed pair. It isessential that A is closed and P -embedded to X . Our aim is to characterize the cochain complex ¯ C ∗ N ( X , A ; G ) as a directlimit. Consequently, we need the notion of a normal covering of a pair.A pair ( α, β ) is said to be a normal covering of a pair ( X , A ) if α and β are normal coverings of X and A , respectively,and β is a refinement of the restriction α | A = (cid:8) U α ∩ A | U α ∈ α (cid:9) . A pair ( α, β ) is said to be a refinement of ( α ′ , β ′ ) if α and β are refinements of α ′ and β ′ , respectively. Let Cov N ( X , A ) be the direct set of all normal coverings ( α, β ) of pair ( X , A ).Let for each ( α, β ) ∈ Cov N ( X , A ), X ( α ) and A ( β ) be abstract simplicial complexes (Vietorisian complexes) whosevertices are points of X and A and whose simplexes are fine subsets F X = { x , x , . . . , x k } of X and F A = { a , a , . . . , a k } of A such that there is some U α ∈ α and V β ∈ β containing F X ⊂ U α and F A ⊂ V β , respectively. By the definition A ( β ) isa subcomplex of X ( α ) . Consider the corresponding cochain complex C ∗ ( α, β ; G ) . Note that an element ϕ α ∈ C n ( α, β ; G )is a function ϕ α : α n + → G which vanishes on β n + ⊂ α n + . Therefore, the system
Cov N ( X , A ) induces a direct system (cid:8) C ∗ ( α, β ; G ) (cid:9) of the cochain complexes and so, we have the limit cochain complexlim −−→ (cid:8) C ∗ ( α, β ; G ) (cid:9) . (11)We need to show that this limit cochain complex is canonically isomorphic to ¯ C ∗ N ( X , A ; G ). If ϕ ∈ C ∗ N ( X , A ; G ) , thenthere is a normal covering γ = n W γ o of A such that ϕ vanishes on γ n + . A is closed and P -embedded in X and so,there exists a normal covering α of X such that β = α | A = (cid:8) U α ∩ A | U α ∈ α (cid:9) is a refinement of γ . Therefore, ( α, β ) is anormal covering of ( X , A ) and the restriction ϕ α : α n + → G of the map ϕ : X n + → G vanishes on β n + ⊂ γ n + and so, ϕ α ∈ C ∗ ( α, β ; G ). Passing limit, we obtain the homomorphism λ : C ∗ ( X , A ; G ) → lim −−→ (cid:8) C ∗ ( α, β ; G ) (cid:9) , (12)which is the canonical cochain map. Theorem 1.
The canonical cochain map λ : C ∗ N ( X , A ; G ) → lim −−→ (cid:8) C ∗ ( α, β ; G ) (cid:9) (13) is an epimorphism and has the kernel equal to C ∗ N ( X ; G ) .Proof. To prove that λ is an epimorphism, let ϕ α ∈ C n ( α, β ; G ) and define ϕ ∈ C n ( X ; G ) by the following formula: ϕ ( x , x , . . . , x n ) = ϕ α ( x , x , . . . , x n ) , i f x , x , . . . , x n ∈ U α , where U α ∈ α , otherwise . (14)The map ϕ : X n + → G vanishes on the β n + ⊂ A n + and so, ϕ ∈ C ∗ N ( X , A ; G ). By definition ϕ α = ϕ | α n + and so, λ is anepimorphism. On the other hand, an element ϕ ∈ C n ( X ; G ) is in the kernel of λ if and only if there is some α such that λ ( ϕ ) = ϕ α = ϕ | α n + =
0. Thus λ ( ϕ ) = ϕ ∈ C nN ( X ; G ) . orollary 1. For the cohomology theory ¯ H ∗ N ( X , A ; G ) there is a canonical isomorphism ¯ H ∗ N ( X , A ; G ) ≃ lim −−→ (cid:8) H n ( C ∗ ( α, β ; G ) (cid:9) , (15) where ( α, β ) ∈ Cov N ( X , A ) . We need to see that the constructed cohomology is isomorphic to the Alexandro ff - ˇCech cohomology based on allnormal coverings [Mor ], [Bar], [Wat], [Ba ], [Ba ]. For this aim, consider any normal covering ( α, β ) of ( X , A ) ∈ K Top and let ( R , R ) be the pair of relations defined in the following way: xR U α ⇐⇒ x ∈ U α , ∀ x ∈ X , U α ∈ α (16) aR V β ⇐⇒ a ∈ V β , ∀ a ∈ A , V β ∈ β. (17)In this case, the pair ( R , R ) defines the pair of abstract simplicial complexes (cid:0) X ( α ) , A ( β ) (cid:1) and (cid:0) N ( α ) , N ( β ) (cid:1) , where (cid:0) X ( α ) , A ( β ) (cid:1) is the pair of Vietorisian complexes and (cid:0) N ( α ) , N ( β ) (cid:1) is the pair of nerves [Dow]. By Theorem 1 [Dow]the homology groups of the obtained complexes are isomorphic and so, by Corollary 1 we obtain: Corollary 2.
For each closed P-pair ( X , A ) ∈ K Top there is an isomorphism ¯ H ∗ N ( X , A ; G ) ≃ ˇ H ∗ N ( X , A ; G ) , (18) where ˇ H ∗ N ( X , A ; G ) is the Alexandro ff - ˇCech cohomology group based on all normal coverings. By Lemma 2 (ii) [Wat] the Alexanrdo ff - ˇCech cohomology theory based all normal coverings satisfies the relativehomeomorphism axiom. Therefore, by Corollary 2 we obtain: Corollary 3.
If f : ( X , A ) → ( Y , B ) is a continuous map of P-pairs such that f maps X \ A homeomorphically ontoY \ B, then it induces an isomorphism: f ∗ : ¯ H ∗ N ( Y , B ; G ) ≃ −→ ˇ H ∗ N ( X , A ; G ) , (19)As it is known, the classical Alexandro ff - ˇCech cohomology theory and consequently, the Alexander-Spanier co-homology theory are characterized on the category of compact pairs K C by the Eilenberg-Spanier axioms and thecontinuous axiom [Sp]. In particular, let F : K C → A b be a contravariant functor.(CA) F is said to be continuous if each inverse limit p : X → X = n X α , p αβ , α ∈ A o induces direct limit F ( p ) : F ( X ) = n F ( X α ) , F ( p αβ ) , α ∈ A o → F ( X ) . If for each integer n ∈ Z a cohomology group H n ( − , − ; G ) is continuous, then H ∗ ( − , − ; G ) is said to be a continuouscohomology theory. In the paper [Wat], is shown that for general topological spaces the classical Alexandro ff - ˇCechcohomology theory does not satisfy the continuous axiom (see Example 3 in [Wat]). Using the resolution, Watanabedefined the continuous axiom and showed that the cohomology theory ˇ H ∗ N ( X , A ; G ) satisfies it on the category K Top .In particular, let K be any subcategory of the category of topological spaces K Top and F : K → A b be a contravariantfunctor. h CA i F is said to be continuous if each resolution p : X → X = n X α , p αβ , α ∈ A o induces direct limit F ( p ) : F ( X ) = n F ( X α ) , F ( p αβ ) , α ∈ A o → F ( X ) . On the category K C of compact Hausdor ff spaces p : X → X = n X α , p αβ , α ∈ A o is a resolution i ff p is an inverselimit (see Theorem 6.1.1, [Mar-Seg]). Therefore, the continuous axiom of the Watanabe sense is consistent with theclassical definition. Using a resolution and the classical method of extension of (co)homology theory from the category K Pol C of pairs of compact polyhedra to the category K C of compact pairs [Mdz ], the extension of a functor from thecategory of K Pol of all pairs of polyhedra to the category K Top of all P -pairs are defined [Wat]. It is shown that theAlexandro ff - ˇCech cohomology theory ˇ H ∗ N ( X , A ; G ) based on all normal coverings on the category K Top is the unique(up to a natural equivalence) continuous extension of the singular cohomology on polyhedral pairs see (see Theorem 7of [Wat]). On the other hand, the singular cohomology theory on the category of K Pol is just an additive cohomologytheory [Mil], [Ber-Mdz ]. Consequently, as a corollary it is obtained that the cohomology theory ˇ H ∗ N ( X , A ; G ) on thecategory K Top is characterized by the Eilenberg-Steenrod Axioms, the continuity axiom and the additivity axiom (seeCorollary 8 ii) in [Wat]. Note that except the continuity axiom, all other axioms are su ffi cient to be fulfilled only thesubcategory K Pol of all pairs of polyhedra.By Corollary 2 and Corollary 8 in [Wat], we have:
Corollary 4.
The Alexander-Spanier normal cohomolgy theory ¯ H ∗ N ( − , − ; G ) defined on the category K Top of all closedP-pairs is continuous in the Watanabe sense. . Alexander-Spanier homology theory based on normal coverings In this section we use the method developed in the paper [Ber-Mdz ] to construct an exact homology theory¯ H N ∗ ( − , − ; G ), the so called Alexander-Spanier normal homology , such that for each closed P -pair there exists theUniversal Coe ffi cient Formula:0 −→ Ext( ¯ H n + N ( X , A ); G ) −→ ¯ H Nn ( X , A ; G ) −→ Hom( ¯ H nN ( X , A ); G ) −→ , (20)where ¯ H ∗ N ( − , − ) is the Alexander-Spanier normal cohomology with the coe ffi cient group Z . We will show that theconstructed homology theory satisfies the Eilenberg-Steenrod axioms on the subcategory K Pol of pairs of polyhedraand by the uniqueness theorem on the category of paracompact spaces given in [San], it is unique homology which isconnected with the Alexandro ff - ˇCech cohomology theory by the Universal Coe ffi cient Formula. On the other hand,we define continuity of an exact homology theory and show that the constructed homology theory is continuous in oursense.Let C ∗ be a cochain complex and 0 → G α −→ G ′ β −→ G ′′ → G . Let β : Hom( C ∗ ; G ′ ) → Hom( C ∗ ; G ′′ ) be the chain map induced by β : G ′ → G ′′ . Consider the cone C ∗ ( β ) = (cid:8) C n ( β ) , ∂ (cid:9) = (cid:8) Hom( C ∗ , β ) , ∂ (cid:9) of the chain map β , i.e. C n ( β ) ≃ Hom( C n ; G ′ ) ⊕ Hom( C n + ; G ′′ ) , (21) ∂ ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ δ, β ◦ ϕ ′ − ϕ ′′ ◦ δ ) , ∀ ( ϕ ′ , ϕ ′′ ) ∈ C n ( β ) . (22)If f : C ∗ −→ C ′∗ is a cochain map, then it induces the chain map f : C ∗ ( β ′ ) −→ C ∗ ( β ). In particular, for each n ∈ Z , f n : C n ( β ′ ) −→ C n ( β ) is defined by the formula f n ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ f n , ϕ ′′ ◦ f n + ) . Lemma 1.
Each short exact sequence of cochain complexes → C ∗ f −→ C ′∗ g −→ C ′′∗ → induces a short exact sequence of chain complexes → C ∗ ( β ′′ ) g −→ C ∗ ( β ′ ) f −→ C ∗ ( β ) → . (24) Proof.
The groups G ′ and G ′′ are injective abelian and so, the map β : G ′ → G ′′ induces the following commutativediagram with the exact rows:0 Hom( C ′′∗ ; G ′ ) Hom( C ′∗ ; G ′ ) Hom( C ∗ ; G ′ ) 00 Hom( C ′′∗ ; G ′′ ) Hom( C ′∗ ; G ′′ ) Hom( C ∗ ; G ′′ ) 0. g f g f β ′′ β ′ β (25)Consequently, the diagram (25) induces the short exact sequence (24).Let ( X , A ) ∈ K Top be a closed P -pair. Consider the short exact sequence of the Alexander-Spanier cochain com-plexes based on all normal coverings:0 → ¯ C ∗ N ( X , A ; Z ) j −→ ¯ C ∗ N ( X ; Z ) i −→ ¯ C ∗ N ( A ; Z ) → . (26)By Lemma 1 the short exact sequence (26) induces a short exact sequence of chain complexes:0 → ¯ C N ∗ ( A ; G ) i −→ ¯ C N ∗ ( X ; G ) j −→ ¯ C N ∗ ( X , A ; G ) → , (27)where ¯ C N ∗ ( A ; G ), ¯ C N ∗ ( X ; G ) and ¯ C N ∗ ( X , A ; G ) are chain cones of the chain maps β A : Hom( ¯ C ∗ N ( A ; Z ); G ′ ) → Hom( ¯ C ∗ N ( A ; Z ); G ′′ ), β X : Hom( ¯ C ∗ N ( X ; Z ); G ′ ) → Hom( ¯ C ∗ N ( X ; Z ); G ′′ ) and β ( X , A ) : Hom( ¯ C ∗ N ( X , A ; Z ); G ′ ) → Hom( ¯ C ∗ N ( X , A ; Z ); G ′′ ), respec-tively. Let the homology groups of the chain complexes ¯ C N ∗ ( A ; G ), ¯ C N ∗ ( X ; G ) and ¯ C N ∗ ( X , A ; G ) define by ¯ H N ∗ ( A ; G ),¯ H N ∗ ( X ; G ) and ¯ H N ∗ ( X , A ; G ) and call the Alexander-Spanier normal homology of A , X and ( X , A ), respectively. By thesequence (27), we obtain that it is an exact homology theory.5 orollary 5. (Exactness Axiom) For each ( X , A ) ∈ K Top closed P-pair, there is a long exact homological sequence: . . . ¯ H Nn ( A ; G ) ¯ H Nn ( X ; G ) ¯ H Nn ( X , A ; G ) ¯ H Nn − ( A ; G ) . . . .i ∗ i ∗ j ∗ E j ∗ (28)Let show that the constructed homology theory satisfies the homotopy axiom. Lemma 2.
Each cochain homotopic maps f , g : C ∗ → C ′∗ induce homotopic chain maps f , g : C ∗ ( β ′ ) → C ∗ ( β ) . Proof.
Let D = { D n } be a cochain homotopy of cochain maps f , g : C ∗ → C ′∗ . Therefore, for each n ∈ Z integer D n : C n → C ′ n − is a homomorphism such that δ ′ ◦ D n + D n + ◦ δ = f n − g n : C n → C ′ n . (29)Let ¯ D = n ¯ D n o be the system of the homomorphisms ¯ D n : C n ( β ′ ) → C n + ( β ) defined by the formula:¯ D n ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ D n , − ϕ ′′ ◦ D n + ) , ∀ ( ϕ ′ , ϕ ′′ ) ∈ C n ( β ′ ) . (30)Let show that ¯ D = n ¯ D n o is degree 1 chain map such that¯ D n ◦ ∂ ′ + ∂ ◦ ¯ D n + = f n − g n : C ∗ ( β ′ ) → C ∗ ( β ) . (31)Indeed, we have (cid:16) ¯ D n ◦ ∂ ′ (cid:17) (cid:16) ϕ ′ , ϕ ′′ (cid:17) = ¯ D n (cid:16) ϕ ′ ◦ δ ′ , β ◦ ϕ ′ − ϕ ′′ ◦ δ ′ (cid:17) = (cid:18) ϕ ′ ◦ δ ′ ◦ D n , − (cid:16) β ◦ ϕ ′ − ϕ ′′ ◦ δ ′ (cid:17) ◦ D n + (cid:19) = (cid:16) ϕ ′ ◦ δ ′ ◦ D n , − β ◦ ϕ ′ ◦ D n + + ϕ ′′ ◦ δ ′ ◦ D n + (cid:17) . (32) (cid:16) ∂ ◦ ¯ D n + (cid:17) (cid:16) ϕ ′ , ϕ ′′ (cid:17) = ∂ (cid:18) ¯ D n + (cid:16) ϕ ′ , ϕ ′′ (cid:17)(cid:19) = ∂ (cid:16) ϕ ′ ◦ D n + , − ϕ ′′ ◦ D n + (cid:17) = (cid:18) ϕ ′ ◦ D n + ◦ δ, β ◦ ϕ ′ ◦ D n + − (cid:16) − ϕ ′′ ◦ D n + ◦ δ (cid:17)(cid:19) = (cid:16) ϕ ′ ◦ D n + ◦ δ, β ◦ ϕ ′ ◦ D n + + ϕ ′′ ◦ D n + ◦ δ (cid:17) . (33)Hence, we have (cid:16) ¯ D n ◦ ∂ ′ + ∂ ◦ ¯ D n + (cid:17) (cid:16) ϕ ′ , ϕ ′′ (cid:17) = (cid:16) ϕ ′ ◦ δ ′ ◦ D n , − β ◦ ϕ ′ ◦ D n + + ϕ ′′ ◦ δ ′ ◦ D n + (cid:17) + (cid:16) ϕ ′ ◦ D n + ◦ δ, β ◦ ϕ ′ ◦ D n + + ϕ ′′ ◦ D n + ◦ δ (cid:17) = (cid:16) ϕ ′ ◦ δ ′ ◦ D n + ϕ ′ ◦ D n + ◦ δ, − β ◦ ϕ ′ ◦ D n + + ϕ ′′ ◦ δ ′ ◦ D n + + β ◦ ϕ ′ ◦ D n + + ϕ ′′ ◦ D n + ◦ δ (cid:17) = (cid:18) ϕ ′ ◦ (cid:16) δ ′ ◦ D n + D n + ◦ δ (cid:17) , ϕ ′′ ◦ (cid:16) δ ′ ◦ D n + + D n + ◦ δ (cid:17)(cid:19) = (cid:18) ϕ ′ ◦ (cid:0) f n − g n (cid:1) , ϕ ′′ ◦ (cid:16) f n + − g n + (cid:17)(cid:19) = (cid:16) ϕ ′ ◦ f n , ϕ ′′ ◦ f n + (cid:17) − (cid:16) ϕ ′ ◦ g n , ϕ ′′ ◦ g n + (cid:17) = f n (cid:16) ϕ ′ , ϕ ′′ (cid:17) − g n (cid:16) ϕ ′ , ϕ ′′ (cid:17) . (34) Corollary 6. (Homotopy Axiom) If f , g : ( X , A ) → ( X , B ) are homotopic continuous maps of closed P-pairs, thenf ∗ = g ∗ : ¯ H N ∗ ( X , A ) → ¯ H N ∗ ( Y , B ) . (35)By Theorem 1 of [Ber-Mdz ], for the homology theory ¯ H N ∗ ( − , − ; G ) there is the Universal Coe ffi cient Formula. Corollary 7. (Universal Coe ffi cient Formula) For each ( X , A ) ∈ K Top closed P-pair and an abelian group G, thereexists a short exact sequence −→ Ext( ¯ H n + N ( X , A ); G ) −→ ¯ H Nn ( X , A ; G ) −→ Hom( ¯ H nN ( X , A ); G ) −→ , (36) where ¯ H n + N ( − , − ) is the Alexander-Spanier normal cohomology with the coe ffi cient group Z . Using the Universal Coe ffi cient Formula we simply obtain the relative homeomorphism and the dimension axioms.6 heorem 2. (Relative Homeomorphism Axiom) The Alexander-Spannier normal homology theory H N ∗ ( − , − ; G ) isinvariant under relative homeomorphism.Proof. Consider the diagram induced by f : ( X , A ) → ( Y , B ):0 Ext (cid:16) ¯ H n + N (cid:0) Y , B ; Z (cid:1) ; G (cid:17) H n ( Y , B ; G ) Hom (cid:16) ¯ H nN (cid:0) Y , B ; Z (cid:1) ; G (cid:17) , (cid:16) ¯ H n + N (cid:0) X , A ; Z (cid:1) ; G (cid:17) H n ( X , A ; G ) Hom (cid:16) ¯ H nN (cid:0) X , A ; Z (cid:1) ; G (cid:17) (cid:0) f ∗ ; G (cid:1) f ∗ Hom (cid:0) f ∗ ; G (cid:1) (37)By Corollary 3 the homomorphisms f ∗ : ¯ H pN (cid:0) Y , B ; Z (cid:1) → ¯ H pN (cid:0) Y , B ; Z (cid:1) is an isomorphism for all p ≥ f ∗ : ¯ H Nn (cid:0) X , A ; Z (cid:1) → ¯ H Nn (cid:0) Y , B ; Z (cid:1) is an isomorphism. Theorem 3. (Dimension Axiom) For each one point topological space P, there is an isomorophism ¯ H Nn ( P ; G ) = G , i f n = , i f n , . (38) Proof.
Consider the corresponding Universal Coe ffi cient Formula:0 −→ Ext( ¯ H n + N ( P ; Z ); G ) −→ ¯ H Nn ( P ; G ) −→ Hom( ¯ H nN ( P ; Z ); G ) −→ . (39)The Alexander-Spanier normal cohomology theory ¯ H ∗ N ( − , − ) satisfies the dimension axiom. Therefore, we have¯ H nN ( P ; Z ) = Z , i f n = , i f n , . (40)Thus, Ext( ¯ H n + N ( P ; Z ); G ) ≃ H Nn ( P ; G ) ≃ Hom( ¯ H nN ( P ; Z ); G ) = Hom( Z ; G ) ≃ G , i f n = G ) ≃ , i f n , . . (41)
3. Some properties of Alexander-Spanier homology theory on the category K Top
The continuity axiom for an exact homology theory H ∗ ( − , − ; G ) on the category of compact spaces is defined inthe paper [Mdz ] (see Definition 1). We can formulate it in the following way:(CA ∗ ) An exact homology theory H ∗ ( − ; G ) : K C → A b is said to be continuous if each inverse limit p : X → X = n X α , p αβ , α ∈ A o induces a long exact sequence: . . . lim ←−− (3) H n + ( X α ; G ) lim ←−− (1) H n + ( X α ; G ) H n ( X ; G ) lim ←−− H n ( X α ; G ) lim ←−− (2) H n + ( X α ; G ) . . . . (42)Using the continuity axiom, an exact homology theory H ∗ ( − , − ; G ) is characterized on the category K C (see Corol-laries 3, 4, 5 in [Mdz ]).Here we formulate three di ff erent properties of an exact homology theory on the category K Top , which are the generalization of the axioms proposed by N. Berikashvili [Ber], L. Mdzinarishvili and Kh. Inasaridze[In-Mdz], L. Mdzinarishvili [Mdz ] and Kh. Inasaridze [In ] on the category K C . Note that for paracompact spaces S.Saneblidze [San] generalized the result obtained by N. Berikashvili [Ber] for compact spaces.Let H ∗ ( − , − ; G ) be a homological functor defined on the category K Top of closed P -pairs.(CEH) ( Continuity for an Exact Homology ) For each resolution p : ( X , A ) → ( X , A ) = n ( X α , A β ) , p αβ , α ∈ A o ofclosed P -pair ( X , A ) ∈ K Top and an abelian group G , there exists a fuctorial long exact sequence: . . . lim ←−− (3) H n + ( X α , A β ; G ) lim ←−− (1) H n + ( X α , A β ; G ) H n ( X , A ; G ) lim ←−− H n ( X α , A β ; G ) lim ←−− (2) H n + ( X α , A β ; G ) . . . . (43)7CIG) ( Continuity for an Injective Group ) For each resolution p : ( X , A ) → ( X , A ) = n ( X α , A β ) , p αβ , α ∈ A o ofclosed P -pair ( X , A ) ∈ K Top and an injective abelian group G , there exists an isomorphism H n ( X , A ; G ) ≈ lim ←−− H n ( X α , A β ; G ) . (44)(UCF) ( Universal Coe ffi cient Formula ): For each ( X , A ) ∈ K Top closed P -pair and an abelian group G , there existsa functorial exact sequence:0 −→ Ext (cid:16) ¯ H n + N ( X , A ); G (cid:17) −→ H n ( X , A ; G ) −→ Hom( ¯ H nN (cid:0) X , A ); G (cid:1) −→ , (45)where ¯ H n + N ( − , − ; G ) is the Alexander-Spanier normal cohomology. Definition 1. (see [Ber-Mdz ]) A direct system C ∗ = { C ∗ α } of cochain complexes C ∗ α is said to be associated with acochain complex C ∗ , if there is a homomorphism C ∗ → C ∗ such that for each n ∈ Z the induced homomorphism lim −−→ H ∗ ( C ∗ α ) → H ∗ ( C ∗ ) (46) is an isomorphism. By Lemma 5.5 [Ber-Mdz ], it is known that if a direct system C ∗ = { C ∗ α } of cochain complexes C ∗ α is associatedwith a cochain complex C ∗ , then there is an infinite exact sequence: · · · −→ lim ←− (2 k + ¯ H n + k + ( C ∗ α ; G ) −→ · · · −→ lim ←− (3) ¯ H n + ( C ∗ α ; G ) −→ lim ←− (1) ¯ H n + ( C ∗ α ; G ) −→−→ ¯ H n ( C ∗ ; G ) π ∗ −→ lim ←− ¯ H n ( C ∗ α ; G ) −→ lim ←− (2) ¯ H n + ( C ∗ α ; G ) −→ · · · −→ lim ←− (2 k ) ¯ H n + k ( C ∗ α ; G ) −→ · · · . (47)where ¯ H ∗ ( C ∗ ; G ) = H ∗ (cid:16) Hom (cid:0) C ∗ ; β (cid:1)(cid:17) = H ∗ (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) and ¯ H ∗ ( C ∗ α ; G ) = H ∗ (cid:16) Hom( C ∗ α ; β ) (cid:17) = H ∗ (cid:16) C α ∗ ( β ) (cid:17) . Theorem 4.
The Alexander-Spanier normal homology theory ¯ H N ∗ ( − ; G ) satisfies CEH property.Proof. Let p : X → X = n X α , p αβ , α ∈ A o be a resolution. Assume that it is a polyhedral resolution. Consider thedirect system C ∗ N = { ¯ C ∗ N ( X α ) } of the cochain complexes, where ¯ C ∗ N ( X α ) is the Alexander-Spanier cochain complexbased on the normal coverings. In this case, p : X → X = n X α , p αβ , α ∈ A o induces the homomorphism p : lim −−→ ¯ C ∗ N ( X α ; G ) → ¯ C ∗ N ( X ; G ) , (48)which itself induces the homomorphism p ∗ : lim −−→ ¯ H ∗ N ( X α ; G ) → ¯ H ∗ N ( X ; G ) . (49)By Corollary 2 we have ¯ H ∗ N ( X ; G ) ≃ ˇ H ∗ N ( X ; G ). On the other hand, for each α , the space X α is a polyhedron and so thecohomology group ˇ H ∗ N ( X α ; G ) is isomorphic to the classical Alexander-Spanier cohomology group ˇ H ∗ ( X α ; G ), whichitself is isomorphic to the singular cohomology group ˇ H ∗ s ( X α ; G ).(see Corollary 6.9.7 [Sp]. Therefore, by Corollary8 ii) in [Wat] (49) is an isomorphism. Consequently, by Lemma 5.5 [Ber-Mdz ], we have the following long exactsequence: . . . lim ←−− (3) ¯ H Nn + ( X α ; G ) lim ←−− (1) ¯ H Nn + ( X α ; G ) ¯ H Nn ( X ; G ) lim ←−− ¯ H Nn ( X α ; G ) lim ←−− (2) ¯ H Nn + ( X α ; G ) . . . .(50)By Theorem of [San], the Alexander-Spanier homology ¯ H N ∗ ( − , − ; G ) is unique (up to a natural equivalence) on thecategory K Pol and therefore, by Theorem 4, we obtain (cf. [Kuz], [Bo-Ku]):
Corollary 8.
If H ∗ ( − , − ; G ) and H ′∗ ( − , − ; G ) are exact homology theories on a subcategory K ⊂ K Top , which areconnected with the Alexander-Spanier normal cohomology theory by the Universal Coe ffi cient Formula for polyhe-dral pair and satisfy CEH property, then any natural transformation (a mapping between corresponding long exactsequences involving derivative limits) T : H ∗ → H ′∗ , which is an isomorphism for the one point topological space, isthe isomorphism for any closed P-pair ( X , A ) ∈ K .
8n cases when homologies ¯ H ∗ and ¯ H ′∗ are generated by cochain complexes C ∗ and C ′∗ , using the method developedin [Ber-Mdz ], any cochain map f ∗ : C ′∗ → C ∗ induces a natural transformation f ∗ : ¯ H ∗ → ¯ H ′∗ that is mentioned inCorollary 8: . . . lim ←−− (3) ¯ H Nn + ( X α , A β ; G ) lim ←−− (1) ¯ H Nn + ( X α , A β ; G ) ¯ H Nn ( X , A ; G ) lim ←−− ¯ H Nn ( X α , A β ; G ) lim ←−− (2) ¯ H Nn + ( X α , A β ; G ) . . .. . .. . . lim ←−− (3) ¯ H ′ Nn + ( X α , A β ; G ) lim ←−− (1) ¯ H ′ Nn + ( X α , A β ; G ) ¯ H ′ Nn ( X , A ; G ) lim ←−− ¯ H ′ Nn ( X α , A β ; G ) lim ←−− (2) ¯ H ′ Nn + ( X α , A β ; G ) . . . . f ∗ f ∗ f ∗ f ∗ f ∗ (51)For example, if C ∗ s ( X ; G ) is the singular cochain complex of topological spaces X and ¯ C s ∗ ( X ; G ) = Hom (cid:16) C ∗ s ( X ); β (cid:17) , then we can construct the homology ¯ H s ∗ ( X ; G ) of the obtained chain complex ¯ C s ∗ ( X ; G ) . Therefore, ¯ H s ∗ ( − ; G ) is thehomology generated by the singular cochain complex C ∗ s ( − ; G ) . It is known that there is a homomorphism j :¯ C ∗ ( X ; G ) → C ∗ s ( X ; G ) from the Alexander-Sapnier cochain complex to the singular cochain complex, which in-duces the isomorphism j ∗ : ¯ H ∗ ( X ; G ) → ¯ H ∗ s ( X ; G ) on the category of manifolds. On the other hand, for a manifold X the Alexander-Spanier cochain complex ¯ C ∗ ( X ; G ) coincides with the Alexander-Spanier normal cochain complex¯ C ∗ N ( X ; G ) . Therefore, by Corollary 8, there is an isomorphism: j ∗ : ¯ H s ∗ ( X ; G ) ≃ −→ ¯ H N ∗ ( X ; G ) . (52)Now, using the method developed in [Ber-Mdz ] we find the relation between (UCF), (CEH) and (CIG) axioms. Theorem 5.
If H ∗ is an exact homological functor defined on the category K Top of closed P-pairs, which is connectedwith the Alexander-Spanier normal cohomology theory by the Universal Coe ffi cient Formula on the subcategory K Pol and satisfies (CEH) property, then it satisfies (CIG) property for all closed P-pairs.Proof.
Let p : ( X , A ) → ( X , A ) = n ( X α , A β ) , p αβ , α ∈ A o be a resolution. The pair ( X , A ) is a closed P -pair and so,the restriction p | A : A → A = n A β , p αα ′ | A α , α ∈ A o is a resolution [Mar-Seg]. Therefore, it is su ffi cient to prove in theabsolute case. Hence, for each resolution p : X → X = (cid:8) X α , p αα ′ , α ∈ A (cid:9) and an injective abelian group G ′ we have toshow that: H n ( X ; G ′ ) ≃ lim ←−− H n ( X α ; G ′ ) . (53)By the condition of the Theorem, for each abelian group G we have the following long exact sequence: . . . lim ←−− (3) H n + ( X α ; G ) lim ←−− (1) H n + ( X α ; G ) H n ( X ; G ) lim ←−− H n ( X α ; G ) lim ←−− (2) H n + ( X α ; G ) . . . . (54)Therefore, we should show that for each injective abelian group G the derivatives are trivial:lim ←−− ( i ) H n + ( X α ; G ) = , i ≥ . (55)Indeed, for each polyhedron X α we have the sequnce:0 → Ext (cid:16) ¯ H n + N ( X α ); G (cid:17) → H n ( X α ; G ) → Hom (cid:16) ¯ H nN ( X α ); G (cid:17) → , (56)which induces the long exact sequence:0 lim ←−− Ext (cid:16) ¯ H n + N ( X α ) , G (cid:17) lim ←−− H n ( X α ; G ) lim ←−− Hom (cid:16) ¯ H nN ( X α ) , G (cid:17) lim ←−− Ext (cid:16) ¯ H n + N ( X α ) , G (cid:17) lim ←−− H n ( X α ; G ) lim ←−− Hom (cid:16) ¯ H nN ( X α ) , G (cid:17) lim ←−− Ext (cid:16) ¯ H n + N ( X α ) , G (cid:17) lim ←−− H n ( X α ; G ) lim ←−− Hom (cid:16) ¯ H nN ( X α ) , G (cid:17) . . . . (57)9or each injective abelian group G the functor Ext( − ; G ) is trivial and so, there is the isomorphism:lim ←−− ( i ) H n ( X α ; G ) ≃ lim ←−− ( i ) Hom (cid:16) H nN ( X α ); G (cid:17) , i ≥ . (58)On the other hand, by Proposition 2 of [H-M], for the direct system ¯ H ∗ N ( X ) = n ¯ H nN ( X α ) , p α,α ′ , A o we have:0 → lim ←−− Hom (cid:16) H n + N ( X α ); G (cid:17) → Ext (cid:18) lim −−→ H nN ( X α ); G (cid:19) → lim ←−− Ext (cid:16) H nN ( X α ); G (cid:17) → lim ←−− Hom (cid:16) H n + N ( X α ); G (cid:17) → . (59)By Lemma 1.3. of [H-M] for each injective abelian group G we obtain:lim ←−− ( i ) Hom (cid:16) H n + ( X α ); G (cid:17) = , i ≥ . (60)Therefore, by (54), (58) and (60) we obtain: H n ( X ; G ) ≈ lim ←−− H n ( X α ; G ) . (61) Theorem 6.
If H ∗ is an exact homological functor defined on the category K Top of closed P-pairs, which is connectedwith the Alexander-Spanier normal cohomology theory by the Universal Coe ffi cient Formula on the subcategory K Pol and satisfies (CIG) property, then it satisfies (UCF) property for all closed P-pairs.Proof.
Let p : ( X , A ) → ( X , A ) = n ( X α , A β ) , p αβ , α ∈ A o be a resolution. The ( X , A ) is a closed P -pair and so, restriction p | A : A → A = n A β , p αα ′ | A α , α ∈ A o is a the resolution [Mar-Seg]. Therefore, it is su ffi cient to prove in the absolutecase. By the condition of the Theorem, for each resolution p : X → X = (cid:8) X α , p αα ′ , α ∈ A (cid:9) and an injective abeliangroup G we have an isomorphism: H n ( X ; G ) ≈ lim ←−− H n ( X α ; G ) . (62)By the condition of the Theorem, for each X α we have the exact sequence:0 → Ext (cid:16) ¯ H n + N ( X α ); G (cid:17) → H n ( X α ; G ) → Hom (cid:16) ¯ H nN ( X α ); G (cid:17) → , (63)which induces the long exact sequence:0 lim ←−− Ext (cid:16) ¯ H n + N ( X α ) , G (cid:17) lim ←−− H n ( X α ; G ) lim ←−− Hom (cid:16) ¯ H nN ( X α ) , G (cid:17) lim ←−− Ext (cid:16) ¯ H n + N ( X α ) , G (cid:17) lim ←−− H n ( X α ; G ) lim ←−− Hom (cid:16) ¯ H nN ( X α ) , G (cid:17) lim ←−− Ext (cid:16) ¯ H n + N ( X α ) , G (cid:17) lim ←−− H n ( X α ; G ) lim ←−− Hom (cid:16) ¯ H nN ( X α ) , G (cid:17) . . . . (64)Note that for each injective abelian group G the functor Ext( − ; G ) is trivial and by (64) we obtain the isomorphism:lim ←−− H n ( X α ; G ) ≈ lim ←−− Hom (cid:16) ¯ H nN ( X α ); G (cid:17) . (65)If we apply the isomorphism lim ←−− Hom (cid:16) ¯ H nN ( X α ); G (cid:17) ≈ Hom (cid:18) lim −−→ ¯ H nN ( X α ); G (cid:19) , then by (65) we obtain:lim ←−− H n ( X α ; G ) ≈ Hom (cid:18) lim −−→ ¯ H n ( X α ); G (cid:19) = Hom (cid:16) ¯ H nN ( X ); G (cid:17) . (66)Therefore, by (62), if G is an injective, then H n ( X ; G ) ≈ lim ←−− ¯ H n ( X α ; G ) ≈ Hom (cid:16) ¯ H nN ( X ); G (cid:17) . (67)10ow consider any abelian group G and the corresponding injective resolution:0 → G → G ′ → G ′′ → . (68)Apply to the sequence (68) by the functor Hom (cid:16) ¯ H nN ( X ); − (cid:17) . The abelian groups G ′ and G ′′ are injective and so wehave: 0 Hom (cid:16) ¯ H nN ( X ); G (cid:17) Hom (cid:16) ¯ H nN ( X ); G ′ (cid:17) Hom (cid:16) ¯ H nN ( X ); G ′′ (cid:17) Ext (cid:16) ¯ H nN ( X ); G (cid:17) . (69)Therefore, for each integer n ∈ N we haveHom (cid:16) ¯ H nN ( X ); G (cid:17) ≃ Ker (cid:16)
Hom( ¯ H nN ( X ); G ′ (cid:17) → Hom (cid:16) ¯ H nN ( X ); G ′′ ) (cid:17) , (70)Ext (cid:16) ¯ H nN ( X ); G (cid:17) ≃ Coker (cid:16)
Hom( ¯ H nN ( X ); G ′ (cid:17) → Hom (cid:16) ¯ H nN ( X ); G ′′ (cid:17) . (71)Now apply sequence (68) by the homological bifunctor ¯ H N ∗ ( X ; − ), which gives the following long exact sequence: · · · → ¯ H Nn + ( X ; G ′ ) → ¯ H Nn + ( X ; G ′′ ) → ¯ H Nn ( X ; G ) → ¯ H Nn ( X ; G ′ ) → ¯ H Nn ( X ; G ′′ ) → . . . . (72)Therefore, for each n ∈ N we obtain the following short exact sequence:0 Coker (cid:16) ¯ H Nn + ( X ; G ′ ) → ¯ H Nn + ( X ; G ′′ ) (cid:17) Hom( ¯ H nN ( X ); G ) Ker (cid:16) ¯ H Nn ( X ; G ′ ) → ¯ H Nn ( X ; G ′′ ) (cid:17) . (73)By (73), (71) and (70) we obtain that for each X ∈ K Top there is a short exact sequence:0 Ext (cid:16) ¯ H n + N ( X ); G (cid:17) H n ( X ; G ) Hom (cid:16) ¯ H nN ( X ); G (cid:17) Corollary 9.
The Alexander-Spanier normal homology theory ¯ H N ∗ ( − , − ; G ) defined on the category K Top satisfies(CEH), (CIG) and (UCF) properties.
4. Uniqueness Theorem
In this section we will follow the approach developed in the paper [In ] to obtain the uniqueness theorem for anexact bifunctor homology theory with the (CIG) property. We will see that the Alexander-Spanier normal homologytheory ¯ H N ∗ ( − , − ; G ) is a bifunctor. On the other hand, by Theorem 4 and 5 it has (CIS) property and so, we obtain theaxiomatic characterization of ¯ H N ∗ ( − , − ; G ) on the category of pairs of general topological spaces K Top . An exact homology functor H ∗ ( − , − ; G ) defined on the category K Top is said to be a bifunctor [In ], if for each pair( X , A ) ∈ K Top and a short exact sequence 0 → G ϕ −→ G ψ −→ G → , (75)there is the functorial natural long exact sequence: . . . ψ ∗ −→ H n + ( X ; G ) d n + −→ H n ( X ; G ) ϕ ∗ −→ H n ( X ; G ) ψ ∗ −→ H n ( X ; G ) d n −→ . . . . (76)In the paper [In ] Kh. Inasaridze described an exact bifunctor homology theory using the continuity property forinfinitely divisible (injective abelian) groups on the subcategory K C of compact Hausdor ff pairs. In particular, it isproved that there exists one and only one exact bifunctor homology theory on the category K C of compact Hausdor ff pairs with coe ffi cients in the category of abelian groups (up to natural equivalence) which satisfies the axioms ofhomotopy, excision, dimension, and continuity for every infinitely divisible (injective abelian) group (see Theorem 1in [In ]). 11 emma 3. Each cochain complex C ∗ and a short exact sequence of abelian groups → G ϕ −→ G ψ −→ G → induces a short exact sequence of chain complexes: → C ∗ ( β ) ϕ −→ C ∗ ( β ) ψ −→ C ∗ ( β ) → . (78) Proof.
Consider an injective resolution of the exact sequence (77):0
G G ′ G ′′
00 0 0 α β G G ′ G ′′ ϕ ϕ ′ ϕ ′′ α β G G ′ G ′′ ,ψ ψ ′ ψ ′′ α β (cid:0) C ∗ ; G ′ (cid:1) Hom (cid:0) C ∗ ; G ′′ (cid:1) β Hom (cid:16) C ∗ ; G ′ (cid:17) Hom (cid:16) C ∗ ; G ′′ (cid:17) ϕ ′ ϕ ′′ β Hom (cid:16) C ∗ ; G ′ (cid:17) Hom (cid:16) C ∗ ; G ′′ (cid:17) . ψ ′ ψ ′′ β β , β and β that is the sequence (78). Theorem 7.
There exists one and only one exact bifunctor homology theory (up to natural equivalence) on the category K Top , which is connected with the Alexander-Spanier normal cohomology theory by the Universal Coe ffi cient Formulafor polyhedron pairs, satisfies the Eilenberg-Steenrod axioms on the subcategory K Pol and has (SIG) property.Proof.
Our aim is to show that any homology theory H ∗ ( − , − ; G ) which satisfies the conditions of the Theorem isisomorphic to the Alexander-Sapanier normal homology theory ¯ H N ∗ ( − , − ; G ) . Let ˜p : X → ˜X = n ˜ X λ , p ˜ λ ˜ λ ′ , ˜ λ ∈ ˜ Λ o be a polyhedral resolution defined by normal covering of X as in [Mar-Seg].Note that in the paper [Mar-Seg] instead of ˜p = ( ˜ p ˜ λ ) : X → ˜X is used p ∗ = ( p ∗ ˜ λ ) : X → X ∗ notation. In particular,let Λ be the set of all finite subsets λ = { α , α , . . . , α n } consisting of di ff erent normal coverings of X . It is clearthat Λ is ordered by inclusion, i.e. λ ≤ λ ′ ⇐⇒ λ ⊂ λ ′ . For each λ ∈ Λ denote by α λ the normal covering α ∧ α ∧· · ·∧ α n = { U α T U α T · · · T U α n | U α i ∈ α i , i = , , . . . , n } . Denote by N λ the nerve N ( α λ ) of covering α λ . Let X λ = | N λ | be the geometric realization with CW-topology of N λ . In this case, if λ ≤ λ ′ , then there is a uniquely definedfunction p λλ ′ , which maps the vertex U α T U α T · · · T U α n T · · · T U α n ′ of N λ ′ to the vertex U α T U α T · · · T U α n of N λ , which itself induces a simplicial mapping p λλ ′ : X λ ′ → X λ . It is known that X = (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) is an inversesystem. Moreover, for each λ ∈ Λ there is the canonical mapping p λ : X → X λ defined by the corresponding partitionof the unity Φ λ = { ϕ α λ } of the covering α λ , such that p λλ ′ p λ ′ = p λ , λ ≤ λ ′ . (81)12onsequently, p = ( p λ ) : X → X is a mapping of the system. For each λ ∈ Λ , let U λ = n U ( λ,µ ) | µ ∈ M o be the systemof all open neighborhoods of closure p λ ( X ) in X λ . Let ˜ λ = ( λ, µ ) and ˜ Λ = n ˜ λ = ( λ, µ ) | λ ∈ Λ , µ ∈ M o . In this case,˜ λ ≤ ∗ ˜ λ ′ if λ ≤ λ ′ and p λλ ′ (cid:18) U ′ ( λ ′ ,µ ′ ) (cid:19) ⊂ U ( λ,µ ) . Let ˜ X ˜ λ = U ( λ,µ ) and ˜ p λ ∗ : X → X ∗ λ ∗ be the mapping p λ : X → U ( λ,µ ) ⊂ X λ .Consequently, denote by ˜ p ˜ λ ˜ λ ′ : ˜ X ˜ λ ′ → X ∗ λ ∗ the mapping p λλ ′ | U ′ ( λ ′ ,µ ′ ) : U ′ ( λ ′ ,µ ′ ) → U ( λ,µ ) . It is known that ˜p = ( ˜ p ˜ λ ) : X → ˜X is a polyhedral resolution of X [Mar-Seg]. Let i = n X λ , i o : X → ˜X be the mapping, where i : ˜ Λ → Λ is givenby i ( ˜ λ ) = i (cid:0) λ, µ (cid:1) = λ, ∀ ˜ λ ∈ ˜ Λ . Consider the direct systems n ¯ C ∗ N ( X λ ) , p λλ ′ , λ ∈ Λ o , (cid:26) ¯ C ∗ N (cid:16) ˜ X ˜ λ (cid:17) , p λ ˜ λ ′ , ˜ λ ∈ ˜ Λ (cid:27) and themapping between the corresponding limit groups i = lim −−→ i λ : lim −−→ ¯ C ∗ N (cid:16) ˜ X ˜ λ (cid:17) → lim −−→ ¯ C ∗ N ( X λ ) . (82)Our aim is to show that (82) induces an isomorphism between the corresponding cohomology groups. Consider thefollowing composition lim −−→ ¯ H ∗ N (cid:16) ˜ X ˜ λ ; G (cid:17) i ∗ −→ lim −−→ ¯ H ∗ N ( X λ ; G ) p ∗ −→ ¯ H ∗ N ( X ; G ) . (83)It is clear that ˜p = i ◦ p : X → ˜X and so, the composition ˜p ∗ = p ∗ ◦ i ∗ : lim −−→ ¯ H ∗ N (cid:16) ˜ X ˜ λ ; G (cid:17) → ¯ H ∗ N ( X ; G ) is an isomorphismby Corollary 4. On the other hand, by Corollary 2 and Theorem 3.2 [Mor ] we have an isomorphism: p ∗ : lim −−→ ¯ H ∗ N ( X λ ; G ) → ¯ H ∗ N ( X ; G ) , (84)and therefore, i ∗ : lim −−→ ¯ H ∗ N (cid:16) ˜ X ˜ λ ; G (cid:17) → lim −−→ ¯ H ∗ N ( X λ ; G ) must be an isomorphism as well. Therefore, by Lemma 5.5[Ber-Mdz ] to study the homology groups ¯ H N ∗ ( X ; G ), instead of the resolution ˜p = ( ˜ p λ ) : X → ˜X = n ˜ X λ , p ˜ λ ˜ λ ′ , ˜ λ ∈ ˜ Λ o , it is su ffi cient to consider the inverse system X = (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) and the corresponding mapping p = ( p λ ) : X → X = (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) .Let N ( X ) = lim ←−− (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) , then the mapping p : X → (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) induces a canonical map p : X →N ( X ), i.e., p λ = p N ( X ) λ ◦ p , ∀ λ ∈ Λ , (85)where p N ( X ) λ : N ( X ) → X λ is a canonical projection. Consider the mapping p N ( X ) = (cid:16) p N ( X ) λ (cid:17) : N ( X ) → X = (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) and let’s show that ¯ H ∗ N ( X ; G ) ≃ ¯ H ∗ N (cid:0) N ( X ); G (cid:1) . (86)For this aim, consider the limit space ˜ X = lim ←−− n ˜ X λ , p ˜ λ ˜ λ ′ , ˜ λ ∈ ˜ Λ o and the corresponding mapping ˜p ˜ X = ( ˜ p ˜ X λ ) : ˜ X → ˜X = n ˜ X λ , p ˜ λ ˜ λ ′ , ˜ λ ∈ ˜ Λ o . Note that all terms ˜ X λ are polyheron and so, topologically complete. Therefore, by Theorem 6.5 and6.16 [Mar], the mapping ˜p ˜ X is a resolution. Hence, by Corollary 3 we have the isomorphisms:¯ H ∗ N ( X ; G ) ≃ lim −−→ ¯ H ∗ N ( ˜ X λ ; G ) , (87)¯ H ∗ N ( ˜ X ; G ) ≃ lim −−→ ¯ H ∗ N ( ˜ X λ ; G ) . (88)Therefore, there is a canonical isomorphism ¯ H ∗ N ( X ; G ) ≃ ¯ H ∗ N ( ˜ X ; G ) . (89)Consider the maps ˜ p : X → ˜ X , p : X → N ( X ), i : N ( X ) → ˜ X and p ˜ X : ˜ X → N ( X ) induced by ˜p = ( ˜ p λ ) : X → ˜X = n ˜ X λ , p ˜ λ ˜ λ ′ , ˜ λ ∈ ˜ Λ o , p = ( p λ ) : X → X = (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) , i = n X λ , i o : (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) → n ˜ X ˜ λ , p ˜ λ ˜ λ ′ , ˜ λ ∈ ˜ Λ o and p ˜ X = ( p λ ) : ˜ X → X = (cid:8) X λ , p λλ ′ , λ ∈ Λ (cid:9) . In this case ˜ p = i ◦ p and p ˜ X ◦ i = N ( X ) , which induce the isomorphisms:¯ H ∗ N ( ˜ X ; G ) i ∗ −→ ¯ H ∗ N ( N ( X ); G ) p ∗ −→ ¯ H ∗ N ( X ; G ) , (90)¯ H ∗ N (cid:0) N ( X ); G (cid:1) (cid:16) p ˜ X (cid:17) ∗ −→ ¯ H ∗ N (cid:16) ˜ X ; G (cid:17) i ∗ −→ ¯ H ∗ N (cid:0) N ( X ); G (cid:1) . (91)13y (89), the composition p ∗ ◦ i ∗ : ¯ H ∗ N (cid:16) ˜ X ; G (cid:17) → ¯ H ∗ N ( X ; G ) is an isomorphism. On the other hand i ∗ ◦ (cid:16) p ˜ X (cid:17) ∗ is theidentity. Therefore, i ∗ : ¯ H ∗ N (cid:16) ˜ X ; G (cid:17) → ¯ H ∗ N (cid:0) N ( X ); G (cid:1) is an isomorphism and by (89), the isomorphism (86) is fulfilled.Therefore, by (84) and (86) we have: ¯ H ∗ N ( N ( X ); G ) ≃ lim −−→ ¯ H ∗ N ( X λ ; G ) . (92)By Theorem 6 for each topological space X , there is the Universal Coe ffi cient Formula:0 Ext (cid:16) ¯ H n + N (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) H n (cid:0) N ( X ) ; G (cid:1) Hom (cid:16) ¯ H nN (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) . (93)For each λ ∈ Λ consider the diagram induced by the natural projection p N ( X ) λ : N ( X ) → X λ :0 Ext (cid:16) ¯ H n + N (cid:0) X λ ; Z (cid:1) ; G (cid:17) H n ( X λ ; G ) Hom (cid:16) ¯ H nN (cid:0) X λ ; Z (cid:1) ; G (cid:17) , (cid:16) ¯ H n + N (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) H n (cid:0) N ( X ) ; G (cid:1) Hom (cid:16) ¯ H nN (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) (cid:18)(cid:16) p N ( X ) λ (cid:17) ∗ ; G (cid:19) (cid:16) p N ( X ) λ (cid:17) ∗ Hom (cid:18)(cid:16) p N ( X ) λ (cid:17) ∗ ; G (cid:19) (94)which induces the following commutative diagram:0 lim ←−− Ext (cid:16) ¯ H n + N (cid:0) X λ ; Z (cid:1) ; G (cid:17) lim ←−− H n ( X λ ; G ) lim ←−− Hom (cid:16) ¯ H nN (cid:0) X λ ; Z (cid:1) ; G (cid:17) lim ←−− Ext (cid:16) ¯ H n + N (cid:0) X λ ; Z (cid:1) ; G (cid:17) . . . . (cid:16) ¯ H n + N (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) H n (cid:0) N ( X ) ; G (cid:1) Hom (cid:16) ¯ H nN (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) (cid:18)(cid:16) p N ( X ) λ (cid:17) ∗ ; G (cid:19) (cid:16) p N ( X ) λ (cid:17) ∗ Hom (cid:18)(cid:16) p N ( X ) λ (cid:17) ∗ ; G (cid:19) (95)Consequently, for an injective abelian group G ′ we obtain the diagram:lim ←−− H n (cid:0) X λ ; G ′ (cid:1) lim ←−− Hom (cid:16) ¯ H nN (cid:0) X λ ; Z (cid:1) ; G ′ (cid:17) . H n (cid:0) N ( X ) ; G ′ (cid:1) Hom (cid:16) ¯ H nN (cid:0) N ( X ) ; Z (cid:1) ; G (cid:17) . (cid:16) p N ( X ) λ (cid:17) ∗ Hom (cid:18)(cid:16) p N ( X ) λ (cid:17) ∗ ; G ′ (cid:19) (96)On the other hand, lim ←−− Hom (cid:16) ¯ H nN (cid:0) X λ ; Z (cid:1) ; G ′ (cid:17) ≃ Hom (cid:18) lim −−→ ¯ H nN (cid:0) X λ ; Z (cid:1) ; G ′ (cid:19) and by (92) and (96) the mapping (cid:16) p N ( X ) λ (cid:17) ∗ is an isomorphism: lim ←−− H n (cid:16) X λ ; G ′ (cid:17) ≃ H n ( N ( X ); G ′ ) . (97)For each injective abelian group G ′ we have H ∗ ( X ; G ′ ) ≃ lim ←−− H ∗ ( X λ ; G ′ ) . (98)Therefore, for each injective abelian group G ′ , the map p : X → N ( X ) induces an isomorphism: p ∗ : H ∗ ( X ; G ′ ) → H ∗ ( N ( X ); G ′ ) . (99)Consider the following diagram induced by p : X → N ( X ) and an injective resolution 0 → G → G ′ → G ′′ → ffi cient group G : . . . H n + ( X ; G ′ ) H n + ( X ; G ′′ ) H n ( X ; G ) H n ( X ; G ′ ) H n ( X ; G ′′ ) . . .. . .. . . H n + ( N ( X ); G ′ ) H n + ( N ( X ); G ′′ ) H n ( N ( X ); G ) H n ( N ( X ); G ′ ) H n ( N ( X ); G ′′ ) . . . . ≃ ≃ p ∗ ≃ ≃ (100)14herefore, by commutative diagram (100) the homomorphism p ∗ : H ∗ ( X ; G ) → H ∗ ( N ( X ); G ) (101)is an isomorphism for any coe ffi cient abelian group G . Our aim is to show that the homology group H ∗ ( N ( X ); G ) doesnot depend on the choice of a homology H ∗ ( − , − ; G ).Consider the filtration of the space N ( X ): N ( X ) ⊂ N ( X ) ⊂ · · · ⊂ N n ( X ) ⊂ . . . . (102)Hence, N n ( X ) = lim ←−− n X n λ , p λλ ′ , λ ∈ Λ o , (103)where X n λ is the n -dimensional skeleton of X λ . Note that, as above, we can show that there is an isomorphism as well:¯ H ∗ N ( N n ( X ); G ) ≃ lim −−→ ¯ H ∗ N ( X n λ ; G ) . (104)Let’s calculate some homology groups of filtrations: (a) H p (cid:0) N n ( X ) ; G (cid:1) = , ∀ p > n . By the Universal Coe ffi cient Formula we have0 Ext (cid:16) ¯ H p + N (cid:0) N n ( X ) ; Z (cid:1) ; G (cid:17) H p (cid:0) N n ( X ) ; G (cid:1) Hom (cid:18) ¯ H pN (cid:16) N n (cid:0) X ; Z (cid:1) ; G (cid:17)(cid:19) . (105)By (104) we obtain that0 Ext (cid:18) lim −−→ ¯ H p + N (cid:16) X n λ ; Z (cid:17) ; G (cid:19) H p (cid:0) N n ( X ) ; G (cid:1) Hom (cid:18) lim −−→ ¯ H pN (cid:16) X n λ ; Z (cid:17) ; G (cid:19) . (106)On the other hand, for polyhedron the Alexander-Spanier and singular cohomology theories are isomorphic [Sp]and so, if n F n α o is the direct system of compact subspaces of X n λ , then by Theorem 3.3 [Ber-Mdz ], there is a shortexact sequence: 0 lim ←−− ¯ H p − N (cid:16) F n α ; Z (cid:17) ¯ H pN (cid:16) X n λ ; Z (cid:17) lim ←−− ¯ H pN (cid:16) F n α ; Z (cid:17) . (107) F n α is a compact polyhedron and so, there is a short exact sequence:0 Ext (cid:16) H p − ( F n α ; Z ); G (cid:17) ¯ H pN (cid:16) F n α ; G (cid:17) Hom (cid:18) H p (cid:16) F n α ; Z (cid:17) ; G (cid:19) . (108)Note that H p ( F n α ; Z ) =
0, if p > n and H n ( F n α ; Z ) is a free abelian group. Therefore, by (108) we have:¯ H pN (cid:16) F n α ; G (cid:17) = , ∀ p ≥ n + . (109)Therefore, by (107), (108) and (109) we have H pN (cid:16) X n λ ; G (cid:17) = , ∀ p ≥ n + . (110)If p = n + , then by (107) and (109) there is an isomorphism¯ H n + N (cid:16) X n λ ; Z (cid:17) = lim −−→ ¯ H nN (cid:16) F n α ; Z (cid:17) . (111)On the other hand, by (108) we have the following long exact sequence:0 lim ←−− Ext (cid:18) H n − (cid:16) F n α ; Z (cid:17) ; G (cid:19) lim ←−− ¯ H nN (cid:16) F n α ; G (cid:17) lim ←−− Hom (cid:18) H n (cid:16) F n α ; Z (cid:17) ; G (cid:19) lim ←−− Ext (cid:18) H n − (cid:16) F n α ; Z (cid:17) ; G (cid:19) lim ←−− ¯ H nN (cid:16) F n α ; G (cid:17) lim ←−− Hom (cid:18) H n (cid:16) F n α ; Z (cid:17) ; G (cid:19) . . . . (112)15he group H n ( F n α ; Z ) is a finitely generated and so, by Corollary 1.5 of [H-M] lim ←−− Hom (cid:18) H n (cid:16) F n α ; Z (cid:17) ; G (cid:19) =
0. ByProposition1.2 and Corollary 1.5 of [H-M] we havelim ←−− Ext (cid:18) H n − (cid:16) F n α ; Z (cid:17) ; G (cid:19) ≃ lim ←−− Hom (cid:18) H n − (cid:16) F n α ; Z (cid:17) ; G (cid:19) = . (113)Therefore, (113), (112), (111), (109) and (106) we obtain H n + (cid:0) N n ( X ); G (cid:1) ≃ . (114) (b) H p (cid:16) N n ( X ) , N n − ( X ); G (cid:17) = , ∀ p , n − , n . Consider the long exact homological sequence of the pair (cid:16) N n ( X ) , N n − ( X ) (cid:17) : · · · → H p ( N n − ( X ); G ) → H p ( N n ( X ); G ) → H p ( N n ( X ) , N n − ( X ); G ) → H p − ( N n − ( X ); G ) → . . . . (115)Therefore, for p > n by the property (a), H p ( N n ( X ); G ) = H p ( N n ( X ) , N n − ( X ); G ) = , ∀ p > n . (116)For the case p < n −
1, we use the same method as in the case (a). In particular, consider the Universal Coe ffi cientFormula for the pair (cid:16) N n ( X ) , N n − ( X ) (cid:17) : (cid:18) ¯ H p + N (cid:16) N n ( X ) , N n − ( X ) , Z (cid:17) ; G (cid:19) H p (cid:16) N n ( X ) , N n − ( X ) ; G (cid:17) Hom (cid:18) ¯ H pN (cid:16) N n ( X ) , N n − ( X ) ; Z (cid:17) ; G (cid:19) . (117)Using the isomorphism (84) the sequence (117) will become:0 Ext (cid:18) lim −−→ ¯ H p + N (cid:16) X n λ , X n − λ ; Z (cid:17) ; G (cid:19) H p (cid:16) N n ( X ) , N n − ( X ) ; G (cid:17) Hom (cid:18) lim −−→ ¯ H pN (cid:16) X n λ , X n − λ ; Z (cid:17) ; G (cid:19) . (118)Consequently, if (cid:26)(cid:16) F n α , E n − α (cid:17)(cid:27) is the direct system of pairs of compact subspaces of X n λ and X n − λ , then we have:0 lim ←−− ¯ H p − N (cid:16) F n α , E n − α ; Z (cid:17) ¯ H pN (cid:16) X n λ , X n − λ ; Z (cid:17) lim ←−− ¯ H pN (cid:16) F n α , E n − α ; Z (cid:17) , (119)and 0 Ext (cid:18) H p − (cid:16) F n α , E n − α ; Z (cid:17) ; G (cid:19) ¯ H pN (cid:16) F n α , E n − α ; G (cid:17) Hom (cid:18) H p (cid:16) F n α , E n − α ; Z (cid:17) ; G (cid:19) . (120)By (120), if p < n −
1, then ¯ H pN (cid:16) F n α , E n − α ; G (cid:17) = H pN (cid:16) X n λ , X n − λ ; G (cid:17) =
0. Therefore, by (118) thegiven property is proven. (c) H n − (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) =
0, for an injective abelian group G ′ . By (117) and (118), in this case we have: H n − (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) ≃ Hom (cid:18) ¯ H nN (cid:16) N n ( X ) , N n − ( X ); Z (cid:17) ; G ′ (cid:19) ≃ Hom (cid:18) lim −−→ ¯ H n − N (cid:16) X n λ , X n − λ ; Z (cid:17) ; G ′ (cid:19) . (121)By (120) ¯ H pN (cid:16) F n α , E n − α ; Z (cid:17) = , ∀ p ≤ n − H n − N (cid:16) X n λ , X n − λ ; Z (cid:17) = . Therefore, by (121) we have H n − (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) = . (122)16et C ∗ (cid:0) N ( X ); G ′ (cid:1) = n C n (cid:0) N ( X ); G ′ (cid:1) , ∂ ′ n o be the chain complex, where C n (cid:0) N ( X ); G ′ (cid:1) = H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) for an injective abelian group G ′ , and ∂ ′ n : C n (cid:0) N ( X ); G ′ (cid:1) → C n − (cid:0) N ( X ); G ′ (cid:1) be the border mapping ∂ ′ n : H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) → H n − (cid:16) N n − ( X ) , N n − ( X ); G ′ (cid:17) . Note that by Universal Coe ffi cient Formula for an injective abelian group G ′ we have H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) ≃ Hom (cid:18) lim −−→ ¯ H nN (cid:16) X n λ , X n − λ ; Z (cid:17) ; G ′ (cid:19) (123)and therefore, the chain complex C ∗ (cid:0) N ( X ); G ′ (cid:1) = n C n (cid:0) N ( X ); G ′ (cid:1) , ∂ n o is independent of choice of a homology theory.By the exact homological sequences for the triples (cid:16) N n + ( X ) , N n ( X ) , N n − ( X ) (cid:17) , (cid:16) N n ( X ) , N n − ( X ) , N n − ( X ) (cid:17) and (cid:16) N n + ( X ) , N n ( X ) , N n − ( X ) (cid:17) for an injective abelian group G ′ , we have H n + (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n + (cid:16) N n + ( X ) , N n ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) ,∂ ′ n + (124) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n − (cid:16) N n − ( X ) , N n − ( X ); G ′ (cid:17) H n − (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) , j ′∗ ∂ ′ n (125) . . . H n + (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n + (cid:16) N n + ( X ) , N n ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) .∂ ′ q ′∗ (126)Let consider the corresponding commutative diagram: H n + (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n + (cid:16) N n + ( X ) , N n ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n − (cid:16) N n − ( X ) , N n − ( X ); G ′ (cid:17) H n − (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n + (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) ... ∂ ′ n + j ′∗ ∂ ′ n ∂ ′ q ′∗ (127)In this case we have H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) ≃ H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) / Im ∂ ′ ≃ Im j ′∗ / Im ( j ′∗ ◦ ∂ ′ ) ≃ Ker ∂ ′ n / Im ∂ ′ n + . (128)On the other, hand Ker ∂ ′ n / Im ∂ ′ n + ≃ H n (cid:16) C ∗ (cid:0) N ( X ); G ′ (cid:1)(cid:17) and so, we have H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) ≃ H n (cid:18) C ∗ (cid:16) N ( X ); G ′ (cid:17)(cid:19) . (129)17onsider the long homological exact sequence of the triple (cid:16) N n + ( X ) , N n + ( X ) , N n − ( X ) (cid:17) : . . . H n + (cid:16) N n + ( X ) , N n + ( X ); G (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) .∂ i ∗ (130)By the property (c), for an injective abelian group G ′ , we have H n + (cid:16) N n + ( X ) , N n + ( X ); G ′ (cid:17) = H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) ≃ H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) . (131)Therefore, by (129) and (131) we have H n (cid:18) C ∗ (cid:16) N ( X ); G ′ (cid:17)(cid:19) ≃ H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) . (132)Consider the long homological exact sequence of the triple (cid:16) N n + ( X ) , N n − ( X ) , N n − ( X ) (cid:17) : . . . H n + (cid:16) N n − ( X ) , N n − ( X ); G (cid:17) H n + (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) H n + (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n − ( X ) , N n − ( X ); G (cid:17) . . . . (133)By the property (a) - H p (cid:0) N n ( X ); G (cid:1) = , ∀ p > n and by the long homological sequences of the pair (cid:16) N n − ( X ) , N n − ( X ) (cid:17) we have H n + (cid:16) N n − ( X ) , N n − ( X ); G (cid:17) = H n (cid:16) N n − ( X ) , N n − ( X ); G (cid:17) = . Therefore, H n + (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) ≃ H n + (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) . (134)By (129) and (134) we have H n + (cid:18) C ∗ (cid:16) N ( X ); G ′ (cid:17)(cid:19) ≃ H n + (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) . (135)Consider the long exact homological sequence of the pair (cid:16) N n ( X ) , N n − ( X ) (cid:17) for the relosution 0 → G α −→ G ′ β −→ G ′′ → G : . . . H n + (cid:16) N n ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′′ (cid:17) H n − (cid:16) N n ( X ) , N n − ( X ); G (cid:17) H n − (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) . . . .α n β n β n (136)By the property (a) H p (cid:0) N n ( X ); G (cid:1) = , ∀ p > n and we have H n + (cid:16) N n ( X ) , N n − ( X ); G (cid:17) = . By the property (c), foreach injective abelian group H n − (cid:16) N n − ( X ) , N n − ( X ); G (cid:17) = . Therefore, by (136) we have the following four-termexact sequence: H n (cid:16) N n ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n ( X ) , N n − ( X ); G ′′ (cid:17) H n − (cid:16) N n ( X ) , N n − ( X ); G (cid:17) ,α n β n (137)where β n : H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) → H n (cid:16) N n ( X ) , N n − ( X ); G ′′ (cid:17) is induced by β : G ′ → G ′′ . Consequently, weobtain the chain map β = (cid:8) β n (cid:9) : C ∗ (cid:16) N ( X ); G ′ (cid:17) → C ∗ (cid:16) N ( X ); G ′′ (cid:17) . (138)Let C ∗ (cid:0) β (cid:1) = n C n (cid:0) β (cid:1) , ˜ ∂ n o be the chain cone of the chain map β , i.e., C n (cid:0) β (cid:1) ≃ C n (cid:16) N ( X ); G ′ (cid:17) ⊕ C n + (cid:16) N ( X ); G ′′ (cid:17) , (139)˜ ∂ n ( c ′ n , c ′′ n + ) = ( ∂ ′ n ( c ′ n ) , β n ( c ′ n ) − ∂ ′′ n + ( c ′′ n + )) , ∀ ( c ′ n , c ′′ n + ) ∈ C n ( β ) . (140)18ur aim is to show that H n (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) ≃ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) . (141)For this aim, consider the following short exact sequence:0 → C ∗ + (cid:16) N ( X ); G ′′ (cid:17) σ −→ C ∗ (cid:0) β (cid:1) τ −→ C ∗ (cid:16) N ( X ); G ′ (cid:17) → , (142)where σ and τ are defined in the following way: σ (cid:16) c ′′ n + (cid:17) = (cid:16) , c ′′ n + (cid:17) , ∀ c ′′ n + ∈ C n + (cid:16) N ( X ); G ′′ (cid:17) , (143) τ (cid:16) c ′ n , c ′′ n + (cid:17) = c ′ n , ∀ (cid:16) c ′ n , c ′′ n + (cid:17) ∈ C n (cid:0) β (cid:1) . (144)Hence, the sequence (142) induces the long exact homological sequence: . . . H n + (cid:0) N ( X ); G ′′ (cid:1) H n (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) H n (cid:0) N ( X ); G ′ (cid:1) H n (cid:0) N ( X ); G ′′ (cid:1) . . . .σ ∗ τ ∗ E (145)On the other hand, we have the homology exact sequence of pair (cid:16) N n + ( X ) , N n − ( X ) (cid:17) with respect to the short exactsequence 0 → G α −→ G ′ β −→ G ′′ → . . . H n + (cid:16) N n + ( X ) , N n − ( X ); G ′′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′′ (cid:17) . . . . d ∗ α ∗ β ∗ (146)By (132) and (135), to obtain the isomorphism (141) it is su ffi cient to define a homomorphism ϕ : H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) → H n (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) such that the following diagram . . . H n + (cid:16) N n + ( X ) , N n − ( X ); G ′′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′′ (cid:17) . . . d ∗ α ∗ β ∗ . . . H n + (cid:0) N ( X ); G ′′ (cid:1) H n (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) H n (cid:0) N ( X ); G ′ (cid:1) H n (cid:0) N ( X ); G ′′ (cid:1) . . .σ ∗ τ ∗ E ≃ ϕ ≃ ≃ (147)is commutative.Consider two diagrams of type (127) corresponding to the coe ffi cient groups G ′ and G ′′ and homomorphimsbetween them induced by β : G ′ → G ′′ . Consequently, homomorphisms and elements will be labeled with ′ and ′′ to distinguish them. Let ¯ h n ∈ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) be an element. By (130) there is an element h n ∈ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) such that i ∗ ( h n ) = ¯ h n . Then, α ∗ ( h n ) ∈ H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) and so by diagram (127)there is an element h ′ n ∈ H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) such that q ′∗ ( h ′ n ) = α ∗ ( h n ). Let c ′ n = j ′∗ ( h ′ n ) ∈ H n (cid:16) N n ( X ) , N n − ( X ); G ′ (cid:17) = C n (cid:0) N ( X ); G ′ (cid:1) and h ′′ n = β ∗ ( h ′ n ) ∈ H n (cid:16) N n ( X ) , N n − ( X ); G ′′ (cid:17) . In this case, q ′′∗ ( h ′′ n ) = q ′′∗ β ∗ ( h ′ n ) = β ∗ q ′∗ ( h ′ n ) = β ∗ α ∗ ( h n ) = . Therefore, h ′′ n ∈ Kerq ′′∗ = Im ∂ ′′ and so, there exists c ′′ n + ∈ H n + (cid:16) N n + ( X ) , N n ( X ); G ′′ (cid:17) = C n + (cid:0) N ( X ); G ′′ (cid:1) such that ∂ ′′ ( c ′′ n + ) = h ′′ n . In this case,˜ ∂ (cid:16) c ′ n , c ′′ n + (cid:17) = (cid:18) ∂ ′ n (cid:16) c ′ n (cid:17) , β ∗ (cid:16) c ′ n (cid:17) − ∂ ′′ n + (cid:16) c ′′ n + (cid:17)(cid:19) = ∂ ′ n (cid:18) j ′∗ (cid:16) h ′ n (cid:17)(cid:19) , β ∗ (cid:18) j ′∗ (cid:16) h ′ n (cid:17)(cid:19) − j ′′∗ (cid:18) ∂ ′′ (cid:16) c ′′ n + (cid:17)(cid:19)! == , j ′′∗ (cid:18) β ∗ (cid:16) h ′ n (cid:17)(cid:19) − j ′′∗ (cid:18) ∂ ′′ (cid:16) c ′′ n + (cid:17)(cid:19)! = , j ′′∗ (cid:18) β ∗ (cid:16) h ′ n (cid:17) − ∂ ′′ (cid:16) c ′′ n + (cid:17)(cid:19)! = , j ′′∗ (cid:18) β ∗ (cid:16) h ′ n (cid:17) − h ′′ n (cid:19)! = (cid:16) , j ′′∗ (0) (cid:17) = . (148)Therefore ( c ′ n , c ′′ n + ) ∈ C ∗ (cid:0) β (cid:1) is a cycle. Our aim is to show that the pair ( c ′ n , c ′′ n + ) ∈ C ∗ (cid:0) β (cid:1) is independent of chose ofrepresentatives h n ∈ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) . Indeed, let ˜ h n ∈ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) be another representative19f ( i ∗ ) − (¯ h n ) , then by (130) there exists h n + ∈ H n + (cid:16) N n + ( X ) , N n + ( X ); G (cid:17) such that ∂ ( h n + ) = h n − ˜ h n . Consider, thefollowing commutative diagram: . . . H n + (cid:16) N n + ( X ) , N n + ( X ); G (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) . . . H n + (cid:16) N n + ( X ) , N n + ( X ); G ′ (cid:17) H n + (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) .∂ i ∗ ∂ ′ i ′∗ α ∗ α ∗ α ∗ (149)By the property (c), we have H n + (cid:16) N n + ( X ) , N n + ( X ); G ′ (cid:17) = α ∗ ◦ ∂ = ∂ ′ ◦ α ∗ =
0. Hence, α ∗ ( h n ) − α ∗ (˜ h n ) = α ∗ (cid:16) h n − ˜ h n (cid:17) = α ∗ (cid:0) ∂ ( h n + ) (cid:1) = ∂ ′ (cid:0) α ∗ ( h n + ) (cid:1) = (cid:16) c ′ n . c ′′ n + (cid:17) . Therefore, we can define ϕ in the following way: ϕ ( h n ) = (cid:16) c ′ n , c ′′ n + (cid:17) + Im ˜ ∂ n + = (cid:20)(cid:16) c ′ n , c ′′ n + (cid:17)(cid:21) , ∀ h n ∈ H n (cid:16) N n + ( X ) , N n − ( X ); G ′ (cid:17) . (151)Consequently, by commutativity of the diagram (147), we have that for each coe ffi cient group G there is an isomor-phism: H n (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) ≃ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) . (152)Using the long exact homological sequence of the pair (cid:16) N n + ( X ) , N n − ( X ) (cid:17) : · · · → H n (cid:16) N n − ( X ); G (cid:17) → H n (cid:16) N n + ( X ); G (cid:17) → H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) → H n − (cid:16) N n − ( X ); G (cid:17) → . . . (153)and the property (a) - H n (cid:16) N n − ( X ); G (cid:17) = H n − (cid:16) N n − ( X ); G (cid:17) = H n (cid:16) N n + ( X ); G (cid:17) ≃ H n (cid:16) N n + ( X ) , N n − ( X ); G (cid:17) . (154)Note that by property (a) - H p (cid:0) N n ( X ); G (cid:1) = , ∀ p > n and the long homological sequence of pair (cid:16) N n + r + ( X ) , N n + r ( X ) (cid:17) , ∀ r ≥ H n (cid:16) N n + r ( X ); G (cid:17) ≃ H n (cid:16) N n + r + ( X ); G (cid:17) , ∀ r ≥ . (155)On the other hand, using the same method, we can prove that H n (cid:16) N ( X ) , N n + ( X ); G (cid:17) = (cid:16) N ( X ) , N n + ( X ) (cid:17) we have H n (cid:16) N n + ( X ); G (cid:17) ≃ H n (cid:0) N ( X ); G (cid:1) . (156)By (156), (155), (154) and (152) we obtain that H n (cid:16) C ∗ (cid:0) β (cid:1)(cid:17) ≃ H n (cid:0) N ( X ); G (cid:1) . (157)For a paracompact space X the limit space N ( X ) is homotopy equivalent to the space X (see Lemma 2.1 in [San]).Therefore, the isomorphism (101) is fulfilled without (CIG) property. Consequently, if we follow the proof of Theorem7 we obtain the following: Corollary 10.
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