On the Universal Coefficient Formula and Derivative \varprojlim ^{(i)} Functor
aa r X i v : . [ m a t h . A T ] F e b On the Universal Coefficient Formula andDerivative lim ←− ( i ) Functor
Anzor BeridzeSchool of MathematicsKutaisi International UniversityYouth Avenue, 5th Lane, Kutaisi, 4600 GeorgiaE-mail: [email protected] MdzinarishviliDepartment of MathematicsFaculty of Informatics and Control SystemsGeorgian Technical University77, Kostava St., Tbilisi, GeorgiaE-mail: [email protected]
Abstract
From the beginning of 1960, there were many approaches to defineexact homology theories using the methods of homological algebra(using an injective resolution) [Bor-Mo], [Mas ], [Mas ], [Kuz], [Skl].These approaches gave us the unique homology theory on the cate-gory of compact Hausdorff spaces [Kuz], [Skl]. Our aim is to developa method of homological algebra which gives opportunity to defineon the category of general topological spaces a unique exact homol-ogy theory, generated by the given cochain complex. If H ∗ is thecohomology of the cochain complex C ∗ = Hom( C ∗ ; G ) , then the co-homology H ∗ is said to be generated by chain complex C ∗ . If a chaincomplex C ∗ is free, then there is a Universal Coefficient Formula of acohomology theory [Eil-St], [Mas ], [Sp]. In the paper [Mdz ], usingthis formula and derivatives of inverse limit, a long exact sequenceis written, which shows a relation of a cohomology of direct limit Mathematics Subject Classification : 55N10
Key words and phrases : Universal Coefficient Formula; inverse limit; derivative limit;tautness of homology. A. Beridze and L. Mdzinarishvili of chain complexes and inverse limit of cohomology groups of corre-sponding cochain complexes. The result for non-free chain complexesis extended in the paper [Mdz-Sp]. On the other hand, to define anexact homology (Steenrod) theory [St], [Ed-Ha ], [Ed-Ha ], it is muchconvenient to obtain a chain complex from a cochain complex [Mas ],[Mas ], [Mil], [St]. In this case, a homology H ∗ is generated by cochaincomplex C ∗ and if the corresponding cochain complex is free, thenthere is an analogous Universal Coefficient Formula of a cohomologytheory (not standard Universal Coefficient Formula of a homologytheory) [Ber], [Ber-Mdz ], [Ber-Mdz ], [Bor-Mo], [Mas ], [Mil], [Skl].In this paper the result is extended for non-free cochain complexesand using of it, the relation of homology groups of the direct limit ofcochain complexes and the inverse limit of homology groups of cor-responding chain complexes is studied. As a corollary, the tautnessproperty of a homology theory is obtained. Moreover, for the de-fined exact homology theory the continuous property (see Definition1 [Mdz ]) is obtained on the category of compact pairs. Let C ∗ be a chain complex and → G α −→ G ′ β −→ G ′′ → be an injectiveresolution of a R -module G over a principal ideal domain R . Let β :Hom( C ∗ ; G ′ ) → Hom( C ∗ ; G ′′ ) be the cochain map induced by β : G ′ → G ′′ and C ∗ ( β ) = { C n ( β ) , δ } be the cone of the cochain map β , i.e., (1.1) C n ( β ) ≃ Hom( C n ; G ′ ) ⊕ Hom( C n − ; G ′′ ) , (1.2) δ ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ ∂, β ◦ ϕ ′ − ϕ ′′ ◦ ∂ ) , ∀ ( ϕ ′ , ϕ ′′ ) ∈ C n ( β ) . It is natural that the cochain complex C ∗ ( β ) = Hom( C ∗ ; β ) is definedby the chain complex C ∗ and R -module G . Consequently, let ¯ H ∗ ( C ∗ ; G ) bea cohomology of the cochain complex C ∗ ( β ) . In the papers [Mdz ] (in thecase of free chain complexes) and [Mdz-Sp] (in the general case) it is shownthat for each direct system C ∗ = { C γ ∗ } of chain complexes there is a naturalexact sequence:(1.3) . . . lim ←− (3) ¯ H n − γ lim ←− (1) ¯ H n − γ ¯ H n (cid:16) lim −→ C γ ∗ ; G (cid:17) lim ←− ¯ H nγ lim ←− (2) ¯ H n − γ . . . where ¯ H ∗ γ = ¯ H ∗ ( C γ ∗ ; G ) = H ∗ ( C ∗ γ ( β )) . Note that, using the methods de-veloped in [Mdz ] and [Mdz ], it is possible to show that if for a cohomology H ∗ there exists a type (2.10) natural sequence, then there is an isomophism ¯ H n (lim −→ C γ ∗ ; G ) ≃ H n (lim −→ C γ ∗ ; G ) . Therefore, it uniquely defines a cohomol-ogy ¯ H ∗ generated by the chain complex C ∗ . On the other hand, our aim isto develop tools which uniquely define a homology theory generated by thegiven cochain complex. Therefore, we will consider the dual case.Let C ∗ be a cochain complex and β : Hom( C ∗ ; G ′ ) → Hom( C ∗ ; G ′′ ) be the chain map induced by β : G ′ → G ′′ . Consider the cone C ∗ ( β ) = { C n ( β ) , ∂ } = { Hom( C ∗ , β ) , ∂ } of the chain map β , i.e., (1.4) C n ( β ) ≃ Hom( C n ; G ′ ) ⊕ Hom( C n +1 ; G ′′ ) , (1.5) ∂ ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ δ, β ◦ ϕ ′ − ϕ ′′ ◦ δ ) , ∀ ( ϕ ′ , ϕ ′′ ) ∈ C n ( β ) . Consequently, the homology group ¯ H n is denoted by ¯ H n = ¯ H n ( C ∗ ; G )= H n ( C ∗ ( β )) and is called a homology with coefficient in G generatedby the cochain complex C ∗ . Note that if f : C ∗ −→ C ′∗ is a cochain map,then it induces the chain map ¯ f : C ′∗ ( β ) −→ C ∗ ( β ) . In particular, foreach n ∈ Z the homomorphism ¯ f n : C ′ n ( β ) −→ C n ( β ) is defined by theformula ¯ f n ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ f n , ϕ ◦ f n +1 ) . Consequently, it induces a homomor-phism of homology groups ¯ f : ¯ H n ( C ′∗ ; G ) −→ ¯ H n ( C ∗ ; G ) . Therefore, ¯ H n isa naturally defined functor.Let C ∗ = { C ∗ γ } be a direct system of cochain complexes. Consider thecorresponding inverse systems C ∗ = { C γ ∗ ( β ) } and H ∗ = { ¯ H ∗ ( C ∗ γ ; G ) } . Inthis paper we have shown that there is a natural exact sequence: (1.6) . . . lim ←− (3) ¯ H γn +2 lim ←− (1) ¯ H γn +1 ¯ H n (cid:0) lim −→ C ∗ γ ; G (cid:1) lim ←− ¯ H γn lim ←− (2) ¯ H γn +1 . . . , where ¯ H γ ∗ = ¯ H ∗ ( C ∗ γ ; G ) = H ∗ ( C γ ∗ ( β )) . On the other hand, to obtain se-quence (1.6), we have shown that for each cochain complex (it is not nec-essary to be free) C ∗ and a R -module G over a principal ideal domain R ,there exists a short exact sequence (Universal Coefficient Formula): (1.7) −→ Ext( H n +1 ( C ∗ ); G ) −→ ¯ H n ( C ∗ ; G ) −→ Hom( H n ( C ∗ ); G ) −→ . At the end we have formulated and studied the tautness property for ahomology theory ¯ H ∗ generated by the Alexander-Spanier cochains ¯ C ∗ onthe category of paracompact Hausdorff spaces. In this section we will prove the Universal Coefficient Formula for a homol-ogy theory ¯ H ∗ generated by the given cochain complex C ∗ . Theorem 2.1 (Universal Coefficient Formula) . For each cochain complex C ∗ and R -module G over a fixed principal ideal domain R , there exists a A. Beridze and L. Mdzinarishvili short exact sequence: (2.1) −→ Ext( H n +1 ( C ∗ ); G ) ¯ χ −→ ¯ H n ( C ∗ ; G ) ¯ ξ −→ Hom( H n ( C ∗ ); G ) −→ . Proof.
We will define a homomorphism ξ : ¯ Z n −→ Hom( H n ( C ∗ ); G ) , whichinduces an epimorphism ¯ ξ : ¯ H n ( C ∗ ; G ) −→ Hom( H n ( C ∗ ); G ) . On the otherhand, we will define a homomorphism χ : Hom( H n +1 ( C ∗ ); G ′′ ) −→ ¯ H n ( C ∗ ; G ) such that χ induces a monomorphism ¯ χ : Ext( H n +1 ( C ∗ ); G ) −→ ¯ H n ( C ∗ ; G ) and the short sequence (2.1) is exact. a. There is a homomorphism ξ : ¯ Z n → Hom( H n ( C ∗ ); G ) . Let ( ϕ ′ , ϕ ′′ ) ∈ ¯ Z n be a cycle, i.e., ϕ ′ : C n → G ′ and ϕ ′′ : C n +1 → G ′′ arehomomorphisms such that ∂ ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ δ, β ◦ ϕ ′ − ϕ ′′ ◦ δ ) = (0 ,
0) = 0 and therefore, the following diagram is commutative:(2.2) C n − C n C n +1 G ′ G ′′ δ δϕ ′ ϕ ′′ β where is the zero map. Consider the groups of coboundaries B n and cocy-cles Z n . Let i : B n → Z n and j : Z n → C n be natural monomorphisms and δ ′ : C n − → B n be an epimophism induced by δ : C n − → C n . Therefore,we have the following sequence: (2.3) C n − δ ′ −→ B n i −→ Z n j −→ C n δ −→ C n +1 , where j ◦ i ◦ δ ′ = δ and consequently δ ◦ j = 0 .Since ( ϕ ′ , ϕ ′′ ) ∈ ¯ Z n , we have the following commutative diagram:(2.4) Z n C n C n +1 G G ′ G ′′ . j δϕ ′ ϕ ′′ α β Hence, ϕ ′′ ◦ δ ◦ j = β ◦ ϕ ′ ◦ j , and by the equality δ ◦ j = 0 , we obtain that β ◦ ϕ ′ ◦ j = 0 . So, Im ( ϕ ′ ◦ j ) ⊂ Kerβ = Imα . Therefore, there is a uniquelydefined map ϕ : Z n → G such that ϕ ′ ◦ j = α ◦ ϕ (see the diagram (2.5).(2.5) Z n C n C n +1 G G ′ G ′′ . j δϕ ϕ ′ ϕ ′′ α β By the commutative diagram (2.2), we have ϕ ′ ◦ δ = ϕ ′ ◦ j ◦ i ◦ δ ′ = 0 .Hence, α ◦ ϕ ◦ i ◦ δ ′ = ϕ ′ ◦ j ◦ i ◦ δ ′ = ϕ ′ ◦ δ = 0 (see the diagram (2.6)). α is a monomorphism and so ϕ ◦ i ◦ δ ′ = 0 . On the other hand, δ ′ is an epi-morphism. Consequently, we have ϕ ◦ i = 0 . Therefore, the homomorphism ϕ : Z n → G induces a homomorphism ¯ ϕ : H n ( C ∗ ) → G which belongs to Hom( H n ( C ∗ ); G ) . Hence, the following diagram is commutative:(2.6) Z n B n C n − C n C n +1 G H n G ′ G ′′ , δ ′ i j δϕ ϕ ′ ϕ ′′ α βp ¯ ϕ where H n = H n ( C ∗ ) . Let ξ : ¯ Z n → Hom( H n ; G ) be the homomorphismdefined by (2.7) ξ ( ϕ ′ , ϕ ′′ ) = ¯ ϕ, ∀ ( ϕ ′ , ϕ ′′ ) ∈ ¯ Z n . b. ξ : ¯ Z n → Hom( H n ( C ∗ ); G ) is an epimorphism. Let ¯ ϕ ∈ Hom( H n ( C ∗ ); G ) be a homomorphism and ϕ = ¯ ϕ ◦ p : Z n → G isthe composition, where p : Z n → H n ( C ∗ ) is a projection. Let ϕ ′ : C n → G ′ be an extension of α ◦ ϕ : Z n → G ′ . In this case ϕ ′ ◦ j = α ◦ ϕ and so β ◦ ϕ ′ ◦ j = β ◦ α ◦ ϕ = 0 . Therefore, β ◦ ϕ ′ : C n → G ′′ vanishes on the subgroup Z n and so it induces a homomorphism ˜ ϕ ′′ : C n /Z n ≃ B n +1 → G ′′ , whichcan be extended to a homomophism ϕ ′′ : C n +1 → G ′′ . Since ˜ ϕ ′′ ◦ δ ′ = β ◦ ϕ ′ ,there is ∂ ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ δ, β ◦ ϕ ′ − ϕ ′′ ◦ δ ) = ( ϕ ′ ◦ j ◦ i ◦ δ ′ , β ◦ ϕ ′ − ϕ ′′ ◦ j ◦ i ◦ δ ′ ) =( α ◦ ϕ ◦ i ◦ δ ′ , β ◦ ϕ ′ − ˜ ϕ ′′ ◦ δ ′ ) = ( α ◦ ¯ ϕ ◦ p ◦ i ◦ δ ′ , β ◦ ϕ ′ − β ◦ ϕ ′ ) = (0 ,
0) = 0 .Hence, ( ϕ ′ , ϕ ′′ ) ∈ ¯ Z n and ¯ ξ ( ϕ ′ , ϕ ′′ ) = ¯ ϕ (see the diagram (2.8)).(2.8) B n C n − Z n C n C n /Z n ≃ B n +1 C n +1 H n G G ′ G ′′ . ¯ ϕpδ ′ i j δ ′ j ◦ iα βϕ ϕ ′ ˜ ϕ ′′ ϕ ′′ c. ξ : ¯ Z n → Hom( H n ( C ∗ ); G ) induces a homomorphism ¯ ξ : ¯ H n ( C ∗ ; G ) → Hom( H n ( C ∗ ); G ) . We have to show that the homomorphism ξ : ¯ Z n → Hom( H n ( C ∗ ); G ) vanishes on the subgroup ¯ B n . Indeed, let ( ψ ′ , ψ ′′ ) ∈ C n +1 ( β ) be an element. For ∂ ( ψ ′ , ψ ′′ ) = ( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ δ ) = ( ϕ ′ , ϕ ′′ ) ∈ ¯ B n ⊂ ¯ Z n we have ϕ ′ ◦ j = 0 . Indeed, ϕ ′ ◦ j = ψ ′ ◦ δ ◦ j = 0 . Therefore, by theconstruction ξ , the homomorphism ϕ : Z n → G corresponding to the pair ( ϕ ′ , ϕ ′′ ) satisfies the equation α ◦ ϕ = ϕ ′ ◦ j = 0 and so ϕ = 0 , because α is A. Beridze and L. Mdzinarishvili a monomorphism. Since ϕ = ¯ ϕ ◦ p and p is an epimorphism, we have ¯ ϕ = 0 .Therefore, ξ∂ ( ψ ′ , ψ ′′ ) = ξ ( ϕ ′ , ϕ ′′ ) = ¯ ϕ = 0 (see the diagram (2.9)).(2.9) Z n C n C n +1 C n +2 G H n G ′ G ′′ . p ¯ ϕ j δ δψ ′ ψ ′′ ϕ ϕ ′ ϕ ′′ α β d. The kernel of ¯ ξ : ¯ H n ( C ∗ ; G ) → Hom( H n ( C ∗ ); G ) is Ext( H n +1 ( C ∗ ); G ) . If we apply the functor
Hom( H n +1 ( C ∗ ); − ) to the short exact sequence → G α −→ G ′ β −→ G ′′ → , then we obtain:(2.10) H n +1 ( C ∗ ); G ) Hom( H n +1 ( C ∗ ); G ′ )Hom( H n +1 ( C ∗ ); G ′′ ) Ext( H n +1 ( C ∗ ); G ) 0 .α ∗ β ∗ β ∗ Therefore, we have the following isomorphism: (2.11)
Ext( H n +1 ( C ∗ ); G ) ≃ Hom( H n +1 ( C ∗ ); G ′′ ) /Imβ ∗ . Our aim is to define such a homomorpism χ : Hom( H n +1 ( C ∗ ); G ′′ ) −→ ¯ H n ( C ∗ ; G ) that the following sequence is exact:(2.12) Hom( H n +1 ( C ∗ ); G ′ ) Hom( H n +1 ( C ∗ ); G ′′ )¯ H n ( C ∗ ; G ) Hom( H n ( C ∗ ); G ) 0 .β ∗ χχ ¯ ξ Indeed, in this case, it is clear that for the homomorphisms ¯ ξ , χ and β ∗ , wehave the following short exact sequences: (2.13) −→ Ker ¯ ξ −→ ¯ H n ( C ∗ ; G ) ¯ ξ −→ Hom( H n ( C ∗ ); G ) −→ , (2.14) −→ Kerχ −→ Hom( H n +1 ( C ∗ ); G ′′ ) χ −→ Imχ −→ , (2.15) −→ Kerβ ∗ −→ Hom( H n +1 ( C ∗ ); G ′ ) β ∗ −→ Imβ ∗ −→ . On the other hand, if we prove exactness of the sequence (2.12), then
Ker ¯ ξ ≃ Imχ and
Kerχ ≃ Imβ ∗ . Therefore, we have:(2.16) Ker ¯ ξ ≃ Imχ ≃ Hom( H n +1 ( C ∗ ); G ′′ ) /Kerχ ≃≃ Hom( H n +1 ( C ∗ ); G ′′ ) /Imβ ∗ ≃ Ext( H n +1 ( C ∗ ); G ) . To define χ , consider an element ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) . Let ϕ ′′ : C n +1 −→ G ′′ be an extension of the composition ¯ ϕ ′′ ◦ p : Z n +1 −→ G ′′ , where p : Z n +1 −→ H n +1 ( C ∗ ) is a natural projection. In this case, ϕ ′′ ◦ δ = ϕ ′′ ◦ j ◦ i ◦ δ ′ = ˜ ϕ ′′ ◦ p ◦ i ◦ δ ′ = 0 and so, if we take ϕ ′ = 0 , then the followingdiagram is commutative:(2.17) C n − C n B n +1 Z n +1 C n +1 G ′ H n +1 G ′′ . p ¯ ϕ ′′ δ δ ′ i jϕ ′ = 0 ϕ ′′ β Therefore, (0 , ϕ ′′ ) ∈ ¯ Z n and so, we can define χ in the following way:(2.18) χ ( ¯ ϕ ′′ ) = (0 , − ϕ ′′ ) + ¯ B n , ∀ ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) . Let check that χ is well defined. Consider two different extensions ϕ ′′ and ϕ ′′ of the map ¯ ϕ ′′ ◦ p : Z n +1 −→ G ′′ and show that (0 , − ϕ ′′ )+ ¯ B n = (0 , − ϕ ′′ )+ ¯ B n .For this, we have to show that (0 , ϕ ′′ − ϕ ′′ ) ∈ ¯ B n . Indeed, by the definition of ϕ ′′ and ϕ ′′ , it is clear that ( ϕ ′′ − ϕ ′′ ) ◦ j = ϕ ′′ ◦ j − ϕ ′′ ◦ j = ¯ ϕ ′′ ◦ p − ¯ ϕ ′′ ◦ p = 0 andso, ϕ ′′ − ϕ ′′ induces a homomorphism ψ : C n +1 /Z n +1 −→ G ′′ . On the otherhand, C n +1 /Z n +1 ≃ B n +2 and so, we have an extension ψ ′′ : C n +2 −→ G ′′ of ψ (see the diagran (2.19)).(2.19) Z n +1 C n +1 C n +1 /Z n +1 ≃ B n +2 C n +2 G ′′ j δ ′ j ◦ i ϕ ′ − ϕ ′′ ψ ψ ′′ In this case, it is easy to see that (2.20) ∂ (0 , ψ ′′ ) = (0 , − ψ ′′ ◦ δ ) = (0 , − ψ ′′ ◦ j ◦ i ◦ δ ′ ) = (0 , − ψ ◦ δ ′ ) = (0 , ϕ ′′ − ϕ ′′ ) . Therefore, it remains to show that
Imχ ≃ Ker ¯ ξ and Imβ ∗ = Kerχ . d . Imχ ≃ Ker ¯ ξ . Let ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) be an element, then ¯ ξ ( χ ( ¯ ϕ ′′ )) = ¯ ξ (cid:0) (0 , − ϕ ′′ ) + ¯ B n (cid:1) = ¯ ϕ . On the other hand, by construction of ξ and the fact that the first coordinate of the pair (0 , − ϕ ′′ ) is zero, it is easyto check that ¯ ϕ = 0 . Therefore, Imχ ⊂ Ker ¯ ξ. Now consider an element ¯ h ∈ Ker ¯ ξ and any of its representatives ( ϕ ′ , ϕ ′′ ) ∈ ¯ Z n . In this case, by thedefinition of ¯ ξ , there exists ϕ : Z n → G such that the following diagram iscommutative:(2.21) Z n C n C n +1 G H n G ′ G ′′ . p ¯ ϕ j δϕ ϕ ′ ϕ ′′ α β A. Beridze and L. Mdzinarishvili
Moreover, ¯ h ∈ Ker ¯ ξ means that the homomorphism ¯ ϕ : H n → G induced by ϕ : Z n → G is zero. Therefore, ϕ = 0 and so ϕ ′ ◦ j = α ◦ ϕ = 0 . Consequently, ϕ ′ : C n → G ′ induces a homomorphism ˜ ψ ′ : C n /Z n ≃ B n +1 → G ′ . Let ψ ′ : C n +1 → G ′ be an extension of ˜ ψ ′ ◦ p : Z n +1 → G ′ and ψ ′′ = β ◦ ψ ′ (see thediagram (2.22)). In this case, the homomorphism ψ = ψ ′′ − ϕ ′′ : C n +1 → G ′′ vanishes on the B n +1 . Indeed, ψ ◦ j ◦ i ◦ δ ′ = ψ ′′ ◦ j ◦ i ◦ δ ′ − ϕ ′′ ◦ j ◦ i ◦ δ ′ = β ◦ ˜ ψ ′ ◦ δ ′ − β ◦ ϕ ′ = β ◦ ϕ ′ − β ◦ ϕ ′ = 0 . On the other hand, δ ′ is an epimorphimand so ψ ◦ j ◦ i = 0 . Therefore, ψ ◦ j : Z n +1 → G ′′ induces a homomorphism ¯ ψ : H n +1 → G ′′ (see the diagram (2.22)).(2.22) Z n C n B n +1 Z n +1 H n +1 C n +1 G G ′ G ′′ . j ψ ′′ ϕ ′′ j δ ′ i pϕ ϕ ′ ¯ ψ ˜ ψ ′ ψ ′ α β Our aim is to show that χ ( ¯ ψ ) = ¯ h = ( ϕ ′ , ϕ ′′ ) + ¯ B n . Indeed, by the definitionof χ , it is easy to see that χ ( ¯ ψ ) = (0 , − ψ ) + ¯ B n . Therefore, we have to showthat ( ϕ ′ , ϕ ′′ ) − (0 , − ψ ) = ( ϕ ′ , ϕ ′′ + ψ ) = ( ϕ ′ , ψ ′′ ) ∈ ¯ B n . Indeed, (2.23) ∂ ( ψ ′ ,
0) = ( ψ ′ ◦ δ, β ◦ ψ ′ ) = ( ψ ′ ◦ j ◦ i ◦ δ ′ , β ◦ ψ ′ ) = ( ˜ ψ ′ ◦ δ ′ , ψ ′′ ) = ( ϕ ′ , ψ ′′ ) . d . Imβ ∗ ≃ Kerχ . Let ¯ ϕ ′ ∈ Hom( H n +1 ( C ∗ ); G ′ ) be an element and ϕ ′′ : C n +1 → G ′′ be an extension of the composition β ◦ ¯ ϕ ′ ◦ p : Z n +1 → G ′′ .In this case we have ∂ (0 , − ϕ ′′ ) = (0 , ϕ ′′ ◦ δ ) = (0 , ϕ ′′ ◦ j ◦ i ◦ δ ′ ) = (0 , β ◦ ¯ ϕ ′ ◦ p ◦ i ◦ δ ′ ) = (0 ,
0) = 0 (see the diagram (2.25)). Therefore, (0 , − ϕ ′′ ) ∈ ¯ Z n and so we have (2.24) ( χ ◦ β ∗ ) ( ¯ ϕ ′ ) = χ ( β ∗ ( ¯ ϕ ′ )) = χ ( β ◦ ¯ ϕ ′ ) = (0 , − ϕ ′′ ) + ¯ B n . Our aim is to show that (0 , − ϕ ′′ ) ∈ ¯ B n . Indeed, let ϕ ′ : C n +1 −→ G ′ be an extension of the composition ¯ ϕ ′ ◦ p : Z n +1 −→ G ′ . In this case ( β ◦ ϕ ′ − ϕ ′′ ) ◦ j = β ◦ ϕ ′ ◦ j − ϕ ′′ ◦ j = β ◦ ¯ ϕ ′ ◦ p − β ◦ ¯ ϕ ′ ◦ p = 0 andso β ◦ ϕ ′ − ϕ ′′ : C n +1 −→ G ′′ induces a homomorphism ˜ ψ ′′ : C n +1 /Z n +1 ≃ B n +2 −→ G ′′ such that β ◦ ϕ ′ − ϕ ′′ = ˜ ψ ′′ ◦ δ ′ . Let ψ ′′ : C n +2 −→ G ′′ be anextension of a homomorphism ˜ ψ ′ : B n +2 −→ G ′′ (see the diagram (2.25) ).(2.25) Z n +1 B n +1 C n C n +1 C n +1 /Z n +1 ≃ B n +2 C n +2 H n +1 G ′ G ′′ δ ′ i j δ ′ j ◦ ip ¯ ϕ ′ ϕ ′ ψ ′′ ϕ ′′ ˜ ψ ′′ β In this case, we have(2.26) ∂ ( − ϕ ′ , − ψ ′′ ) = ( − ϕ ′ ◦ δ, − β ◦ ϕ ′ + ψ ′′ ◦ δ ) = ( − ϕ ′ ◦ j ◦ i ◦ δ ′ , − β ◦ ϕ ′ + ˜ ψ ′′ ◦ δ ′ ) == ( − ¯ ϕ ′ ◦ p ◦ i ◦ δ ′ , − β ◦ ϕ ′ + ( β ◦ ϕ ′ − ϕ ′′ )) = (0 , − ϕ ′′ ) . Therefore, (0 , − ϕ ′′ ) ∈ ¯ B n and so, χ ◦ β ∗ = 0 . Hence, Imβ ∗ ⊂ Ker χ . Nowconsider an element ¯ ϕ ′′ ∈ Kerχ . Let ϕ ′′ : C n +1 → G ′′ be an extension ofthe composition ¯ ϕ ′′ ◦ p : Z n +1 → G ′′ (see the diagram (2.28)). Then, by χ ( ¯ ϕ ′′ ) = (0 , − ϕ ′′ ) + ¯ B n = 0 , there exists ( ψ ′ , ψ ′′ ) ∈ C n +1 ( β ) such that (2.27) ∂ ( ψ ′ , ψ ′′ ) = ( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ δ ) = (0 , − ϕ ′′ ) . Therefore, ψ ′ ◦ δ = ψ ′ ◦ j ◦ i ◦ δ ′ = 0 . Since δ ′ : C n → B n +1 is an epimorphism,we have ψ ′ ◦ j ◦ i = 0 and so ψ ′ ◦ j : Z n +1 → G ′ induces a homomorphism ¯ ψ ′ : H n +1 ( C ∗ ) → G ′ . On the other hand, by β ◦ ψ ′ − ψ ′′ ◦ δ = − ϕ ′′ , wehave − ϕ ′′ ◦ j = β ◦ ψ ′ ◦ j − ψ ′′ ◦ δ ◦ j = β ◦ ψ ′ ◦ j (see the diagram (2.28)).Therefore, β ∗ ( ¯ ψ ′ ) = ¯ ϕ ′′ and so, Kerχ ⊂ Imβ ∗ : (2.28) C n +1 C n +2 Z n +1 H n +1 G ′ G ′′ δβj p ¯ ψ ′ ψ ′ ψ ′′ ¯ ϕ ′′ Since for each injective abelian group G , a group of extensions Ext( − ; G ) is trivial, by the exact sequence (2.1) we obtain the following corollary (cf.Lemma VII.4.4 [Mas ]) Corollary 2.2. If G is an injective, then there is an isomorphism (2.29) ¯ H n ( C ∗ ; G ) ≃ Hom( H n ( C ∗ ); G ) . Let C ∗ = Hom( C ∗ ; G ) be a chain complex, where C n = Hom( C n ; G ) and ∂ are defined by ∂ ( ϕ ) = ϕ ◦ δ, for ϕ ∈ Hom( C n ; G ) . In this case, there is amap α ∗ : Hom( C ∗ ; G ) −→ Hom( C ∗ ; β ) defined by: (2.30) α ∗ ( ϕ ) = ( α ◦ ϕ, , ∀ ϕ ∈ Hom( C n ; G ) . Let H n ( C ∗ ; G ) be a homology group of chain complex Hom( C ∗ ; G ) . Theorem 2.3.
If a cochain complex C ∗ is free, then the homomorphism α ∗ : Hom( C ∗ ; G ) −→ Hom( C ∗ ; β ) induces an isomoprhism (2.31) ¯ α ∗ : H n ( C ∗ ; G ) −→ ¯ H n ( C ∗ ; G ) . A. Beridze and L. Mdzinarishvili
Proof.
Since C ∗ is a free cochain complex, there is a short exact sequence: (2.32) −→ Ext( H n +1 ( C ∗ ); G ) ˜ χ −→ H n (Hom( C ∗ ; G )) ˜ ξ −→ Hom( H n ( C ∗ ); G ) −→ . Let review how the morphisms ˜ ξ and ˜ χ are defined according to W. Massey’s[Mas ] approach. Note that Massey has considered a free chain complexcase and consequently, he has obtained Universal Coefficient Formula forcohomology theory and not homology theory. a. For each ¯ ϕ ∈ H n (Hom( C ∗ ; G )) element let ˜ ξ ( ¯ ϕ ) : H n ( C ∗ ) −→ G behomomorphism given by (2.33) ˜ ξ ( ¯ ϕ )(¯ c ) = h ϕ, c i = ϕ ( c ) , ∀ c ∈ ¯ c, ¯ c ∈ H n ( C ∗ ) , where ϕ is a representative of ¯ ϕ [Mas ]. b. To define the homomorphism ˜ χ : Ext( H n +1 ( C ∗ ); G ) −→ H n (Hom( C ∗ ; G )) ,we need to use the isomorphism (2.11). Consequently, the homomorphism ˜ χ is the homomorphism induced by χ : Hom( H n +1 ( C ∗ ); G ′′ ) −→ H n (Hom( C ∗ ; G )) , where χ is defined in the following way. Let ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) beany element. Since G ′′ is an injective, there is an extension ϕ ′′ : C n +1 −→ G ′′ of the composition ¯ ϕ ′′ ◦ p : Z n +1 −→ G ′′ . In this case ∂ ( ϕ ′′ ) = ϕ ′′ ◦ δ = ϕ ′′ ◦ j ◦ i ◦ δ ′ = ¯ ϕ ′′ ◦ p ◦ i ◦ δ ′ = 0 (see diagram (2.34)). (2.34) C n − C n B n +1 Z n +1 C n +1 H n +1 G G ′ G ′′ . p ¯ ϕ ′′ j δ ′ i jϕ ϕ ′ ϕ ′′ ψ ′ α β Therefore, it defines homology class [ ϕ ′′ ] ∈ H n +1 (Hom( C ∗ ; G ′′ )) . Let E : H n +1 (Hom( C ∗ ; G )) −→ H n (Hom( C ∗ ; G )) be a boundary homomorphisminduced by the following exact sequence: (2.35) −→ Hom( C ∗ ; G ) α −→ Hom( C ∗ ; G ′ ) β −→ Hom( C ∗ ; G ′′ ) −→ . Define a homomorphism χ by the formula (2.36) χ ( ¯ ϕ ′′ ) = E ([ ϕ ′′ ]) , ∀ ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) . Note that the homomorphism χ : Hom( H n +1 ( C ∗ ); G ) −→ H n (Hom( C ∗ ; G )) is a composition of the isomorphism Hom( H n +1 ( C ∗ ); G ) ≃ −→ H n +1 (Hom( C ∗ ; G )) and the homomorphism E : H n +1 (Hom( C ∗ ; G )) −→ H n (Hom( C ∗ ; G )) . Towrite the explicit formula for χ , consider such a map ψ ′ : C n +1 → G ′ that β ◦ ψ ′ = ϕ ′′ (this is possible, because the cochain complex C ∗ is free). Let ϕ ′ = ψ ′ ◦ δ : C n → G ′ , then we have β ◦ ψ ′ = ϕ ′′ ◦ δ = 0 . Therefore, ϕ ′ ∈ Kerβ = Imα and so, there exists a unique map ϕ : C n → G such that1 α ◦ ϕ = ϕ ′ . In this case, we have α ◦ ϕ ◦ δ = ϕ ′ ◦ δ = ψ ′ ◦ δ ◦ δ = 0 and so, ϕ ◦ δ = 0 because α is a monomorphism. On the other hand, ∂ ( ϕ ) = ϕ ◦ δ = 0 . Con-sequently, ϕ defines a homology class [ ϕ ] ∈ H n ((Hom( C ∗ ; G )) (see diagram(2.34)). Finally, by the formula (2.36) we have (2.37) χ ( ¯ ϕ ′′ ) = E ([ ϕ ′′ ]) = [ ϕ ] , ∀ ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) . In this case, the sequence (2.32) is induced by the following sequence(2.38)
Hom( H n +1 ( C ∗ ); G ′ ) Hom( H n +1 ( C ∗ ); G ′′ )¯ H n ( C ∗ ; G ) Hom( H n ( C ∗ ); G ) 0 .β ∗ χ χ ¯ ξ Therefore, by (2.12) and (2.38), it is sufficient to show that the followingdiagram is commutative:(2.39)
Hom( H n +1 ( C ∗ ); G ′ ) Hom( H n +1 ( C ∗ ); G ′′ ) H n (Hom( C ∗ ; G )) Hom( H n ( C ∗ ); G ) 0Hom( H n +1 ( C ∗ ); G ′ ) Hom( H n +1 ( C ∗ ); G ′′ ) ¯ H n (Hom( C ∗ ; β )) Hom( H n ( C ∗ ); G ) 0 . β ∗ χ ˜ ξβ ∗ χ ¯ ξ α ∗ Indeed, let ¯ ϕ ′′ ∈ Hom( H n +1 ( C ∗ ); G ′′ ) be an element and ϕ ′′ : C n +1 −→ G ′′ be an extension of the composition ¯ ϕ ′′ ◦ p : Z n +1 −→ G ′′ . Then, bydefinition of χ and method of snake lemma, we must take an element ϕ ′ ∈ Hom( C n +1 ; G ′ ) , such that β ( ϕ ′ ) = ϕ ′′ . Note that this is possible becauseof exactness of the sequence (2.35). Then, there is a cycle ϕ ∈ Hom( C n ; G ) ,such that α ( ϕ ) = ∂ ( ϕ ′ ) . Let [ ϕ ] = ϕ + B n be the corresponding element inthe homology group H n (Hom( C ∗ ; G )) , then χ ( ¯ ϕ ′′ ) = [ ϕ ] . By the definitionof the map ˜ α : H n (Hom( C ∗ ); G ) −→ ¯ H n (Hom( C ∗ ); β ) , we have (2.40) ( ˜ α ∗ ◦ χ ) ( ¯ ϕ ′′ ) = ˜ α ∗ ( χ ( ¯ ϕ ′′ )) = ˜ α ∗ ([ ϕ ]) = ( α ◦ ϕ,
0) + ¯ B n . On the other hand, by the definition of χ : Hom( G n +1 ( C ∗ ); G ′′ ) −→ ¯ H n (Hom( C ∗ ); β ) , we have (2.41) χ ( ¯ ϕ ′′ ) = (0 , − ϕ ′′ ) + ¯ B n , ∀ ¯ ϕ ′′ ∈ Hom( H n +1 ; G ′′ ) . Therefore, we have to show that ( α ◦ ϕ, − (0 , − ϕ ′′ ) = ( α ◦ ϕ, ϕ ′′ ) ∈ ¯ B n . Indeed, by the equality α ( ϕ ) = ∂ ( ϕ ′ ) and β ( ϕ ′ ) = ϕ ′′ , we have α ◦ ϕ = ϕ ′ ◦ δ and β ◦ ϕ ′ = ϕ ′′ . Therefore, (2.42) ∂ ( ϕ ′ ,
0) = ( ϕ ′ ◦ δ, β ◦ ϕ ′ ) = ( α ◦ ϕ, ϕ ′′ ) . A. Beridze and L. Mdzinarishvili
By (2.40), (2.41), and (2.42), we obtain that ˜ α ∗ ◦ χ = χ . So, it remains toshow that ¯ ξ ◦ ˜ α = ˜ ξ .Let [ ϕ ] ∈ H n (Hom( C ∗ ; G )) be an element and ϕ is its representative.Then, by the definitions of ¯ ξ and ˜ α ∗ we have (2.43) (cid:0) ¯ ξ ◦ ˜ α ∗ (cid:1) ([ ϕ ]) = ¯ ξ (˜ α ∗ ([ ϕ ])) = ¯ ξ ( α ◦ ϕ,
0) = ( ϕ ) + ¯ B n . Therefore, if we take an element ¯ c ∈ H n ( C ∗ ) and any of its representatives c ∈ ¯ c , then by (2.43) we have (2.44) (cid:0) ¯ ξ ◦ ˜ α (cid:1) ([ ϕ ]) (¯ c ) = (cid:0) ϕ + ¯ B n (cid:1) (¯ c ) = ϕ ( c ) . Therefore, by (2.33) , (2.43) and (2.44), we obtain that (2.45) ¯ ξ ◦ ˜ α = ˜ ξ. Note that by the commutative diagram (2.39), we obtain the followingcommutative diagram:(2.46) H n +1 ( C ∗ ); G ) H n (Hom( C ∗ ; G )) Hom( H n ( C ∗ ); G ) 00 Ext( H n +1 ( C ∗ ); G ) ¯ H n ( C ∗ ; G ) Hom( H n ( C ∗ ); G ) 0 . ˜ χ ˜ ξ ¯ χ ¯ ξ α ∗ Therefore, if a cochain complex C ∗ is free, then the classical Univer-sal Coefficient Formula is isomorphic to the Universal Coefficient Formuladefined in this paper. Hom and inverse limit func-tors
As we have seen in the previous section, there exists an epimorphism ξ :¯ Z n −→ Hom( H n ( C ∗ ); G ) which induces a homomorphism: (3.1) ¯ ξ : ¯ H n ( C ∗ ; G ) −→ Hom( H n ( C ∗ ); G ) and the following diagram is commutative:(3.2) ¯ Z n Hom( H n ( C ∗ ); G )¯ H n ( C ∗ ; G ) . ξ ¯ p ¯ ξ Kerξ we construct a homomorphism (3.3) ω : Hom( C n +1 ; G ′ ) ⊕ Hom( C n +1 /B n +1 ; G ′′ ) −→ Kerξ by ω ( ψ ′ , ψ ′′ ) = ( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ q ) , where q : C n +1 −→ C n +1 /B n +1 isthe quotient map. Let show that ω ( ψ ′ , ψ ′′ ) ∈ Kerξ . Indeed, ∂ω ( ψ ′ , ψ ′′ ) = ∂ ( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ q ) = ( ψ ◦ δ ◦ δ, β ◦ ψ ′ ◦ δ − ( β ◦ ψ ′ − ψ ′′ ◦ q ) ◦ δ ) = (0 , β ◦ ψ ′ ◦ δ − β ◦ ψ ′ ◦ δ + ψ ′′ ◦ q ◦ δ ) = (0 , ψ ′′ ◦ q ◦ δ ) = (0 , , because q ◦ δ = 0 . Hence, ω ( ψ ′ , ψ ′′ ) ∈ ¯ Z n . By the definition of ξ : ¯ Z n −→ Hom( H n ( C ∗ ); G ) , thereexists a uniquely defined map ϕ : Z n −→ G such that α ◦ ϕ = ψ ′ ◦ δ ◦ j . Onthe other hand, δ ◦ j = 0 and so, α ◦ ϕ = 0 , which induces a homomorphism ¯ ϕ : H n ( C ∗ ) −→ G . Note that α is a monomorphism and α ◦ ϕ = 0 impliesthat ϕ = 0 and consequently ( ω ( ψ ′ , ψ ′′ )) = ( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ q ) = ¯ ϕ = 0 .Therefore, we obtain that ω ( ψ ′ , ψ ′′ ) ∈ Kerξ.
Lemma 3.1.
For each integer n ∈ N , there exists the following short exactsequence (3.4) C n +1 /B n +1 ; G ′ ) Hom( C n +1 ; G ′ ) ⊕ Hom( C n +1 /B n +1 ; G ′′ ) Kerξ ,σ ω where σ : Hom( C n +1 /B n +1 ; G ′ ) −→ Hom( C n +1 ; G ′ ) ⊕ Hom( C n +1 /B n +1 ; G ′′ ) is defined by the formula (3.5) σ ( ϕ ) = ( ϕ ◦ q, β ◦ ϕ ) , ∀ ϕ ∈ Hom( C n +1 /B n +1 ; G ′ ) . Proof. a. ω is an epimorphism. If ( ϕ ′ , ϕ ′′ ) ∈ Kerξ, then ξ ( ϕ ′ , ϕ ′′ ) = ¯ ϕ = 0 and so ϕ : Z n −→ G is zero as well. On the other hand, α ◦ ϕ = ϕ ′ ◦ j = 0 .Therefore, there is a unique homomorphism ˜ ϕ ′ : B n +1 −→ G ′ , such that ϕ ′ = ˜ ϕ ′ ◦ δ ′ . Let ψ ′ : C n +1 −→ G ′ be an extension of the map ˜ ϕ ′ : B n +1 −→ G ′ (see the diagram (3.6)).(3.6) Z n C n B n +1 C n +1 j δ ′ j ◦ i G G ′ G ′′ . α βϕ ϕ ′ ¯ ϕ ′ ψ ′ ϕ ′′ If we consider the map β ◦ ψ ′ − ϕ ′′ : C n +1 −→ G ′′ , then by ( ϕ ′ , ϕ ′′ ) ∈ ¯ Z n , we have ( β ◦ ψ ′ − ϕ ′′ ) ◦ δ = β ◦ ψ ′ ◦ δ − ϕ ′′ ◦ δ = β ◦ ϕ ′ − ϕ ′′ ◦ δ = 0 . Since ( β ◦ ψ ′ − ϕ ′′ ) ◦ δ = ( β ◦ ψ ′ − ϕ ′′ ) ◦ j ◦ i ◦ δ ′ = 0 and δ ′ is an epimorphism,there is ( β ◦ ψ ′ − ψ ′′ ) ◦ j ◦ i = 0 . Therefore, there is a homomorphism ψ ′′ : C n +1 /B n +1 −→ G ′′ such that β ◦ ψ ′ − ϕ ′′ = ψ ′′ ◦ q and so ϕ ′′ = β ◦ ψ ′ − ψ ′′ ◦ q (see the diagram (3.7)).4 A. Beridze and L. Mdzinarishvili (3.7) B n +1 C n +1 C n +1 /B n +1 j ◦ i qG ′′ β ◦ ψ ′ − ϕ ′′ ψ ′′ Hence, ( ψ ′ , ψ ′′ ) ∈ Hom( C n +1 ; G ′ ) ⊕ Hom( C n +1 /B n +1 ; G ′′ ) and ω ( ψ ′ , ψ ′′ ) =( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ q ) = ( ϕ ′ , ϕ ′′ ) . So, ω is an epimorphism. b. There is an equality Imσ = Kerω . By the definition, we have ( ω ◦ σ )( ϕ ) = ω ( σ ( ϕ )) = ω ( ϕ ◦ q, β ◦ ϕ ) = ( ϕ ◦ q ◦ δ, β ◦ ϕ ◦ q − β ◦ ϕ ◦ q ) =(0 ,
0) = 0 , because q ◦ δ = 0 . Therefore, Imσ ⊂ Kerω.
On the other hand,if ( ψ ′ , ψ ′′ ) ∈ Kerω, then ω ( ψ ′ , ψ ′′ ) = ( ψ ′ ◦ δ, β ◦ ψ ′ − ψ ′′ ◦ q ) = 0 and so, ψ ′ ◦ δ = 0 and β ◦ ψ ′ = ψ ′′ ◦ q. On the other hand, ψ ′ ◦ δ = ψ ′ ◦ j ◦ i ◦ δ ′ = 0 .Therefore, we have ψ ′ ◦ j ◦ i = 0 , because δ ′ is an epimorphism. So, there is aunique homomorphism ϕ : C n +1 /B n +1 −→ G ′ such that ψ ′ = ϕ ◦ q. In thiscase, β ◦ ϕ ◦ q = β ◦ ψ ′ = ψ ′′ ◦ q and since q is an epimorphism, β ◦ ϕ = ψ ′′ . Therefore, σ ( ϕ ) = ( ψ ′ , ψ ′′ ) and so, Kerω ⊂ Imσ. c. σ is a monomorphism. If σ ( ϕ ) = ( ϕ ◦ q, β ◦ ϕ ) = 0 , i.e. ϕ ◦ q = 0 and since q is an epimorphism, we have ϕ = 0 .Let C ∗ = { C ∗ γ } be a direct system of cochain complexes. Consider thecorresponding inverse system C ∗ = { C γ ∗ ( β ) } = { Hom( C ∗ γ ; β ) } of chaincomplexes. Lemma 3.2.
For each direct system C ∗ = { C ∗ γ } of cochain complexes, thereis an isomorphism (3.8) Hom(lim −→ C ∗ γ ; β ) ≃ lim ←− Hom( C ∗ γ ; β ) . Proof.
Consider a chain complex (3.9)
Hom(lim −→ C ∗ γ ; β ) = { Hom(lim −→ C nγ ; G ′ ) ⊕ Hom(lim −→ C n +1 γ ; G ′′ ) , ∂ } , where ∂ ( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ δ, β ◦ ϕ ′ − ϕ ′′ ◦ δ ) . Note that δ = lim −→ δ γ : lim −→ C n − γ −→ lim −→ C nγ , where δ γ : C n − γ −→ C nγ is the coboundary map of the cochaincomplex C ∗ γ . Since for any G there is an isomorphism Hom(lim −→ C ∗ γ ; G ) ≃ lim ←− Hom( C ∗ γ ; G ) , we have (3.10) Hom(lim −→ C nγ ; G ′ ) ⊕ Hom(lim −→ C n +1 γ ; G ′′ ) ≃ lim ←− Hom( C nγ ; G ′ ) ⊕ lim ←− Hom( C n +1 γ ; G ′′ ) . Lemma 3.3. If f : C ∗ −→ C ′∗ is a homomorphism of cochain complexes,then there is a commutative diagram: (3.11) C ′ n +1 /B ′ n +1 ; G ′ ) Hom( C ′ n +1 ; G ′ ) ⊕ Hom( C ′ n +1 /B ′ n +1 ; G ′′ ) Kerξ ′ τ ′ µ ′ C n +1 /B n +1 ; G ′ ) Hom( C n +1 ; G ′ ) ⊕ Hom( C n +1 /B n +1 ; G ′′ ) Kerξ .τ µ ˜ f ( f , ˜ f ) ˜ f Proof.
Note that homomorphisms ˜ f : Hom( C ′ n +1 /B ′ n +1 ; G ′ ) −→ Hom( C n +1 /B n +1 ; G ′ ) and ( f , ˜ f ) : Hom( C ′ n +1 ; G ′ ) ⊕ Hom( C ′ n +1 /B ′ n +1 ; G ′′ ) −→ Hom( C n +1 ; G ′ ) ⊕ Hom( C n +1 /B n +1 ; G ′′ ) are naturally defined by ˜ f ( ϕ ′ ) = ϕ ′ ◦ ˜ f n +1 and ( f , ˜ f )( ϕ ′ , ϕ ′′ ) = ( ϕ ′ ◦ f n +1 , ϕ ′′ ◦ ˜ f n +1 ) , where ˜ f n +1 : C n +1 /B n +1 −→ C ′ n +1 /B ′ n +1 is induced by f n +1 : C n +1 −→ C ′ n +1 . a. τ ◦ ˜ f = ( f , ˜ f ) ◦ τ ′ . By the definition, we have (cid:16) τ ◦ ˜ f (cid:17) ( ϕ ′ ) = τ (cid:16) ˜ f ( ϕ ′ ) (cid:17) = τ (cid:16) ϕ ′ ◦ ˜ f n +1 (cid:17) = (cid:16) ϕ ′ ◦ ˜ f n +1 ◦ q, β ◦ ϕ ′ ◦ ˜ f n +1 (cid:17) and (cid:16) ( f , ˜ f ) ◦ τ ′ (cid:17) ( ϕ ′ ) = (cid:16) f , ˜ f (cid:17) ( τ ′ ( ϕ ′ )) = (cid:16) f , ˜ f (cid:17) ( ϕ ′ ◦ q ′ , β ◦ ϕ ′ ) = ( ϕ ′ ◦ q ′ ◦ f n +1 , β ◦ ϕ ′ ◦ ˜ f n +1 ) . Since ˜ f n +1 ◦ q = q ′ ◦ f n +1 , we have ϕ ′ ◦ ˜ f n +1 ◦ q = ϕ ′ ◦ q ′ ◦ f n +1 . Hence, τ ◦ ˜ f = ( f , ˜ f ) ◦ τ ′ . b. µ ◦ ( f , ˜ f ) = ˜ f ◦ µ ′ . By the definition, we have (cid:16) µ ◦ ( f , ˜ f ) (cid:17) ( ϕ ′ , ϕ ′′ ) = µ (cid:16) ( f , ˜ f )( ϕ ′ , ϕ ′′ ) (cid:17) = µ (cid:16) ϕ ′ ◦ f n +1 , ϕ ′′ ◦ ˜ f n +1 (cid:17) = ( ϕ ′ ◦ f n +1 ◦ δ, β ◦ ϕ ′ ◦ f n +1 − ϕ ′′ ◦ ˜ f n +1 ◦ q ) and (cid:16) ˜ f ◦ µ ′ (cid:17) ( ϕ ′ , ϕ ′′ ) = ˜ f ( µ ′ ( ϕ ′ , ϕ ′′ )) = ˜ f ( ϕ ′ ◦ δ ′ , β ◦ ϕ ′ − ϕ ′′ ◦ q ′ ) =( ϕ ′ ◦ δ ′ ◦ f n , β ◦ ϕ ′ ◦ f n +1 − ϕ ′′ ◦ q ′ ◦ f n +1 ) . Since ˜ f n +1 ◦ q = q ′ ◦ f n +1 and δ ′ ◦ f n = f n +1 ◦ δ, there are qualities ϕ ′ ◦ f n +1 ◦ δ = ϕ ′ ◦ δ ′ ◦ f n and ϕ ′′ ◦ ˜ f n +1 ◦ q = ϕ ′′ ◦ q ′ ◦ f n +1 . Hence, µ ◦ ( f , ˜ f ) = ˜ f ◦ µ ′ .Let { Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) } be an inverse system gen-erated by the direct system { C n +1 γ } . It is clear that for each γ there is anexact sequence(3.12) C n +1 γ ; G ′ ) Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) Hom( C n +1 γ /B n +1 γ ; G ′′ ) 0 . τ µ Hence, by the main property of the derived functors lim ←− ( i ) there is a longexact sequence:(3.13) . . . lim ←− ( i ) Hom( C n +1 γ ; G ′ ) lim ←− ( i ) (cid:0) Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) (cid:1) lim ←− ( i ) Hom( C n +1 γ /B n +1 γ ; G ′′ ) . . . . ˜ τ ˜ µ ˜ µ A. Beridze and L. Mdzinarishvili
On the other hand, since for each injective abelian groups G , lim ←− ( i ) { Hom( C n +1 ; G ) } =0 , i ≥ (see Lemma 1.3 [Hub-Mei]), we obtain the following result. Corollary 3.4.
For each injective abelian groups G ′ and G ′′ , there is thefollowing equality (3.14) lim ←− ( i ) (cid:0) Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) (cid:1) = 0 , i ≥ . Using the obtained result, we will prove the following lemma.
Lemma 3.5.
For each integer i ≥ , there is an equality (3.15) lim ←− ( i ) Kerξ γ = 0 . Proof.
By Lemma 1, for each ξ γ , there is a short exact sequence(3.16) C n +1 γ /B n +1 γ ; G ′ ) Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) Kerξ γ . σ γ ω γ By the main property of a derived functor lim ←− ( i ) , there is a long exactsequence(3.17) . . . lim ←− ( i ) Hom( C n +1 γ /B n +1 γ ; G ′ ) lim ←− ( i ) (cid:0) Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) (cid:1) lim ←− ( i ) Ker ξ γ . . . . By Lemma 1.3 [Hub-Mei] for each i ≥ , there is an equality lim ←− ( i ) Hom( C n +1 γ /B n +1 γ ; G ′ ) =0 and by Corollary 2, for each i ≥ we obtain (3.18) lim ←− ( i ) (cid:0) Hom( C n +1 γ ; G ′ ) ⊕ Hom( C n +1 γ /B n +1 γ ; G ′′ ) (cid:1) = 0 . Hence, by the long exact sequence (3.17) we obtain that lim ←− ( i ) Kerξ γ = 0 , i ≥ . Corollary 3.6.
For each integer i ≥ , there is an isomorphism (3.19) lim ←− ( i ) ¯ Z γn ≃ lim ←− ( i ) Hom( H n ( C ∗ γ ); G ) . Proof. By a. of Theorem 1, there is an epimorphism ξ γ : ¯ Z γn −→ Hom( H n ( C ∗ γ ); G ) . Therefore, the following sequence is exact(3.20) Kerξ γ ¯ Z γn Hom( H n ( C ∗ γ ); G ) 0 . ξ γ Consequently, it induces the following long exact sequence(3.21) . . . lim ←− ( i ) Kerξ γ lim ←− ( i ) ¯ Z γn lim ←− ( i ) Hom( H n ( C ∗ γ ); G ) lim ←− ( i +1) Kerξ γ . . . . On the other hand, by Lemma 4, lim ←− ( i ) { Kerξ γ } = 0 , i ≥ . Therefore, for i ≥ , we have an isomorphism lim ←− ( i ) ¯ Z γn ≃ lim ←− ( i ) Hom( H n ( C ∗ γ ); G ) . γ , there is a natural commutative triangle(3.22) ¯ Z γn Hom( H n ( C ∗ γ ); G )¯ H n ( C ∗ γ ; G ) . ξ γ ¯ p γ ¯ ξ γ Therefore, if we take lim ←− ( i ) of this diagram, then by Corollary 2, we obtainthe following result. Corollary 3.7.
For each integer i ≥ , lim ←− ( i ) ¯ Z γn is a direct summand of lim ←− ( i ) ¯ H n ( C ∗ γ ; G ) and the projection of lim ←− ( i ) ¯ H n ( C ∗ γ ; G ) onto lim ←− ( i ) ¯ Z γn is nat-ural. Finally, we obtain the following important property of lim ←− ( i ) functor. Theorem 3.8.
For each integer i ≥ , there is a short exact sequence: (3.23) ←− ( i ) Ext( H n +1 ( C ∗ γ ); G ) lim ←− ( i ) ¯ H n ( C ∗ γ ; G ) lim ←− ( i ) Hom( H n ( C ∗ γ ); G ) 0 ,ξ γ and this sequence splits naturally for i ≥ .Proof. Using the commutative diagram (3.22), for each γ we have a com-mutative diagram with exact rows:(3.24) Kerξ γ ¯ Z γn Hom( H n ( C ∗ γ ); G ) 0 ξ γ H n +1 ( C ∗ γ ); G ) ¯ H n ( C ∗ γ ; G ) Hom( H n ( C ∗ γ ); G ) 0 . ¯ ξ γ ¯ p γ This induces the following commutative diagram with exact rows:(3.25) . . . lim ←− ( i ) Kerξ γ lim ←− ( i ) ¯ Z γn lim ←− ( i ) Hom( H n ( C ∗ γ ); G ) lim ←− ( i +1) Kerξ γ . . . lim ←− ( i ) ξ γ . . . lim ←− ( i ) Ext( H n +1 ( C ∗ γ ); G ) lim ←− ( i ) ¯ H n ( C ∗ γ ; G ) lim ←− ( i ) Hom( H n ( C ∗ γ ); G ) lim ←− ( i +1) Ext( H n +1 ( C ∗ γ ); G ) . . . . lim ←− ( i ) ¯ ξ γ lim ←− ( i ) ¯ p γ By Lemma 4, lim ←− ( i ) Kerξ γ = 0 , for i ≥ , and so the beginning of thediagram (3.25) of the following form:8 A. Beridze and L. Mdzinarishvili (3.26) ←− Kerξ γ lim ←− ¯ Z γn lim ←− Hom( H n ( C ∗ γ ); G ) 0lim ←− ξ γ ←− Ext( H n +1 ( C ∗ γ ); G ) lim ←− ¯ H n ( C ∗ γ ; G ) lim ←− Hom( H n ( C ∗ γ ); G ) . . . . lim ←− ¯ ξ γ lim ←− ( i ) ¯ p γ Therefore, the following sequence is exact:(3.27) ←− Ext( H n +1 ( C ∗ γ ); G ) lim ←− ¯ H n ( C ∗ γ ; G ) lim ←− Hom( H n ( C ∗ γ ); G ) 0 ξ γ and the lim ←− (1) { Ext( H n +1 ( C ∗ γ ); G ) } −→ lim ←− (1) { ¯ H n ( C ∗ γ ; G ) } is a monomor-phism. Therefore, we obtain the result for i = 0 . On the other hand, for i ≥ , the result follows from the commutativity of the diagram (3.25) andCorollary 2 and 3. Here we have formulated the main result of the paper.
Theorem 4.1.
Let C ∗ = { C ∗ γ } be a direct system of cochain complexes.Then, there is a natural exact sequence: (4.1) . . . lim ←− (3) ¯ H γn +2 lim ←− (1) ¯ H γn +1 ¯ H n (cid:16) lim −→ C ∗ γ ; G (cid:17) lim ←− ¯ H γn lim ←− (2) ¯ H γn +1 . . . where ¯ H γ ∗ = ¯ H ∗ ( C ∗ γ ; G ) . Proof.
By Proposition 1.2 of [Hub-Mei], for the inverse system { H n +1 γ } wehave an exact sequence:(4.2) ←− (1) Hom( H n +1 γ ; G ) Ext(lim −→ H n +1 γ ; G ) lim ←− Ext( H n +1 γ ; G ) lim ←− (2) Hom( H n +1 γ ; G ) 0 and (4.3) lim ←− ( i ) Ext( H n +1 γ ; G ) ≃ lim ←− ( i +2) Hom( H n +1 γ ; G ) , for i ≥ . Since cohomology commutes with direct limits, we have H ∗ (lim −→ C ∗ γ ; G ) ≃ lim −→ H ∗ ( C ∗ γ ; G ) . Therefore, if C ∗ ≃ lim −→ C ∗ γ , then H n +1 ( C ∗ ) ≃ lim −→ H n +1 γ , where H n +1 γ = H n +1 ( C ∗ γ ; G ) . So, we obtain an exact sequence:(4.4) ←− (1) Hom( H n +1 γ ; G ) Ext( H n +1 ( C ∗ ); G ) lim ←− Ext( H n +1 γ ; G ) lim ←− (2) Hom( H n +1 γ ; G ) 0 . i γ : C ∗ γ −→ C ∗ is a natural map, then it induces π γ ; ¯ H ∗ ( C ; G ) −→ ¯ H n ( C ∗ γ ; G ) map. On the other hand, by Theorem 1, the following diagramis commutative:(4.5) H n +1 ( C ∗ ); G ) ¯ H n ( C ∗ ; G ) Hom( H n ( C ∗ ); G ) 00 Ext( H n +1 ( C ∗ γ ); G ) ¯ H n ( C ∗ γ ; G ) Hom( H n ( C ∗ γ ); G ) 0 . ˜ π γ π γ ¯ π γ The diagram (4.5) generates the following diagram:(4.6) H n +1 ( C ∗ ); G ) ¯ H n ( C ∗ ; G ) Hom( H n ( C ∗ ); G ) 00 lim ←− Ext( H n +1 γ ; G ) lim ←− ¯ H γn lim ←− Hom( H nγ ; G ) 0 . ˜ π π ≃ Therefore, we have
Ker ˜ π ≃ Kerπ and
Coker ˜ π ≃ Cokerπ and so, thefollowing diagram is commutative:(4.7) H n +1 ( C ); G ) ¯ H n ( C ; G ) Hom( H n ( C ); G ) 0 Ker ˜ π Kerπ ≃ ←− Ext( H n +1 γ ; G ) lim ←− ¯ H γn lim ←− Hom( H nγ ; G ) 0 . ˜ π π ≃ Coker ˜ π Cokerπ ≃ Using the exact sequence (4.4) and diagram (4.7), we obtain a four-termexact sequence:(4.8) ←− (1) Hom( H n +1 γ ; G ) ¯ H n ( C ; G ) lim ←− ¯ H γn lim ←− (2) Hom( H n +1 γ ; G ) 0 . Using the exact sequence (4.8), Theorem 3 and isomorphism (4.3), we ob-tain the following diagram, which contains the long exact sequence of the0
A. Beridze and L. Mdzinarishvili theorem:(4.9) ←− (1) Hom( H n +1 γ ; G ) ¯ H n ( C ; G ) lim ←− ¯ H γn lim ←− (2) Hom( H n +1 γ ; G ) 0 . ←− (2) Ext( H n +2 γ ; G ) lim ←− (2) ¯ H γn +1 lim ←− (2) Hom( H n +1 γ ; G ) 00 lim ←− (4) Hom( H n +2 γ ; G ) lim ←− (4) ¯ H γn +2 lim ←− (4) Ext( H n +3 γ ; G ) 0 ... ←− (1) Hom( H n +1 γ ; G ) lim ←− (1) ¯ H γn +1 lim ←− (1) Ext( H n +2 γ ; G ) 00 lim ←− (3) Ext( H n +3 γ ; G ) lim ←− (3) ¯ H γn +2 lim ←− (3) Hom( H n +2 γ ; G ) 0 ... ≃≃ ≃≃ Corollary 4.2.
Let C ∗ = { C ∗ γ } be a direct system of cochain complexes.Then, for each injective abelian group G, there is an isomorphism (4.10) ¯ H n (cid:16) lim −→ C ∗ γ ; G (cid:17) ≃ lim ←− ¯ H ∗ ( C ∗ γ ; G ) . Let C ∗ c ( X, G ) be the cochain complex of Massey [Mas ]. It is knownthat for each locally compact Hausdorff space X and each integer n thecochain group C nc ( X, Z ) with integer coefficient is a free abelian group (the-orem 4.1 [Mas ]). Using the cochain complex C ∗ c ( X, G ) , Massey defined anexact homology H M ∗ , the so called Massey homology on the category oflocally compact spaces and proper maps as a homology of the chain com-plex C ∗ ( X, G ) = Hom( C ∗ c ( X ) , G ) . Consequently, for the given category theUniversal Coefficient Formula is obtained (see theorem 4.1, corollary 4.18[Mas ] and theorem of universal coefficient 4.1 [Mac]): (5.1) −→ Ext( H n +1 c ( X ) , G ) −→ H Mn ( X, G ) −→ Hom( H nc ( X ) , G ) −→ . Let C ∗ c = C ∗ c ( X ; G ) be the cochain complex of Massey. Consider the chaincomplex ¯ C M ∗ ( X ; G ) = Hom( C ∗ ( X ); β ) . Let ¯ H M ∗ ( X ; G ) be homology of the1chain complex ¯ C M ∗ ( X ; G ) . In this case, by Theorem 1 we will obtain theUniversal Coefficient Formula: (5.2) −→ Ext( H n +1 c ( X ) , G ) −→ ¯ H Mn ( X, G ) −→ Hom( H nc ( X ) , G ) −→ . Note that by Theorem 2, for the category of locally compact spaces thehomologies ¯ H Mn ( X, G ) and H Mn ( X, G ) are isomorphic.Note that for the Massey homology theory ¯ H M ∗ ( − ; G ) our constructiongives the following result: Corollary 5.1.
Let X be a locally compact Hausdorff space, thena) if { N α } is the system of closed neighborhoods N α of closed subspace A of X , directed by inclusion, then it induces the following exact sequence: (5.3) . . . lim ←− (2 k +1) ¯ H Mn ++ k +1 ( N α ) . . . lim ←− (3) ¯ H Mn +2 ( N α ) lim ←− (1) ¯ H Mn +1 ( N α )¯ H Mn ( A ; G ) lim ←− ¯ H Mn ( N α ) lim ←− (2) ¯ H Mn +1 ( N α ) . . . lim ←− (2 k ) ¯ H Mn + k ( N α ) . . . .i ∗ b) if { U α } is the system of open subspaces of X , such that ¯ U α is compactand X = S U α directed by inclusion, then it induces the following exactsequence: (5.4) . . . lim ←− (2 k +1) ¯ H Mn ++ k +1 ( U α ) . . . lim ←− (3) ¯ H Mn +2 ( U α ) lim ←− (1) ¯ H Mn +1 ( U α )¯ H Mn ( A ; G ) lim ←− ¯ H Mn ( U α ) lim ←− (2) ¯ H Mn +1 ( U α ) . . . lim ←− (2 k ) ¯ H Mn + k ( U α ) . . . .i ∗ Note that the formula ( ?? ) is a generalization of Theorem 1.4.2 of [Mas ]. Let G be an R -module over a principal ideal domain R and let X be atopological space. Denote by ¯ C ∗ ( X ; G ) the cochain complex of Alexander-Spanier [Sp] and by ¯ H ∗ ( X ; G ) the Alexander-Spanier cohomology. let A bea subspace of a topological space X and { U α } be the family of all neigh-borhoods of A in X directed downward by inclusion. Hence, { ¯ H n ( U α ; G ) } is a direct system. The restriction maps ¯ H n ( U α ; G ) −→ ¯ H n ( A ; G ) define anatural homomorphism (5.5) i : lim −→ ¯ H n ( U α ; G ) −→ ¯ H n ( A ; G ) . By Theorem 6.6.2 [Sp], if A is a closed subspace of a paracompact Hausdorffspace X , then (5.5) is an isomorphism. In this case, A is called a tautsubspace relative to the Alexander-Spanier cohomology theory. In the caseof homology theory, we have a natural homomorphism (5.6) i : H n ( A ; G ) −→ lim ←− H n ( U α ; G ) . A. Beridze and L. Mdzinarishvili
The question whether the homomorphism (5.6) is an isomorphism or notwas open.Let ¯ C ∗ = ¯ C ∗ ( X ; G ) be the cochain complex of Alexander-Spanier. Con-sider the chain complex ¯ C ∗ ( X ; G ) = Hom( ¯ C ∗ ( X ); β ) . Let ¯ H ∗ ( X ; G ) bethe homology of the chain complex ¯ C ∗ ( X ; G ) . In this case, we will say thatthe homology ¯ H ∗ ( X ; G ) is generated by the Alexander-Spanier cochains ¯ C ∗ ( X ; G ) . By Theorem 4 we have the long exact sequence, which containsthe homomorphisms (5.6). Corollary 5.2. If A is a closed subspace of a paracompact Hausdorff space X and { U α } is the family of all neighborhoods of A in X , then there is along exact sequence: (5.7) . . . lim ←− (2 k +1) ¯ H n ++ k +1 ( U α ) . . . lim ←− (3) ¯ H n +2 ( U α ) lim ←− (1) ¯ H n +1 ( U α )¯ H n ( A ; G ) lim ←− ¯ H n ( U α ) lim ←− (2) ¯ H n +1 ( U α ) . . . lim ←− (2 k ) ¯ H n + k ( U α ) . . . .i ∗ It is clear that there is a natural inclusion i : C ∗ c ( X ; G ) → ¯ C ∗ ( X ; G ) from the Massey cochain complex to the Alexander-Spanier cochain com-plex, which induces the corresponding homomorphism i ∗ : ¯ H ∗ ( − ; G ) → ¯ H M ∗ ( − ; G ) , where ¯ H ∗ ( − ; G ) and ¯ H M ∗ ( − ; G ) are homologies generated bythe Alexander-Spanier and the Massey cochains, respectively. Therefore, ¯ H ∗ ( − ; G ) and ¯ H M ∗ ( − ; G ) are homologies of the chain complexes ¯ C ∗ ( − ; G ) =Hom( ¯ C ∗ ( − ); β ) and ¯ C M ∗ ( − ; G ) = Hom( C ∗ c ( − ); β ) . On the other hand, onthe category of compact Hausdorff spaces, the Alexsander-Spanier and theMassey cohomology are isomorphic and by the Universal Coefficient For-mula, we will obtain that for each compact Hausdorff space the there is anisomorphism: (5.8) i ∗ : ¯ H ∗ ( X ; G ) ≃ −→ ¯ H M ∗ ( X ; G ) . On the other hand, since on the category of compact metric spaces theSteenrod homology H St ∗ [St], [Ed-Ha ], [Ed-Ha ] and the Massey homologyare isomorphic, using the isomorphism (5.8), we will obtain that (5.9) ¯ H ∗ ( X ; G ) ≃ H St ∗ ( X ; G ) . The same way, on the category of compact Hausdorff spaces, the Milnorhomology H Mil ∗ and the Massey homology are isomorphic and consequently,we have (5.10) ¯ H ∗ ( X ; G ) ≃ H Mil ∗ ( X ; G ) . H BM ∗ ( − ; G ) is the Borel-Moore homology with coefficient in G , then byTheorem 3 [Kuz], we have the isomororphism (5.11) ¯ H ∗ ( X ; G ) ≃ H BM ∗ ( X ; G ) . Let K C be the category of compact pairs ( X, A ) and continuous mapsand H ∗ be an exact homology theory. Let { ( X α , A α ) } be an inverse systemof compact pairs ( X α , A α ) and ( X, A ) = lim ←− ( X α , A α ) . The inverse system { ( X α , A α ) } generates an inverse system { H ∗ ( X α , A α ) } and the projection π α : ( X, A ) → ( X α , A α ) induces the homomorphism π α, ∗ : H ∗ ( X, A ) → H ∗ ( X α , A α ) , which induces the homomorphism (5.12) π ∗ : H ∗ ( X, A ) → lim ←− H ∗ ( X α , A α ) . Definition 5.3.
An exact homology theory H ∗ is said to be a continuous onthe category K C , if for each inverse system { ( X α , A α ) } of the given category,there is an infinite exact sequence:(5.13) . . . lim ←− (2 k +1) ¯ H Mn ++ k +1 ( X α , A α ) . . . lim ←− (3) ¯ H Mn +2 ( X α , A α ) lim ←− (1) ¯ H Mn +1 ( X α , A α )¯ H Mn ( X, A ; G ) lim ←− ¯ H Mn ( X α , A α ) lim ←− (2) ¯ H Mn +1 ( X α , A α ) . . . lim ←− (2 k ) ¯ H Mn + k ( X α , A α ) . . . .i ∗ Definition 5.4.
A direct system C ∗ = { C ∗ α } of the cochain complexes C ∗ α issaid to be associated with a cochain complex C ∗ , if there is a homomorphism C ∗ → C ∗ such that for each n ∈ Z the induced homomorphism (5.14) lim −→ H ∗ ( C ∗ α ) → H ∗ ( C ∗ ) is an isomorphism. Lemma 5.5.
If a direct system C ∗ = { C ∗ α } of the cochain complexes C ∗ α is associated with a cochain complex C ∗ , then there is an infinite exactsequence: (5.15) . . . lim ←− (2 k +1) ¯ H Mn ++ k +1 ( C ∗ α ; G ) . . . lim ←− (3) ¯ H Mn +2 ( C ∗ α ; G ) lim ←− (1) ¯ H Mn +1 ( C ∗ α ; G )¯ H Mn ( C ∗ ; G ) lim ←− ¯ H Mn ( C ∗ α ; G ) lim ←− (2) ¯ H Mn +1 ( C ∗ α ; G ) . . . lim ←− (2 k ) ¯ H Mn + k ( C ∗ α ; G ) . . . .i ∗ where ¯ H ∗ ( C ∗ ) = H ∗ ((Hom( C ∗ ; β )) and ¯ H ∗ ( C ∗ α ) = H ∗ (Hom( C ∗ α ; β )) . Proof.
By theorem 4, There is a natural exact sequence: (5.16) . . . lim ←− (3) ¯ H αn +2 lim ←− (1) ¯ H αn +1 ¯ H n (cid:16) lim −→ C ∗ α ; G (cid:17) lim ←− ¯ H αn lim ←− (2) ¯ H αn +1 . . . , A. Beridze and L. Mdzinarishvili where ¯ H α ∗ = ¯ H ∗ ( C ∗ α ; G ) . Since the direct system C ∗ = { C ∗ α } of cochain com-plexes C ∗ α is associated with a cochain complex C ∗ , there is an isomorphism (5.17) ¯ H ∗ (lim −→ C ∗ α ; G ) ≃ lim −→ ¯ H ∗ ( C ∗ α ; G ) ≃ −→ ¯ H ∗ ( C ∗ ; G ) . On the other hand, by Universal Coefficient Formula, we have the followingcommutative diagram with exact rows:(5.18) H n +1 ( C ∗ ); G ) ¯ H n ( C ∗ ; G ) Hom( H n ( C ∗ ); G ) 00 Ext( H n +1 (lim −→ C ∗ α ); G ) ¯ H n (lim −→ C ∗ α ; G ) Hom( H n (lim −→ C ∗ α ); G ) 0 . ≃ ¯ π n ≃ Hence, the homomorphism ¯ π n is an isomorphism for all n ∈ Z . Using theexact sequence (5.16) and the isomorphism ¯ π n , we obtain an infinite exactsequence (5.15). Corollary 5.6.
Let { ( X α , A α ) } be an inverse system of pairs of compactspaces ( X α , A α ) and ( X, A ) = lim ←− ( X α , A α ) . If ¯ H ∗ is the homology theorygenerated by the Alexander-Spanier cochains, then there is an infinite exactsequence: (5.19) . . . lim ←− (2 k +1) ¯ H n + k +1 ( X α , A α ) . . . lim ←− (3) ¯ H n +2 ( X α , A α ) lim ←− (1) ¯ H n +1 ( X α , A α )¯ H n ( X, A ; G ) lim ←− ¯ H n ( X α , A α ) lim ←− (2) ¯ H n +1 ( X α , A α ) . . . lim ←− (2 k ) ¯ H n + k ( X α , A α ) . . . .π ∗ Corollary 5.7.
Let { ( X i , A i ) } i ∈ Z be an inverse sequnce of compact metricspaces ( X i , A i ) and ( X, A ) = lim ←− ( X i , A i ) . If ¯ H ∗ is the homology theory gen-erated by the Alexander-Spanier cochains, then there is an exact sequence: (5.20) −→ lim ←− (1) ¯ H n +1 ( X i , A i ) −→ ¯ H n ( X, A ; G ) π ∗ −→ lim ←− ¯ H n ( X i , A i ) −→ . Corollary 5.8. If ¯ H ∗ is the homology theory generated by the Alexander-Spanier cochains, then there is an exact sequence: (5.21) −→ lim ←− (1) ¯ H n +1 ( K α , L α ) −→ ¯ H n ( X, A ; G ) π ∗ −→ lim ←− ¯ H n ( K α , L α ) −→ , where ( X, A ) = lim ←− ( K α , L α ) and ( K α , L α ) are finite polyhedral pairs. Let C ∗ s ( X ; G ) be the singular cochain complex of topological spaces X and ¯ C s ∗ ( X ; G ) = Hom( C ∗ s ( X ); β ) . Let ¯ H s ∗ ( X ; G ) be the homology ofthe obtained chain complex ¯ C s ∗ ( X ; G ) . Therefore, ¯ H s ∗ ( − ; G ) is the homol-ogy generated by the singular cochain complex C ∗ s ( − ; G ) . It is known that5there is a natural homomorphism j : ¯ C ∗ ( X : G ) → C ∗ s ( X ; G ) from theAlexander-Sapnier cochain complex to the singular cochain complex, whichinduces the isomorphism j ∗ : ¯ H ∗ ( X ; G ) → ¯ H ∗ s ( X ; G ) on the category ofmanifolds. Therefore, by the Universal Coefficient Formula, we will obtainthat if X is manifold, then there is an isomorphism: (5.22) j ∗ : ¯ H s ∗ ( X ; G ) ≃ −→ ¯ H ∗ ( X ; G ) . References [Ber-Mdz ] A. Beridze, L. Mdzinarishvili,
On the axiomatic systems of sin-gular cohomology theory. Topology Appl. 275 (2020)[Ber-Mdz ] A. Beridze, L. Mdzinarishvili,
On the axiomatic systems ofSteenrod homology theory of compact spaces. Topology Appl. 249(2018), 73–82[Ber]
N. Berikashvili,
Axiomatics of the Steenrod-Sitnikov homology the-ory on the category of compact Hausdorff spaces.(Russian) Topol-ogy (Moscow, 1979). Trudy Mat. Inst. Steklov. 154 (1983), 24–37.[Bor-Mo]
A. Borel, J. C. Moore,
Homology theory for locally compactspaces. Michigan Math. J. 7 (1960), 137–159[Ed-Ha ] D. A. Edwards and H. M. Hastings, Čech theory: its past, present,and future. Rocky Mountain J. Math. 10 (1980), no. 3, 429–468[Ed-Ha ] D. A. Edwards and H. M. Hastings,
ȸ ech and Steenrod homotopytheories with applications to geometric topology. Lecture Notes inMathematics, Vol. 542. Springer-Verlag, Berlin-New York, 1976[Eil-St]
S. Eilenberg, N. Steenrod,
Foundations of algebraic topology.Princeton, New Jersey: Princeton University Press, 1952.[Hub-Mei]
M. Huber, W. Meier,
Cohomology theories and infinite CW -complexes. Comment. Math. Helv. 1978. V. 53, no. 2. P. 239–257.[Kuz] Kuzminov, V. I.
Equivalence of homology theories on categories ofbicompacta. (Russian) Sibirsk. Mat. Zh. 21 (1980), no. 1, 125–129,237.[Mac]
Mac Lane S.,
Homology. Die Grundlehren der mathematischen Wis-senschaften, Bd. 114. New York: Academic Press, Inc., Publishers;Berlin-Göttingen-Heidelberg: Springer-Verlag, 1963.6
A. Beridze and L. Mdzinarishvili [Mas ] W. S. Massey,
Singular homology theory. Graduate Texts in Math-ematics, 70. Springer-Verlag, New York-Berlin, 1980[Mas ] W. S. Massey,
Homology and Cohomology Theory. An ApproachBased on Alexander-Spanier Cochains. Monographs and Textbooksin Pure and Applied Mathematics, Vol. 46. New York-Basel: MarcelDekker, Inc., 1978.[Mas ] W. S. Massey,
How to give an exposition of the Čech-Alexander-Spanier type homology theory. Amer. Math. Monthly 85 (1978), no.2, 75–83[Mas ] W. S. Massey,
Notes on homology and cohomology theory, YaleUniversity, 1964 (mimeographed)[Mdz ] L. Mdzinarishvili,
On the Continuity Property of the Exact Homol-ogy Theories, Top. Proc. 56 (2020) pp. 237-247[Mdz ] L. Mdzinarishvili,
The uniqueness theorem for cohomologies on thecategory of polyhedral pairs, Trans. A. Razmadze Math. Inst. 2018.V. 172, no. 2. P. 265–275.[Mdz ] L. Mdzinarishvili,
Universelle Koeffizientenfolgen für den lim ←− -Funktor und Anwendungen. (German) Manuscripta Math. 48(1984), no. 1-3, 255–273.[Mdz ] L. Mdzinarishvili,
On homology extensions. Glas. Mat. Ser. III21(41) (1986), no. 2, 455–482.[Mdz-Sp]
L. Mdzinarishvili, E. Spanier,
Inverse limits and cohomology.Glas. Mat. Ser. III 28(48) (1993), no. 1, 167–176.[Mel]
S. Melikhov , Algebraic topology of Polish spaces. II: Axiomatic ho-mology, arXiv:1808.10243[Mil]
J. Milnor,
On the Steenrod homology theory, Mimeographed Note,Princeton, 1960, in: Novikov Conjectures, Index Theorems andRigidity, vol. 1, in: Lond. Math. Soc. Lect. Note Ser., vol. 226,Oberwolfach, 1993, pp. 79–96.[Skl]
E. G. Skljarenko,
On the homology theory associated withAleksandrov-Čech cohomology. (Russian) Uspekhi Mat. Nauk 34(1979), no. 6(210), 90–118.7[Sp]
E. H. Spanier,
Algebraic Topology. Corrected reprint of the 1966original. New York: Springer-Verlag, 1966.[St]