TAUTNESS AND APPLICATIONS OF THE ALEXANDER-SPANIER COHOMOLOGY OF K-TYPES

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TAUTNESS AND APPLICATIONS OF THE ALEXANDER-SPANIER COHOMOLOGY OF K-TYPES

ABD EL-SATTAR A. DABBOUR and ROLA A. HIJAZI (Received 5 February 1999)

Abstract.The aim of the present work is centered around the tautness property for the twoK-types of Alexander-Spanier cohomology given by the authors. A version of the con- tinuity property is proved, and some applications are given.

2000 Mathematics Subject Classification. Primary 55N05, 55N20, 55N35.

1. Introduction. It is well known that in the Alexander-Spanier cohomology theory [17,18] or in the isomorphictheory of ˇCech [9], if the coefficient groupGis topological then either the theory does not take into account the topology onG[9,18], or considers only the case whenGis compact to obtain a compact cohomology [4,1]. Continuous cohomology naturally arises when the coefficient group of a cohomology theory is topological [2,3,11]. The partially continuous Alexander-Spanier cohomology theory [14] can be considered as a variant of the continuous cohomology of a space with two topologies in the sense of Bott-Haefliger [15]; also it is isomorphicto the continuous cohomology of a simplicial space defined by Brown-Szczarba [2].

The idea ofK-groups [5,6], whereKis a locally-finite simplicial complex, is used to introduce theK-types of Alexander-Spanier cohomology with coefficients in a pair (G,G)of topological abelian groups [7,8]; namely,K-Alexander-Spanier and partially continuousK-Alexander-Spanier cohomologies ¯H_K^∗,H˜_K^∗. It is proved that theseK-types satisfied the seven Eilenberg-Steenrod axioms [9]; the excision axiom for the second K-type is verified for compact Hausdorff spaces when(G,G)are absolutely retract.

Therefore, the uniqueness theorem of the cohomology theory on the category of com- pact polyhedral pairs [9] asserts that our Alexander-Spanier K-types over a pair of absolute retract coefficient abelian groups are naturally isomorphic.

In the present work, we prove that theK-Alexander-Spanier cohomology of a closed subset in a paracompact space is isomorphic to the direct limit of theK-Alexander- Spanier cohomology of its neighborhoods, and that the partially continuous K- Alexander-Spanier cohomology of a neighborhood retract closed subspace of a Haus- dorff space is isomorphic to the direct limit of the partially continuousK-Alexander- Spanier cohomology of its neighborhoods. Also a version of the continuity property is proved. Moreover, we study some applications of theK-type cohomologies.

2. Alexander-Spanier cohomology of K-types. Here we mention the notations which we used throughout [7,8].

For an object(X,A)of the categoryQof the pairs of topological spaces and their continuous maps, denote byΩ(X,A)[Ω(X,A)]˜ the set of the pairs ¯α=(α,α), where

αis an open covering ofXandαis a subcollection ofαcoveringA[α=α∩A]; it is directed with respect to the refinement relation ¯α <β, that is,¯ α < βandα< β[9].

Denote byC^q(τ)(X)˜ the group of functionsϕ^τ: ˜X^q(τ)+1→G, whereτis a simplex in K,q(τ)=q+dimτ,q≥0, and ˜Xdenotes either a spaceXorα∈Ω(X). LetC^q(τ)(X)˜ be the subgroup of the direct product

τ∈KC^q(τ)(X)˜ consisting of suchϕ= {ϕ^τ} for which the condition(k)is satisfied, which states that there is a cofinite subset τ(ϕ)˘ ofK, that is,K−τ(ϕ)˘ is finite such that(ϕ^τ)⁻¹(G)=X˜^q(τ)+1,∀τ∈τ(ϕ). The˘ coboundaryδ^q:C^q(X)˜ →C^q+1(X)˜ is given by

δ^qϕ_τ

=

q(τ)+1

i=1

(−1)ⁱϕ^τp^(q(τ)+1)_i +(−1)^q(τ)+1

σ∈st(τ)

[σ:τ]ϕ^σ, (2.1)

where st(τ)= {σ ∈K:τ is(dimσ−1)-face ofσ},p^(τ)_i :X^τ+1→X^τ is the projection defined by: if ˆti is theτ-tuple consisting oft=(x0,...,xτ)∈X^τ+1withxi omitted, thenp1^(τ)(t)=ˆti, 0≤i≤τ. The cohomology groups of the cochain complexC^≠(X)= {C^q(X),δ^q}is, in general, uninteresting, as shown in the following theorem [8].

Theorem2.1. IfdimK=0, thenH^q(C^≠(X))G^∗K(the subgroup ofG^K=

τ∈KG^τ, G^τ=G, consisting of those elements having all but a finite number of theirτ-coordinates inG), andH^q(C^≠(X))=0, whenq≠0.

To pass to more interesting cohomology groups, the topology of the spaceXwill be used to define thatϕ∈C^q(X)is said to be K-locally zero on M⊆X if there is α∈ΩX(M) (the set of external covering ofM by open subsets of X) such thatϕ vanishes on α∩M, that is, eachϕ^τ vanishes on (α∩M)^q(τ)+1, whereα^τ= ∪{u^τ_α: uα∈α}. The subgroups ofC^q(X)consisting of those elements which areK-locally zero onX, A, respectively, are denoted byC₀^q(X),C^q(X,A). TheK-Alexander-Spanier cohomology of(X,A)over (G,G), denoted by ¯H_K^∗(X,A), is the cohomology of the quotient cochain complex ¯C_K^≠(X,A)=C^≠(X,A)/C₀^≠(X). Iff:(X,A)→(Y ,B)is inQ, β¯∈Ω(Y ,B)and ¯α=f⁻¹(β), then¯ f defines a cochain map ¯f^≠: ¯C_K^≠(Y ,B)→C¯_K^≠(X,A), where ˘τ(f^qϕ)=τ(ϕ)˘ for eachϕ∈C^q(Y ). In turn, ¯f^≠induces the homomorphism f¯^∗: ¯H_K^∗(Y ,B)→H¯_K^∗(X,A).

On the other hand, for ¯α∈Ω(X,A), denote by C_α^q_¯ the subgroup of Cα^q=C^q(α) consisting of thoseϕthat vanish onα∩A. Then we obtain a direct system{C_α^≠_¯}Ω(X,A)

such that any mapf∈Qconstitutes a mapF :{Cβ^≠¯}Ω(Y ,B)→ {Cα^≠¯}Ω(X,A)[9]; its limit isF^∞.

Theorem2.2. TheK-Alexander-Spanier cohomologyfunctor{H¯_K^∗,f¯^∗}is naturally isomorphic to the functor{lim→{H^∗(C_α^≠_¯)}Ω(X,A),F^∞∗}[7].

In the previous part, the topology on(G,G)plays no role; to pass to the second coho- mology ofK-type we characterize an elementϕ∈C^q(X)to beK-partially continuous if it is continuous on someα∈Ω(X), that is,ϕ^τ|α^q(τ)+1are continuous functions. Let L^q(X)be the group of all such elements, andM_K^≠(X)=L^≠(X)/C₀^≠(X). The subgroup ofCα^q, whereα∈Ω(X), consisting of theK-continuous elementsϕ, that is, ϕ^τ are continuous, is denoted byMα^q. Leti:AX, defineM_K^≠(X,A)to be the mapping cone of i^≠:M_K^≠(X)→M_K^≠(A), (see [13,18]), assuming thatM_K^q(X,A)=M_K^q(X)⊕M_K^q−1(A), and

the coboundary is^q(ϕ,ψ)=(−δ^qϕ,i^qϕ+δ^q−1ψ). The cohomology ofMK^≠(X,A)is the partially continuousK-Alexander-Spanier cohomology of(X,A)over the topolog- ical pair(G,G)of coefficient groups; it is denoted by ˜H_K^∗(X,A).

On the other hand, if ¯α∈Ω(X,A), then˜ idefines a cochain mapi^≠α:Mα^≠→M_α^≠; its mapping cone is denoted byM_α^≠_¯.

Theorem2.3. For a pair(X,A)∈QwithAclosed,M_K^≠(X,A)is naturallyisomorphic tolim→{M_α^≠_¯}Ω(X,A)˜ [7].

Theorem2.4. For a discrete space, andq≥0,H˜_K^q(X)H¯_K^q(X).

Proof. SinceX^q(τ)+1admits a discrete topology, it follows that eachτ-coordinate ϕ^τofϕ∈CK^q(X)is continuous [16]. ThenϕisK-partially continuous with respect to anyα∈Ω(X). Therefore,L^q(X)=C_K^q(X)andM_K^≠(X)=C¯_K^≠(X).

3. Tautness and continuity properties. This article is devoted to study the taut- ness property for both Alexander-Spanier cohomology ofK-types. One of its applica- tions is the continuity property.

The star of a subsetAin a spaceXwith respect toα∈Ω(X)is st(A,α)= ∪

Uα∈α:Ud∩A≠∅

. (3.1)

The star ofαis

α^∗= st

Uα,α

:uα∈α

. (3.2)

Definition3.1. Letα,β∈Ω(X), thenβis a star-refinement ofα, writtenα <^∗β ifα < β^∗.

Denote byᏺ(A)the collections of neighborhoods{N}ofAinX; it is directed down- ward by inclusion. IfN1< N2, then the inclusionπN1N2:N2N1induces the homo- morphisms ¯πN^∗1N2: ¯HK^q(N1)→H¯^qK(N2). AlsoiN :ANinduces ¯i^∗N : ¯HK^q(N)→H¯K^q(A), and they define a homomorphism

I^∞: lim→

H¯_K^q(N),π¯_N^∗_1N2

ᏺ(A) →H¯_K^q(A). (3.3) Theorem3.2(Tautness). A closed subspace of a paracompact space is a taut sub- space relative to theK-Alexander-Spanier cohomology, that is,I^∞ is an isomorphism for eachqand anypair(G,G)of coefficient groups.

Proof. (1)I^∞is an epimorphism. Leth∈H¯^qK(A)with representative ¯ϕ∈C¯K^q(A), written ash=[ϕ]. Let¯ ϕ∈C^q(A)such thatϕ∈ϕ. Then there is¯ α= {uα=να∩A: να⊆Xis open} ∈Ω(A)such that

δ^qϕ

|α^q(τ)+2=0. (3.4)

SinceAis closed, it follows thatβ= {να}∪{X−A} ∈Ω(X). The paracompactness ofXis equivalent to the existence of suchγ∈Ω(X)thatβ <^∗γ, and a neighborhood NofAand an extensionf:N→A(not necessarily continuous) of the identity map idA

ofA, that is,f iN=idA, such thatf (uγ∩N)⊆st(uγ,γ)for eachuγ∈γ[18]. One can show thatfdefines a cochain mapf^≠:C^≠(A)→C^≠(N)by(f^qϕ)^τ=ϕ^τf^(q(τ)+1)with

τ(f˘ ^qϕ)=τ(ϕ), where˘ f^(τ):N^τ→A^τgiven byf (x0,...,xτ−1)=(f (x0),...,f (xτ−1)).

The relationβ < γ^∗yields that for eachuγ∈γthere isuβ∈βsuch thatf (uγ∩N)⊆ st(uγ,γ)⊆uβ. Becausef (N)=A, thenf (uγ∩N)⊆uβ∩A⊆uα for someuα∈α.

By using (3.4), we get(δ^qf^qϕ)^τ|(γ∩N)^q(τ)+2=0, that is,δ^q(f^qϕ)∈C₀^q+1(N). Then f^qϕrepresents a cocyclef^qϕ∈C¯_K^q(N)which, in turn, defineshN ∈H¯_K^q(N), that is, hN=[f^qϕ]. Lett∈A^q(τ)+1, then

i^q_N

f^qϕτ(t)=ϕ^τf^(q(τ)+1)i^(q(τ)+1)_N (t)=ϕ^τ(t), (3.5) and therefore, ¯i^∗_NhN=[(f iN)^qϕ]=[ϕ]¯ =h.

(2)I^∞is a monomorphism. Leth1∈H¯_K^q(N1), ¯ϕ1∈C¯_K^q(N1)andϕ1∈C^q(N1)such thatϕ1∈ϕ¯1, ¯ϕ1∈h1, and[h1]∈KerI^∞.

First, one can consider that the neighborhoodN1ofAis a paracompact subset of X. For, ifN1is not so, then there is a paracompact subsetM1ofXsuch thatM1< N1

(e.g., takeM1=X) [10]. The inclusionπM1N1 induces an epimorphism ¯πM^≠1N1 [8], let

¯

π_M^q_1N1ψ¯1=ϕ¯1. Thus the cohomology class of ¯H_K^q(M1) represented by ¯ψ1 is [h1], which shows thatN1can be taken paracompact.

Now, ¯ϕ1∈Kerδ^q, or equivalently, there isα= {uα=να∩N1:να⊆Xis open} ∈

Ω(N1)such that

δ^qϕ1τ α^q(τ)+2=0. (3.6)

On the other hand, the assumption ¯i^∗_N1h1=0 asserts that there exists ¯ϕ∈C¯_K^q−1(A) such thati^q_N1ϕ1−δ^q−1ϕ∈C₀^q(A), whereϕ∈ϕ. This means that there is¯ β= {uβ= ωβ∩A:ωβ⊆Xis open} ∈Ω(A)such that

i^qN1ϕ1_τ

=

δ^q−1ϕ_τ

onβ^q(τ)+1. (3.7)

Assume thatβ1= {uβ1=ωβ∩N1} ∪ {N1−A}. The paracompactness ofN1asserts the existence ofγ1,γ2∈Ω(N1)for whichα <^∗γ1andβ1<^∗γ2. The directedness of Ω(N1)implies that there isγ∈Ω(N1)for whichγ1,γ2< γ; and so for eachuγ∈γ there areuγi∈γi,i=1,2 anduα∈α,uβ1∈β1such that

uγ⊂uγi⊆st uγi,γi

⊆uα∩uβ1. (3.8)

Then

st uγ,γ

⊆uα∩uβ1, (3.9)

that is,α,β1<^∗γ. According to [18, Lemma 6.6.1], there is a neighborhoodN2ofN1

andf:N2→A(not necessarily continuous) such thatf iN2=idA, anduβ1∈β1such that

f

uγ∩N2

⊆st uγ,γ

⊆uβ1⊆uβ1∩A=uβ. (3.10) Then, by (3.7), we get

δ^q−1f^q−1ϕ_τ

=

f^qi^q_N1ϕ1 on

γ∩N2_q(τ)+1

. (3.11)

DefineD^q:C^q+1(N1)→C^q(N2)by ift=

x0,...,xq(τ)

∈N₂^q(τ)+1andψ1∈C^q+1 N1

(3.12)

then

D^qψ1_τ (t)=

q(τ)

r=0

(−1)^γψ^τ₁

y0,...,yτ,zτ,...,zq(τ)

, (3.13)

where

yj=πN1N2

xj

, zj= iN1f

xj

=f (xj), (3.14)

and ˘τ(D^qψ1)=τ(ψ˘ 1). By a similar calculation as given in [7], we get δ^q−1D^q−1ϕ1_τ

=

f^qi^q_N1ϕ1_τ

−

π_N^q_1N2ϕ1_τ

−

D^qδ^qϕ1_τ

. (3.15)

By (3.9), (3.10) for eachuγ∈γ, there isuα∈αsuch that uγ∩N2

∪f

uγ∩N2

⊆uα. (3.16)

Then, by (3.6), (3.11), and (3.15) consequently, we have δ^q−1D^q−1ϕ1τ=

f^qi^q_N1ϕ1τ−

π_N^q_1N2ϕ1τ on

γ∩N2q(τ)+1, (3.17) and so

π_N^q_1N2ϕ1τ= δ^q−1

f^q−1ϕ−D^q−1ϕ1τ on

γ∩N2q(τ)+1. (3.18) Therefore

ψ2=f^q−1ϕ−D^q−1ϕ1∈C^q−1 N2

(3.19) such that

π_N^q_1N2ϕ1τ=

δ^q−1ψ2τ on

γ∩N2q(τ)+1, (3.20) that is, ¯πN1N2h1=0 which completes the proof.

Corollary3.3. Anyone-point subset of a paracompact is a taut subspace relative toH¯_K^∗.

The next part is devoted to studying the tautness property for ˜H^∗_K, which is also valid for ¯H_K^∗. The idea and results ofα-β-contiguous maps, introduced in [7] plays an essential role in this study.

The inclusionsπN1N2:N2N1, corresponding to the relationsN1< N2inᏺ(A), define the direct system {H˜_K^q(N),π˜_N^∗_1N2}. Also the inclusioniN :AN, where N∈ ᏺ(A), defines a map of direct systems [9]:

IN: H^q

M_α^≠ ,π˜_αβ^∗

Ω(N) → H^q

M_α^≠_˜ ,π˜_α^∗_˜β˜

Ω(A), (3.21)

whereα∈Ω(N), ˜α=i⁻¹N (α)=α∩A. On the other hand,{i˜^∗N}defines a homomor- phism

I˜^∞: lim→

H˜_K^q(N),π˜_N^∗_1N2

ᏺ(A) →H˜_K^qA. (3.22) Theorem3.4(Tautness). IfAis a closed subset in a Hausdorff spaceXsuch thatA is a neighborhood retract, thenAis a taut subspace relative to the cohomologyH˜_K^∗.

Proof. (1) ˜I^∞is an epimorphism. Leth∈H˜K^q(A), without loss of generality, the neighborhood retractness ofAinXyields thatAhas an open neighborhoodU(inX) such thatU⊆N and a retractionτ1:U→A(ifU1is an open neighborhood ofAof whichAis retract butU1N, takeU=U1∩IntN). LetiU:AUthen, ˜I^∞[˜τ1^∗(h)]=

˜i^∗_U(˜τ₁^∗h)=id^∗_A(h)=h.

(2) ˜I^∞is a monomorphism. Let[h]∈Ker˜I^∞. It is sufficient to constructV∈ᏺ(A) satisfyingN < V and ˜π_NV^∗ h=0. Since the cohomology functor commutes with the direct limit [18]. Theorem 2.3 asserts that one may assume that h belongs to lim→{H^q(Mα^≠),π˜_αβ^∗ }Ω(N)with representativehα∈H^q(Mα^≠), where

α=

uα=ωα∩N:ωα⊆Xis open

∈Ω(N). (3.23) Letα1= {ωα}∪{X−A}, ˜α=α1∩A,

β=

uβ=τ₁⁻¹ uα˜

∩ uα∩U

:φ≠uα˜∈α˜

, (3.24)

V = ∪uβ, τ=τ1|V:VA, andα=α1∩V. Then ˜α∈Ω(A),α=α∩V ∈Ω(V ), uα˜⊆uβ for eachuα˜≠φ,βis a family of open subsets inUand so open inX,V is an open neighborhood ofAsuch thatV ⊆U, andβ∈Ω(V ). Since uβ=uβ∩uα⊆ V∩uα=uα, it follows thatα< β. Alsoα∩A=α∩A=α˜and j⁻¹β=α, where˜ j:AV. If4:VN, and[ϕ]∈H^q(M_α^≠), then

j˜^∗_βπ˜_α^∗β4˜^∗_α[ϕ]=j˜_β^∗ ϕ^τα^q(τ)+1β^q(τ)+1

= ϕ^τα˜^q(τ)+1

, (3.25)

that is,

j˜^∗_βπ˜_α^∗β4˜^∗_α=˜i^∗_N,α, (3.26) where ˜i^≠_N,α:M_α^≠→M_α^≠_˜ is induced byiN:AN.

On the other hand,(jτ)uβ⊆uβand sojτ, idV:V→Vareβ−β-contiguous [7].

It follows that(idV)^q_β−β,(jr ) ^q_β−β:M_β^q→M_β^qare cochain homotopic [7]. Then(idV)^∗_β−β

=(jr ) ^∗_β−β =r˜_α−β˜^∗ j˜^∗_β, which yields that ˜j_β^∗ is a monomorphism. Because ˜i^∗_N,αhα=0, equation (3.26) yields that ˜π_α^∗β4˜^∗_αhα=0. Since ˜4^∗_αhα,π˜_α^∗β(4˜αhα)represent the zero element of lim→{H^q(M_α^≠),π˜_αβ^∗}Ω(N), it follows that ˜π_NV^∗ h=[4˜^∗_αhα]=0.

The rest of this article is centered around a special case of the continuity property for ¯H_K^∗. As an application of the continuity property the cohomology groups satisfy a much stronger form of the excision axiom.

The following results can be deduced from those given in [9].

Lemma3.5. LetXbe the intersection of a nested system{Xα,πβα}Λ, then (i) Xandlim←{Xα,πβα}Λare homeomorphic.

(ii) If the nested system consists of compact Hausdorff spaces thenX is a closed subset of eachXα.

(iii) IfNis an open neighborhood ofXinXα(for someα∈Λ), then there isβ > αin Λsuch thatXβ⊆N.

The inclusionsiα:XXαdefine a map I:H¯_K^q

Xα ,π¯_αβ^∗

Λ →H¯_K^q(X), (3.27)

its direct limit is denoted by¯I^∞.

Theorem 3.6(weak continuity). If X is the intersection of a nested system {Xα, πβα}Λof compact Hausdorff spaces, then¯I^∞is an isomorphism.

Proof. Since eachXαis a paracompact Hausdorff space [10] andXαis closed inX (Lemma 3.5), it follows, byTheorem 3.2, thatXis a taut subspace inXαrelative to ¯H_K^∗. (1) ¯I^∞ is an epimorphism. Leth∈H¯_K^q(X), then, according to Theorem 3.2, there exists an open neighborhoodNofXinXαandhN∈H¯_K^q(N), such that ¯i^∗_N(hN)=h. By Lemma 3.5, there isβ > αinΛsuch thatXβ⊆N. Letiβ:XXβ,jβ:XβN. Because

¯i^∗_β(j¯_β^∗hN)=(jβiβ)^∗hN=¯i^∗_NhN=h, then ¯I^∞[j¯_β^∗hN]=h.

(2) ¯I^∞is a monomorphism. Let[hα]∈Ker¯I^∞, that is, ¯i^∗_αhα=0. The tautness ofX inXαyields, byTheorem 3.2, an open neighborhoodNofXinXαsuch thathNis the unique element for which ¯i^∗_NhN=0, wherei_N:XN. Because ¯i^∗_N(¯i^∗_Nhα)=¯i^∗_αhα=0, then ¯i^∗_Nhα=0. Letβ > αinΛsuch thatXβ⊆N, then ¯π_αβ^∗ hα=(iNiβ)^∗hα=j¯_β^∗(¯i^∗_Nhα)= 0, that is,[hα]=0.

4. Applications. One of the good applications of the Alexander-Spanier cohomol- ogy ofK-types is the study of the 0-dimensional cohomology groups and their relation with the connectedness of the space [7]. In this article, two applications are given. In a next work, we hope to give more applications. As a first application, we define the par- tially continuousK-Alexander-Spanier cohomology of an excision map and calculate its value for some dimensions.

Let ˜f^≠:MK^≠(Y ,B)→MK^≠(X,A)be the cochain map induced by the mapfinQ. Define M_K^≠(f )to be the mapping cone of ˜f^≠by

M_K^q(f )=M_K^q(Y ,B)⊕M_K^q−1(X,A)

=MK^q(Y )⊕MK^q−1(B)⊕MK^q−1(X)⊕MK^q−2(A), (4.1) and the coboundary is

˜^q

ϕ2,ψ2,ϕ1,ψ1

=

−˜^q ϕ2,ψ2

,^q ϕ1,ψ1

+f˜^q

ϕ2,ψ2

=

δ^qϕ2,−i˜^qϕ2−δ^q−1ψ2,−δ^q−1ϕ1+f˜^qϕ2,˜i^q−1ϕ1+δ^q−2ψ1+_f|A^q−1_ψ2. (4.2) Then there is a short exact sequence

0 →M^+≠_K(X,A) ^λ≠→M_K^≠(f ) ^x≠→M¯_K^≠(Y ,B) →O₂, (4.3) whereλ^≠,χ^≠are injection, projection, respectively;M^+≠(X,A)is the complexMK^≠(X,A) with the dimensions all raised by one, and ¯M^≠(Y ,B)is the complexM^≠(Y ,B)with the

sign of the coboundary changed [12]. Note thatH^q(M¯K^≠(Y ,B))=H˜K^q(Y ,B). LetVbe an open subset ofXsuch that ¯V⊆IntA,B=X−V, andC=A−V. Put the excision map e:(B,C)(X,A)in (4.3) instead off, and then apply the cohomology functor to get the long exact sequence

··· →H˜_K^q(e) ^χ˜∗→H˜_K^q(X,A) ^˜e∗→H˜_K^q(B,C) ^λ˜∗→H˜_K^q+1(e)→ ···. (4.4) Thus the groups ˜H_K^q(e),H˜_K^q+1(e)measure how much the cohomological groups deviate from the excision axiom.

Theorem4.1. IfdimK=0,e:(B,C)(X,A)is an excision map, whereAis closed and(G,G)anypair of topological abelian groups, thenH˜K^q(e)=0whenq=0orq=1.

Proof. (1) Caseq=0. We have

M_K⁰(e)=M_K⁰(X,A)=M_K⁰(X)=L⁰_K(X). (4.5) Letϕ∈M_K⁰(e)such that ˜ϕ =0, then ˜i⁰ϕ=0, ˜eϕ=0. Then ϕ=0 [7], which means that Ker ˜⁰=0.

(2) Caseq=1. We have

MK¹(e)=MK(X)⊕L⁰(A)⊕L⁰(B). (4.6) It is sufficient to show that Ker ˜¹⊆Im ˜⁰. Let(ϕ2,ψ2,ϕ1,0)∈Ker ˜¹, then

δ¹ϕ=0, ˜iϕ2= −δ⁰ψ2, (4.7)

˜

e¹ϕ2=δ⁰ϕ1, (4.8)

˜ e⁰₁

−ψ2

=jϕ˜ 1, (4.9)

wherei:AX,j:CBande1=e|C.

By (4.9), there existsϕ∈M_K⁰(X)[7] such that

˜i⁰ϕ= −ψ2, e˜⁰ϕ=ϕ1. (4.10) By (4.8), (4.9), and (4.10), we get

˜i¹

δ⁰ϕ−ϕ2

=0, ˜e¹

δ⁰ϕ−ϕ2

=0. (4.11)

Thenδ⁰ϕ=ϕ2 [7], which together with (4.11) yield (ϕ,0,0,0)∈M_K⁰(e)such that ˜⁰(ϕ,0,0,0)=(ϕ2,ψ2,ϕ1,0).

Combining the sequence (4.4) and the above theorem, we get the following result.

Corollary4.2. Under the assumptions ofTheorem 4.1, the mape˜^∗0: ˜H_K⁰(X,A)→ H˜_K⁰(B,C)is an isomorphism bute˜^∗1is a monomorphism.

Next we give a second application to the work introduced in this paper.

Letη:(G,G)→(F,F)be a homeomorphism of pairs of (discrete) abelian groups, which is an epimorphism,(L,L)=Kerηandλ:(L,L)(G,G). Then for each ¯α∈ Ω(X,A), the mapsη,λdefine naturally a short exact sequence

0 →C^q

¯ α,L,L

→C^q

¯ α;G,G

→C^q

¯ α;F,F

→0; (4.12)

its cohomology is a long exact sequence [12] denoted by Sα¯. One can show that {Sα¯}Ω(X,A)is a direct system, its direct limit [7,8] is

··· →H¯_K^q−1

X,A;F,F

→H¯_K^q

X,A;L,L

→H¯_K^q

X,A;G,G

→H¯_K^q

X,A,F,F

→H¯_K^q+1

X,A;L,L

→ ···. (4.13) Now instead ofF take the factor groupG/Gand soFwill be the null subgroup of G/G. Then the above sequence yields the following result.

Theorem4.3. Consider(X,A)has a trivial(q−1)-dimensional spaceK-Alexander- Spanier cohomologygroup with finite cochains, and a trivial (q+1)-dimensionalK- Alexander-Spanier cohomologywith infinite cochains, taken over the coefficient groups G/GandG, respectively. Then the groupH¯K^q(X,A;G,G)defined over an arbitrary pair(G,G)of coefficient groups is the extension of the cohomologygroupH¯_K^q(X,A;G) with infinite cochains overGbythe groupH¯_K^q(X,A,G/G)with finite cochains over G/G.

Acknowledgement. The first author would like to thank the Abdus Salam Inter- national Centre for Theoretical Physics, Trieste, Italy, for hospitality.

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Abd El-Sattar A. Dabbour: Department of Mathematics, Faculty of Science, Ain-Shams University, Abbasia, Cairo, Egypt

Current address:The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

E-mail address:[email protected]

Rola A. Hijazi: Department of Mathematics, Faculty of Science, KingAbdulaziz Uni- versity, P.O. Box14466, Jeddah21424, Saudi Arabia