THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS TAUTNESS AND APPLICATIONS OF THE ALEXANDER-SPANIER COHOMOLOGY OF K-TYPES

The aim of the present work is centered around the tautness property for the two K-types of Alexander-Spanier cohomology given by the authors. A version of the continuity property is proved, and some applications are given. MIRAMARE TRIESTE September 1998 Regular Associate of the Abdus Salam ICTP. E-mail (c/o): sherif@sunet.shams.eg


Introduction
It is well-known that in the Alexander-Spanier cohomology theory [17], [18] or in the isomorphic theory of Cech [9], if the coefficient group G is topological then either the theory does not take into account the topology on G [9], [18], or considers only the case when G is compact to obtain a compact cohomology [5], [8]. Continuous cohomology naturally arises when the coefficient group of a cohomology theory is topological [6], [7], [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 isomorphic to the continuous cohomology of a simplicial space defined by Brown-Szczarba [6].
The idea of K-groups [1], [2] where K is a locally-finite simplicial complex, is used to introduce the K-types of Alexander-Spanier cohomology with coefficients in a pair (G, G') of topological abelian groups [3], [4]; namely, K-Alexander-Spanier and partially continuous K-Alexander-Spanier cohomologies H* K , H* K . It is proved that these K-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 compact polyhedral pairs [9], asserts that our Alexander-Spanier K-types over a pair of absolute retract coefficient abelian groups are naturally isomorphic.
The present work is centered around the tautness property for the Alexander-Spanier Ktypes cohomology. Roughly speaking, we prove that the K-Alexander-Spanier cohomology of a closed subset in a paracompact space is isomorphic to the direct limit of the K-Alexander-Spanier cohomology of its neighborhoods; and that the partially continuous K-Alexander-Spanier cohomology of a neighborhood retract closed subspace of a Hausdorff space is isomorphic to the direct limit of the partially continuous K-Alexander-Spanier cohomology of its neighborhoods.
Also a version of the continuity property is proved. Moreover, we study some application of the if-type cohomologies.

Alexander-Spanier Cohomology of K Types
Here we mention the notations which will be used throughout the present work [3], [4].
The cohomology groups of the cochain complex C^(X) = {C q (X), δ q } is, in general, uninteresting, as shown in the following theorem [3]. To pass to more interesting cohomology groups, the topology of the space X will be used Proof. Since X q(τ ) +1 admits a discrete topology, it follows that each τ-coordinate <p T of <p G CK q (X) is continuous [16]. Then t £> is K-partially continuous with respect to any α G Vt{X).

Tautness and Continuity Properties
This article is devoted to study the tautness property for both Alexander-Spanier cohomology of K-types. One of its applications is the continuity property.
The star of a subset A in a space X with respect to α G Q(X) is The star of α is Theorem 2.1 (Tautness). A closed subspace of a paracompact space is a taut subspace relative to the K-Alexander-Spanier cohomology, i.e. /°° is an isomorphism for each q and any pair

Corollary 2.2. Any one-point subset of a paracompact is a taut subspace relative to H^.
The next part is devoted to study the tautness property for H* K , which is also valid for H* K .
The idea and results of α -β-contiguous maps, introduced in [4] plays an essential role in this study. Since the cohomology functor commutes with the direct limit [18]. Theorem 1.3 asserts that one may assume that h belongs to \im{H 9  where i^ a : Mα ->• M^ is induced by iN : A "-^ N.
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]. Lemma 2.4. Let X be the intersection of a nested system {X Α ,Π Β Α}Λ, then (i) X and lim{X α ,π β α}Λ are homeomorphic (ii) If the nested system consists of compact Hausdorff spaces then X is a closed subset of each X α .
(iii) If N is an open neighborhood of X in X α (for some aeA), then there is β > α in Λ such that Xβ QN.
The inclusions i α : X <->• X α define a map its direct limit is denoted by J°°. Proof. Since each X α is a paracompact Hausdorff space [10] and X α is closed in X (Lemma 2.4), it follows, by Theorem 2.1, that X is a taut subspace in X α relative to H^.

Applications
One of the good applications of the Alexander-Spanier cohomology of K-types is the study of the 0-dimensional cohomology groups and their relation with the connectedness of the space [4].
In this article two applications are given. In a next work we hope to give more applications.
The first application is concentrated to define the partially continuous K-Alexander-Spanier cohomology of an excision map and calculate its value for some dimensions. in (3.1) instead of f, and then apply the cohomology functor, we get the long exact sequence: Thus the groups ii q K (e), H^l(e) measure how much the cohomological groups deviate from the excision axiom. Then 6°<p = tp 2 [4], which with (3.6) yield that (</?, 0,0,0) e MK0(e) such that A°(<p, 0,0,0) = Combining the sequence (3.2) and the above theorem, we get the following result. is an isomorphism but e* 1 is a monomorphism: The second application is to give attention in our work to use a pair of coefficients groups, an arbitrary locally-finite simplicial complex K, and the condition (k).
Let η : (G, G') )> (F, F') be a homeomorphism of pairs of (discrete) abelian groups which its cohomology is a long exact sequence [12] denoted by S&-One can show that {Sa}ci(x,A) is a direct system, its direct limit [3], [4] .