ALEXANDER-SPANIER COHOMOLOGY OF K-TYPES

Abstract

It is well-known that it has been given many definitions for extend­ ing the cohomology theory of complexes to arbitrary topological spaces; the two theories most commonly used are the C ech and singular theories, [19],[27], [37]. A different definition was given by J.W.Alexander, [9], who constructed cohomology group for a compact metric space using functions of sets of point in the space. With a suitable choice of coef­ ficient groups, the cohomology groups obtained by Alexander were the character groups of the Vietories homology groups, [12], [39]. Since for a compact metric space the C ech groups and the Vietories groups are identical, Alexander's construction gives the Cech cohomology groups for such a space. As the definition involving functions is simpler than the definitions in terms of covers usually used to obtain the Cech groups, a generalization of Alexander's definition applicable to arbitrary space is desirable. A.D.Wallace ,[40], has suggested such definition which is valid for any space and is even some what simpler than Alexander's definitions. E.Spanier in his paper, [36], has concerned with the question of how the groups defined by Wallace compare with the cohomology groups ob­ tained by different constructions. The new groups agree with the C ech cohomology groups on the category of compact Hausdorff spaces. This implies that they differ from the singular cohomlogy groups as it is known that there exists compact Hausdorff space on which the C ech and singular theories differ. For compact spaces the Spanier's groups don't agree with the Cech groups. Since the singular groups of infinite locally finite com­ plexes agree with those based on infinite cochains, [36], the new groups coinicide with the singular ones for such spaces. Thus the Spanier's definition gives rise to an extension of cohomology from polyhedra to general spaces which is distinct from the known extensions. At the present time the resulting cohomology theory is called Alexan- der -Spanier cohomology theory, denoted by ff* . For deeper properties •• of the Alexander-Spanier theory, E.Spanier has introduced the cohomol- ogy of a space with coefficients in a presheaf, [37]. By using the gen- eral properties of this cohomology it is proved that on the category of paracompact spaces the Alexander-Spannier and C ech cohomologies are isomorphic.