Kolmogorov duality

A duality in algebraic topology consisting in the isomorphism between the -dimensional homology group of a closed set in a locally compact Hausdorff space with zero - and -dimensional homology groups, and the -dimensional homology group of the complement , with Abelian coefficient group , as well as the isomorphism between the corresponding cohomology groups, with and .

The homology and cohomology groups involved in these isomorphisms are defined as follows. An -dimensional chain is taken to be any function of subsets of the space having compact closures that is skew-symmetric and additive in each of its arguments, takes values in , and is equal to zero when the intersection is empty. The boundary operator is defined by the formula where is any open subset of with compact closure containing . The cycles are the chains with zero boundaries, , and the cycles homologous to zero are the chains that are boundaries, . The group of all -dimensional cycles with the usual addition of functions contains the group of all -dimensional boundaries as a subgroup. The quotient group is the group . A.N. Kolmogorov always considered the group to be compact, and compactly topologized the homology group as well. However, the topology of the coefficient group has no effect on the structure of the homology group, and the homology can be taken over any group.

For the definition of cochains one considers the skew-symmetric functions of points of the space with values in such that there exists for each a finite system of pairwise-disjoint subsets of with compact closures and satisfying the condition if and belong to the same element of the system for any ; if at least one of the is not contained in any element of . The coboundary operator is defined by the formula  Two functions and are regarded as equivalent if each point in has a neighbourhood such that whenever all the belong to , and a cochain is taken to be an equivalence class of functions. The boundary of a cochain is defined as the class of coboundaries entering into this cochain of functions. A cocycle is a cochain with zero coboundary and a cocycle is cohomologous to zero if it is a coboundary, . The group of all -dimensional coboundaries is a subgroup of the group of all -dimensional cocycles; the quotient group then defines the group .

The homology and cohomology groups defined in this way, which is often referred to as the functional way, were introduced by Kolmogorov . He then went on to prove that, apart from the indicated duality isomorphism, there is a duality between the homology and cohomology groups and in the sense of the Pontryagin theory of characters, when the compact group is dual to the group , and the Poincaré duality where is an open -dimensional manifold, and are functional groups over a compact (or discrete) group (or ), and and are the cohomology (homology) groups of infinite cochains (finite chains) of an arbitrary cellular decomposition of .

In the case when is the -dimensional Euclidean space, one obtains from the above dualities the Pontryagin duality theorem (see Alexander duality). A special case of these dualities is the Steenrod duality theorem (see Duality in algebraic topology), since the Kolmogorov duality for homology is valid also for arbitrary coefficient groups .

The functional homology groups are isomorphic: to the Vietoris groups (see Vietoris homology) in the case of compact metric spaces and compact coefficient groups ; to the Aleksandrov spectral homology groups with respect to singular subcomplexes  in the case of locally compact spaces and compact coefficient groups  and, consequently, to the Aleksandrov–Čech homology groups of the one-point compactification of a given locally compact space ; and to the Steenrod homology groups  in the case of compact metric spaces and arbitrary groups . Thus, the Kolmogorov homology groups introduced four years earlier than those of Steenrod are a generalization of the latter to a wider class of spaces.

The functional homology and cohomology groups satisfy all the Steenrod–Eilenberg axioms on the category of locally compact spaces with admissible mappings (that is, when the pre-image of each compact set is compact)  and in addition the two Milnor axioms on the category of compact metric spaces .