Kolmogorov duality

A duality in algebraic topology consisting in the isomorphism

$$ H _ {r} ( A , G ) \sim \ H _ {r+1} ( R \setminus A , G ) $$

between the $ r $- dimensional homology group $ H _ {r} ( A , G ) $ of a closed set $ A $ in a locally compact Hausdorff space $ R $ with zero $ r $- and $ ( r + 1 ) $- dimensional homology groups, and the $ ( r + 1 ) $- dimensional homology group of the complement $ R \setminus A $, with Abelian coefficient group $ G $, as well as the isomorphism

$$ H ^ {r } ( A , G ) \sim \ H ^ {r+ 1 } ( R \setminus A , G ) $$

between the corresponding cohomology groups, with $ H ^ {r } ( R , G ) = 0 $ and $ H ^ {r+1} ( R , G ) = 0 $.

The homology and cohomology groups involved in these isomorphisms are defined as follows. An $ r $- dimensional chain is taken to be any function $ c _ {r} ( e _ {0} \dots e _ {r} ) $ of $ r + 1 $ subsets of the space $ R $ having compact closures that is skew-symmetric and additive in each of its arguments, takes values in $ G $, and is equal to zero when the intersection $ \overline{e}\; _ {0} \cap \dots \cap \overline{e}\; _ {r} $ is empty. The boundary operator $ \partial $ is defined by the formula

$$ \partial c _ {r} ( e _ {0} \dots e _ {r-1} ) = \ c _ {r} ( U , e _ {0} \dots e _ {r-1} ) , $$

where $ U $ is any open subset of $ R $ with compact closure containing $ \overline{e}\; _ {0} \cup \dots \cup \overline{e}\; _ {r-1}] $. The cycles are the chains $ c _ {r} $ with zero boundaries, $ \partial c _ {r} = 0 $, and the cycles homologous to zero are the chains $ c _ {r} $ that are boundaries, $ c _ {r} = \partial c _ {r+1} $. The group $ Z _ {r} ( R , G ) $ of all $ r $- dimensional cycles with the usual addition of functions contains the group $ B _ {r} ( R , G ) $ of all $ r $- dimensional boundaries as a subgroup. The quotient group $ Z _ {r} ( R , G ) / B _ {r} ( R , G ) $ is the group $ H _ {r} ( R , G ) $. A.N. Kolmogorov always considered the group $ G $ 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 $ f ^ { r } ( x _ {0} \dots x _ {r} ) $ of $ r + 1 $ points $ x _ {0} \dots x _ {r} $ of the space $ R $ with values in $ G $ such that there exists for each $ f ^ { r } $ a finite system $ S _ {f ^ { r } } $ of pairwise-disjoint subsets of $ R $ with compact closures and satisfying the condition $ f ^ { r } ( x _ {0} \dots x _ {r} ) = f ^ { r } ( \widetilde{x} _ {0} \dots \widetilde{x} _ {r} ) $ if $ x _ {i} $ and $ \widetilde{x} _ {i} $ belong to the same element of the system $ S _ {f ^ { r } } $ for any $ i $; $ f ^ { r } ( x _ {0} \dots x _ {r} ) = 0 $ if at least one of the $ x _ {i} $ is not contained in any element of $ S _ {f ^ { r } } $. The coboundary operator $ \delta $ is defined by the formula

$$ \delta f ^ { r } ( x _ {0} \dots x _ {r+1} ) = $$

$$ = \ \sum_{i=0}^ { r+1} ( - 1 ) ^ {i} f ^ { r } ( x _ {0} \dots x _ {i-1} , x _ {i+1} \dots x _ {r+1} ) . $$

Two functions $ f _ {1} ^ { r } $ and $ f _ {2} ^ { r } $ are regarded as equivalent if each point in $ R $ has a neighbourhood $ U $ such that $ f _ {1} ^ { r } ( x _ {0} \dots x _ {r} ) = f _ {2} ^ { r } ( x _ {0} \dots x _ {r} ) $ whenever all the $ x _ {i} $ belong to $ U $, 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 $ c ^ {r } $ with zero coboundary and a cocycle is cohomologous to zero if it is a coboundary, $ c ^ {r } = \delta c ^ {r- 1 } $. The group $ B ^ {r } ( R , G) $ of all $ r $- dimensional coboundaries is a subgroup of the group $ Z ^ {r } ( R , G ) $ of all $ r $- dimensional cocycles; the quotient group $ Z ^ {r} ( R , G ) / B ^ {r } ( R , G ) $ then defines the group $ H ^ {r } ( R , G ) $.

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 $ H _ {r} ( R , G ^ {*} ) $ and $ H ^ {r } ( R , G ) $ in the sense of the Pontryagin theory of characters, when the compact group $ G ^ {*} $ is dual to the group $ G $, and the Poincaré duality

$$ H _ {r} ( R , G ^ {*} ) \sim \ \widetilde{H} {} ^ {n-r} ( R , G ^ {*} ) ,\ \ H ^ {r } ( R , G ) \sim \ \widetilde{H} _ {n-r} ( R , G ) , $$

where $ R $ is an open $ n $- dimensional manifold, $ H _ {r} ( R , G ^ {*} ) $ and $ H ^ {r } ( R , G ) $ are functional groups over a compact (or discrete) group $ G ^ {*} $( or $ G $), and $ \widetilde{H} {} ^ {n-r} ( R , G ^ {*} ) $ and $ \widetilde{H} _ {n-r} ( R , G ) $ are the cohomology (homology) groups of infinite cochains (finite chains) of an arbitrary cellular decomposition of $ R $.

In the case when $ R $ is the $ n $- 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 [2].

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 [3] in the case of locally compact spaces and compact coefficient groups [4] and, consequently, to the Aleksandrov–Čech homology groups of the one-point compactification of a given locally compact space [5]; and to the Steenrod homology groups [8] in the case of compact metric spaces and arbitrary groups [7]. 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) [6] and in addition the two Milnor axioms on the category of compact metric spaces [7].

References

[1a]	A.N. Kolmogorov, "Les groupes de Betti des espaces localement bicompacts" C.R. Acad. Sci. , 202 (1936) pp. 1144–1147
[1b]	A.N. Kolmogorov, "Propriétés des groupes de Betti des espaces localement bicompacts" C.R. Acad. Sci. , 202 (1936) pp. 1325–1327
[1c]	A.N. Kolmogorov, "Les groupes de Betti des espaces métriques" C.R. Acad. Sci. , 202 (1936) pp. 1558–1560
[1d]	A.N. Kolmogorov, "Cycles relatifs. Théorème de dualité de M. Alexander" C.R. Acad. Sci. , 202 (1936) pp. 1641–1643
[2]	L.D. Mdzinarishvili, "On Kolmogorov's duality law" Soviet Math. Dokl. , 15 (1974) pp. 840–843 Dokl. Akad. Nauk SSSR , 216 : 3 (1974) pp. 502–504
[3]	P.S. Aleksandrov, "General theory of homology" Uchen. Zap. Moscov. Gos. Univ. , 45 (1940) pp. 3–60 (In Russian)
[4]	G.S. Chogoshvili, "On the equivalence of functional and spectral homology theory" Izv. Akad. Nauk SSSR. Ser. Mat. , 15 : 5 (1951) pp. 421–438 (In Russian)
[5]	N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[6]	M.B. Balavadze, "On Kolmogorov's duality theory" Trudy Tbilisi Mat. Inst. , 41 (1972) pp. 5–40 (In Russian)
[7]	L.D. Mdzinarishvili, "The equivalence of Kolmogorov's and Steenrod's homology theories" Trudy Tbilisi Mat. Inst. , 41 (1972) pp. 143–163 (In Russian) (French and Georgian summaries)
[8]	N. Steenrod, "Regular cycles of compact metric spaces" Ann. of Math. , 41 (1940) pp. 833–851

Comments

The homology and cohomology groups defined above are also known as the Kolmogorov–Alexander, or Alexander–Kolmogorov, homology groups (, [a1]) and the Kolmogorov–Spanier cohomology groups (, [a2]).

References

[a1]	J.W. Alexander, "On the chains of a complex and their duals" Proc. Nat. Acad. Sci. USA , 21 (1935) pp. 509–512
[a2]	E.H. Spanier, "Cohomology theory for general spaces" Ann. of Math. , 49 (1948) pp. 407–427
[a3]	J.W. Milnor, "On axiomatic homology theory" Pacific J. Math. , 12 (1962) pp. 337–341