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Weak homology

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An equivalence relation between cycles leading to the definition of the spectral homology groups . It is known that the Steenrod–Sitnikov homology groups of a compact space map epimorphically onto , and that the kernel of this epimorphism is isomorphic to the first derived functor of the inverse limit of the homology groups of the nerves of the open coverings of the space . The groups were originally defined in terms of Vietoris cycles, and the cycles giving the elements of the subgroup were called weakly homologous to zero. On the other hand, Vietoris cycles homologous to zero in the above definition of the groups are sometimes called strongly homologous to zero (and the corresponding equivalence relation between them is called strong homology). In the case when is a compact group or a field, the kernel is equal to zero, and the concepts of strong and weak homology turn out to be equivalent.

References

[1] P.S. Aleksandrov, "Topological duality theorems II. Non-closed sets" Trudy Mat. Inst. Steklov. , 54 (1959) pp. 3–136 (In Russian)
[2] W.S. Massey, "Notes on homology and cohomology theory" , Yale Univ. Press (1964)
How to Cite This Entry:
Strong homology. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Strong_homology&oldid=38812