Weak homology

An equivalence relation between cycles leading to the definition of the spectral homology groups $ \check{H} _ {p} ( C; G) $. It is known that the Steenrod–Sitnikov homology groups $ H _ {p} ( C; G) $ of a compact space map epimorphically onto $ \check{H} _ {p} ( C; G) $, and that the kernel $ K $ of this epimorphism is isomorphic to the first derived functor $ \lim\limits _ \leftarrow {} ^ {1} $ of the inverse limit of the homology groups $ H _ {p} ( \alpha ; G) $ of the nerves of the open coverings $ \alpha $ of the space $ C $. The groups $ H _ {p} $ were originally defined in terms of Vietoris cycles, and the cycles giving the elements of the subgroup $ K \subset H _ {p} ( C; G) $ were called weakly homologous to zero. On the other hand, Vietoris cycles homologous to zero in the above definition of the groups $ H _ {p} $ are sometimes called strongly homologous to zero (and the corresponding equivalence relation between them is called strong homology). In the case when $ G $ is a compact group or a field, the kernel $ K $ 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)