# 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) |

**How to Cite This Entry:**

Weak homology.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Weak_homology&oldid=49181