View source for Weak homology

An equivalence relation between cycles leading to the definition of the [[Spectral homology|spectral homology]] groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972101.png" />. It is known that the Steenrod–Sitnikov homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972102.png" /> of a compact space map epimorphically onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972103.png" />, and that the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972104.png" /> of this epimorphism is isomorphic to the first derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972105.png" /> of the inverse limit of the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972106.png" /> of the nerves of the open coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972107.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972108.png" />. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w0972109.png" /> were originally defined in terms of Vietoris cycles, and the cycles giving the elements of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w09721010.png" /> were called weakly homologous to zero. On the other hand, Vietoris cycles homologous to zero in the above definition of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w09721011.png" /> are sometimes called strongly homologous to zero (and the corresponding equivalence relation between them is called strong homology). In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w09721012.png" /> is a compact group or a field, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097210/w09721013.png" /> is equal to zero, and the concepts of strong and weak homology turn out to be equivalent.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Topological duality theorems II. Non-closed sets"  ''Trudy Mat. Inst. Steklov.'' , '''54'''  (1959)  pp. 3–136  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.S. Massey,   "Notes on homology and cohomology theory" , Yale Univ. Press  (1964)</TD></TR></table>