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Difference between revisions of "Weak homology"

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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.
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An equivalence relation between cycles leading to the definition of the [[Spectral homology|spectral homology]] groups $  \check{H}  _ {p} ( C;  G) $.  
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It is known that the Steenrod–Sitnikov homology groups $  H _ {p} ( C;  G) $
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of a compact space map epimorphically onto $  \check{H}  _ {p} ( C;  G) $,  
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and that the kernel $  K $
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of this epimorphism is isomorphic to the first derived functor $  \lim\limits _  \leftarrow  {}  ^ {1} $
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of the inverse limit of the homology groups $  H _ {p} ( \alpha ;  G) $
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of the nerves of the open coverings $  \alpha $
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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====
 
====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>
 
<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>

Latest revision as of 08:28, 6 June 2020


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=13822
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article