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The inverse limit
 
The inverse limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864301.png" /></td> </tr></table>
+
$$
 +
H ^ { {size }  - 3 \lor } {} _ {n} ( X; G)  = \lim\limits _  \leftarrow  H _ {n} ( \alpha ; G)
 +
$$
  
of homology groups with coefficients in the Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864302.png" /> of nerves of open coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864303.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864304.png" /> (also called Čech homology, or Aleksandrov–Čech homology). For a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864305.png" />, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864306.png" /> can be defined in a similar way using the subsystems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864307.png" /> of all those subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864308.png" /> having non-empty intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s0864309.png" />. The inverse limit of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643010.png" /> is called the spectral homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643011.png" /> of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643012.png" />.
+
of homology groups with coefficients in the Abelian group $  G $
 +
of nerves of open coverings $  \alpha $
 +
of a topological space $  X $(
 +
also called Čech homology, or Aleksandrov–Čech homology). For a closed set $  A\subset  X $,  
 +
the groups $  H ^ { {size }  - 3 \lor } {} _ {n} ( A;  G) $
 +
can be defined in a similar way using the subsystems $  \alpha  ^  \prime  \subset  \alpha $
 +
of all those subsets of $  \alpha $
 +
having non-empty intersection with $  A $.  
 +
The inverse limit of the groups $  H _ {n} ( \alpha , \alpha  ^  \prime  ;  G) $
 +
is called the spectral homology group $  H ^ { {size }  - 3 \lor } {} _ {n} ( X, A;  G) $
 +
of the pair $  ( X, A) $.
  
Since the inverse limit functor does not preserve exactness, the homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643013.png" /> is, in general, not exact. It is semi-exact, in the sense that the composite of any two mappings in the sequence is equal to zero. For a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643014.png" /> the sequence turns out to be exact in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643015.png" /> is a compact group or field (or, more generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643016.png" /> is algebraically compact). The spectral homology of compact spaces is continuous in the sense that
+
Since the inverse limit functor does not preserve exactness, the homology sequence of the pair $  ( X, A) $
 +
is, in general, not exact. It is semi-exact, in the sense that the composite of any two mappings in the sequence is equal to zero. For a compact space $  X $
 +
the sequence turns out to be exact in the case when $  G $
 +
is a compact group or field (or, more generally, if $  G $
 +
is algebraically compact). The spectral homology of compact spaces is continuous in the sense that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643017.png" /></td> </tr></table>
+
$$
 +
H ^ { {size }  - 3 \lor } {} _ {n} \left ( \lim\limits _  \leftarrow  X _  \lambda  ; G
 +
\right )  = \lim\limits _  \leftarrow  H ^ { {size }  - 3 \lor } {} _ {n} ( X _  \lambda  ; G).
 +
$$
  
Lack of exactness is not the only deficiency of spectral homology. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643018.png" /> turn out to be non-additive, in the sense that the homology of a discrete union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643019.png" /> can be different from the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643020.png" />. This deficiency disappears if one considers the spectral homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643021.png" /> with compact support, defined as the direct limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643022.png" /> taken over all compact subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643023.png" />. It is natural to consider the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643024.png" />, in view of the fact that all the usual homologies (simplicial, cellular and singular) are homologies with compact support.
+
Lack of exactness is not the only deficiency of spectral homology. The groups $  H ^ { {size }  - 3 \lor } {} _ {n} $
 +
turn out to be non-additive, in the sense that the homology of a discrete union $  X = \cup _  \lambda  X _  \lambda  $
 +
can be different from the direct sum $  \sum _  \lambda  H ^ { {size }  - 3 \lor } {} _ {n} ( X _  \lambda  ;  G) $.  
 +
This deficiency disappears if one considers the spectral homology groups $  H _ {n}  ^ {c} ( X;  G) $
 +
with compact support, defined as the direct limit $  \lim\limits H ^ { {size }  - 3 \lor } {} _ {n} ( C;  G) $
 +
taken over all compact subsets $  C\subset  X $.  
 +
It is natural to consider the functor $  H ^ { {size }  - 3 \lor } {} _ {n}  ^ {c} $,  
 +
in view of the fact that all the usual homologies (simplicial, cellular and singular) are homologies with compact support.
  
The difference between the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643026.png" /> is one of the examples of how homology groups react to small changes in their initial definition (on the other hand, cohomology groups exhibit significant stability in this respect). Among the logically possible variants of the definition of homology groups in general categories of topological spaces, the correct one was not the first to be selected. The theory of the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643027.png" /> associated with the Aleksandrov–Čech cohomology achieved great recognition only in the 1960's (although the first definitions were given in the 1940's and 1950's). The theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643028.png" /> satisfies all the [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]] (and is a theory with compact supports). For compact spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643029.png" /> the following sequence is exact:
+
The difference between the functors $  H ^ { {size }  - 3 \lor } {} _ {n} $
 +
and $  H ^ { {size }  - 3 \lor } {} _ {n}  ^ {c} $
 +
is one of the examples of how homology groups react to small changes in their initial definition (on the other hand, cohomology groups exhibit significant stability in this respect). Among the logically possible variants of the definition of homology groups in general categories of topological spaces, the correct one was not the first to be selected. The theory of the homology groups $  H _ {*}  ^ {c} $
 +
associated with the Aleksandrov–Čech cohomology achieved great recognition only in the 1960's (although the first definitions were given in the 1940's and 1950's). The theory of $  H _ {*}  ^ {c} $
 +
satisfies all the [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]] (and is a theory with compact supports). For compact spaces $  X $
 +
the following sequence is exact:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643030.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  \lim\limits _  \leftarrow  {}  ^ {1}  H _ {n+} 1 ( \alpha ; G)  \rightarrow  H _ {n} ( X; G)  \rightarrow  H ^ { {size }  - 3 \lor } {} ( X; G)  \rightarrow  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643031.png" /> is the derived inverse limit functor. In general there is an epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643032.png" /> whose kernel is zero for any algebraically compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643033.png" />. For any locally compact space that is also homologically locally connected (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643034.png" />), the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086430/s08643037.png" /> are isomorphic.
+
where $  \lim\limits _  \leftarrow  ^ {1} $
 +
is the derived inverse limit functor. In general there is an epimorphism $  H _ {n}  ^ {c} ( X;  G) \rightarrow H ^ { {size }  - 3 \lor } {} _ {n}  ^ {c} ( X;  G) $
 +
whose kernel is zero for any algebraically compact group $  G $.  
 +
For any locally compact space that is also homologically locally connected (with respect to $  H _ {*}  ^ {c} $),  
 +
the functors $  H ^ { {size }  - 3 \lor } {} _ {n} $,  
 +
$  H ^ { {size }  - 3 \lor } {} _ {n}  ^ {c} $,  
 +
$  H _ {n}  ^ {c} $
 +
are isomorphic.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Eilenberg,  N.E. Steenrod,  "Foundations of algebraic topology" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.G. Sklyarenko,  "On homology theory associated with the Aleksandrov–Čech cohomology"  ''Russian Math. Surveys'' , '''34''' :  6  (1979)  pp. 103–137  ''Uspekhi Mat. Nauk'' , '''34''' :  6  (1979)  pp. 90–118</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.S. Massey,  "Homology and cohomology theory" , M. Dekker  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Eilenberg,  N.E. Steenrod,  "Foundations of algebraic topology" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.G. Sklyarenko,  "On homology theory associated with the Aleksandrov–Čech cohomology"  ''Russian Math. Surveys'' , '''34''' :  6  (1979)  pp. 103–137  ''Uspekhi Mat. Nauk'' , '''34''' :  6  (1979)  pp. 90–118</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.S. Massey,  "Homology and cohomology theory" , M. Dekker  (1978)</TD></TR></table>

Revision as of 08:22, 6 June 2020


The inverse limit

$$ H ^ { {size } - 3 \lor } {} _ {n} ( X; G) = \lim\limits _ \leftarrow H _ {n} ( \alpha ; G) $$

of homology groups with coefficients in the Abelian group $ G $ of nerves of open coverings $ \alpha $ of a topological space $ X $( also called Čech homology, or Aleksandrov–Čech homology). For a closed set $ A\subset X $, the groups $ H ^ { {size } - 3 \lor } {} _ {n} ( A; G) $ can be defined in a similar way using the subsystems $ \alpha ^ \prime \subset \alpha $ of all those subsets of $ \alpha $ having non-empty intersection with $ A $. The inverse limit of the groups $ H _ {n} ( \alpha , \alpha ^ \prime ; G) $ is called the spectral homology group $ H ^ { {size } - 3 \lor } {} _ {n} ( X, A; G) $ of the pair $ ( X, A) $.

Since the inverse limit functor does not preserve exactness, the homology sequence of the pair $ ( X, A) $ is, in general, not exact. It is semi-exact, in the sense that the composite of any two mappings in the sequence is equal to zero. For a compact space $ X $ the sequence turns out to be exact in the case when $ G $ is a compact group or field (or, more generally, if $ G $ is algebraically compact). The spectral homology of compact spaces is continuous in the sense that

$$ H ^ { {size } - 3 \lor } {} _ {n} \left ( \lim\limits _ \leftarrow X _ \lambda ; G \right ) = \lim\limits _ \leftarrow H ^ { {size } - 3 \lor } {} _ {n} ( X _ \lambda ; G). $$

Lack of exactness is not the only deficiency of spectral homology. The groups $ H ^ { {size } - 3 \lor } {} _ {n} $ turn out to be non-additive, in the sense that the homology of a discrete union $ X = \cup _ \lambda X _ \lambda $ can be different from the direct sum $ \sum _ \lambda H ^ { {size } - 3 \lor } {} _ {n} ( X _ \lambda ; G) $. This deficiency disappears if one considers the spectral homology groups $ H _ {n} ^ {c} ( X; G) $ with compact support, defined as the direct limit $ \lim\limits H ^ { {size } - 3 \lor } {} _ {n} ( C; G) $ taken over all compact subsets $ C\subset X $. It is natural to consider the functor $ H ^ { {size } - 3 \lor } {} _ {n} ^ {c} $, in view of the fact that all the usual homologies (simplicial, cellular and singular) are homologies with compact support.

The difference between the functors $ H ^ { {size } - 3 \lor } {} _ {n} $ and $ H ^ { {size } - 3 \lor } {} _ {n} ^ {c} $ is one of the examples of how homology groups react to small changes in their initial definition (on the other hand, cohomology groups exhibit significant stability in this respect). Among the logically possible variants of the definition of homology groups in general categories of topological spaces, the correct one was not the first to be selected. The theory of the homology groups $ H _ {*} ^ {c} $ associated with the Aleksandrov–Čech cohomology achieved great recognition only in the 1960's (although the first definitions were given in the 1940's and 1950's). The theory of $ H _ {*} ^ {c} $ satisfies all the Steenrod–Eilenberg axioms (and is a theory with compact supports). For compact spaces $ X $ the following sequence is exact:

$$ 0 \rightarrow \lim\limits _ \leftarrow {} ^ {1} H _ {n+} 1 ( \alpha ; G) \rightarrow H _ {n} ( X; G) \rightarrow H ^ { {size } - 3 \lor } {} ( X; G) \rightarrow 0 , $$

where $ \lim\limits _ \leftarrow ^ {1} $ is the derived inverse limit functor. In general there is an epimorphism $ H _ {n} ^ {c} ( X; G) \rightarrow H ^ { {size } - 3 \lor } {} _ {n} ^ {c} ( X; G) $ whose kernel is zero for any algebraically compact group $ G $. For any locally compact space that is also homologically locally connected (with respect to $ H _ {*} ^ {c} $), the functors $ H ^ { {size } - 3 \lor } {} _ {n} $, $ H ^ { {size } - 3 \lor } {} _ {n} ^ {c} $, $ H _ {n} ^ {c} $ are isomorphic.

References

[1] S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[2] E.G. Sklyarenko, "On homology theory associated with the Aleksandrov–Čech cohomology" Russian Math. Surveys , 34 : 6 (1979) pp. 103–137 Uspekhi Mat. Nauk , 34 : 6 (1979) pp. 90–118
[3] W.S. Massey, "Homology and cohomology theory" , M. Dekker (1978)
How to Cite This Entry:
Spectral homology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_homology&oldid=48760
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article