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The homology groups (cf. [[Homology group|Homology group]])
 
The homology groups (cf. [[Homology group|Homology group]])
  
<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/l/l060/l060150/l0601501.png" /></td> </tr></table>
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$$
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H _ {p}  ^ {x}  = H _ {p}  ^ {c} ( X , X \setminus  x ; G ) ,
 +
$$
  
defined at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l0601502.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l0601503.png" /> is [[Homology with compact support|homology with compact support]]. These groups coincide with the direct limits
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defined at points $  x \in X $,  
 +
where $  H _ {p}  ^ {c} $
 +
is [[Homology with compact support|homology with compact support]]. These groups coincide with the direct limits
  
<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/l/l060/l060150/l0601504.png" /></td> </tr></table>
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$$
 +
\lim\limits _  \rightarrow  H _ {p}  ^ {c} ( X , X \setminus  U ; G )
 +
$$
  
over open neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l0601505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l0601506.png" />, and for homologically locally connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l0601507.png" /> they also coincide with the inverse limits
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over open neighbourhoods $  U $
 +
of $  x $,  
 +
and for homologically locally connected $  X $
 +
they also coincide with the inverse limits
  
<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/l/l060/l060150/l0601508.png" /></td> </tr></table>
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$$
 +
\lim\limits _  \leftarrow  H _ {p-} 1  ^ {c} ( U \setminus  x ; G ) .
 +
$$
  
The homological dimension of a finite-dimensional metrizable locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l0601509.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015010.png" /> (cf. [[Homological dimension of a space|Homological dimension of a space]]) coincides with the largest value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015011.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015012.png" />, and the set of such points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015013.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015014.png" />.
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The homological dimension of a finite-dimensional metrizable locally compact space $  X $
 +
over $  G $(
 +
cf. [[Homological dimension of a space|Homological dimension of a space]]) coincides with the largest value of $  n $
 +
for which $  H _ {n}  ^ {x} \neq 0 $,  
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and the set of such points $  x \in X $
 +
has dimension $  n $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015015.png" /> be the differential sheaf over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015016.png" /> defined by associating with each open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015017.png" /> the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015018.png" />. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015019.png" /> are the fibres of the derived sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015020.png" />. For generalized manifolds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015022.png" />. In this case the homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015023.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015024.png" /> coincides with the cohomology of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015025.png" /> with coefficients in the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060150/l06015026.png" /> (Poincaré–Lefschetz duality). The similar facts for the local cohomology of locally compact spaces do not hold.
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Let $  {\mathcal C} _ {*} $
 +
be the differential sheaf over $  X $
 +
defined by associating with each open set $  U \subset  X $
 +
the chain complex $  C _ {*} ( X , X \setminus  U ;  G ) $.  
 +
The groups $  H _ {p}  ^ {x} $
 +
are the fibres of the derived sheaves $  {\mathcal H} _ {p} = H _ {p} ( {\mathcal C} _ {*} ) $.  
 +
For generalized manifolds, $  H _ {p}  ^ {x} = 0 $
 +
for $  p \neq n = \mathop{\rm dim}  X $.  
 +
In this case the homology sequence of the pair $  ( X , A ) $
 +
with coefficients in $  G $
 +
coincides with the cohomology of the pair $  ( X , X \setminus  A ) $
 +
with coefficients in the sheaf $  {\mathcal H} _ {n} $(
 +
Poincaré–Lefschetz duality). The similar facts for the local cohomology of locally compact spaces do not hold.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Sklyarenko,  "On the theory of generalized manifolds"  ''Math. USSR Izv.'' , '''5''' :  4  (1971)  pp. 845–858  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35'''  (1971)  pp. 831–843</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.E. [A.E. Kharlap] Harlap,  "Local homology and cohomology, homology dimension and generalized manifolds"  ''Math. USSR Sb.'' , '''25''' :  3  (1975)  pp. 323–349  ''Mat. Sb.'' , '''96'''  (1975)  pp. 347–373</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Sklyarenko,  "On the theory of generalized manifolds"  ''Math. USSR Izv.'' , '''5''' :  4  (1971)  pp. 845–858  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35'''  (1971)  pp. 831–843</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.E. [A.E. Kharlap] Harlap,  "Local homology and cohomology, homology dimension and generalized manifolds"  ''Math. USSR Sb.'' , '''25''' :  3  (1975)  pp. 323–349  ''Mat. Sb.'' , '''96'''  (1975)  pp. 347–373</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1972)  pp. Sect. IV.3</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1972)  pp. Sect. IV.3</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


The homology groups (cf. Homology group)

$$ H _ {p} ^ {x} = H _ {p} ^ {c} ( X , X \setminus x ; G ) , $$

defined at points $ x \in X $, where $ H _ {p} ^ {c} $ is homology with compact support. These groups coincide with the direct limits

$$ \lim\limits _ \rightarrow H _ {p} ^ {c} ( X , X \setminus U ; G ) $$

over open neighbourhoods $ U $ of $ x $, and for homologically locally connected $ X $ they also coincide with the inverse limits

$$ \lim\limits _ \leftarrow H _ {p-} 1 ^ {c} ( U \setminus x ; G ) . $$

The homological dimension of a finite-dimensional metrizable locally compact space $ X $ over $ G $( cf. Homological dimension of a space) coincides with the largest value of $ n $ for which $ H _ {n} ^ {x} \neq 0 $, and the set of such points $ x \in X $ has dimension $ n $.

Let $ {\mathcal C} _ {*} $ be the differential sheaf over $ X $ defined by associating with each open set $ U \subset X $ the chain complex $ C _ {*} ( X , X \setminus U ; G ) $. The groups $ H _ {p} ^ {x} $ are the fibres of the derived sheaves $ {\mathcal H} _ {p} = H _ {p} ( {\mathcal C} _ {*} ) $. For generalized manifolds, $ H _ {p} ^ {x} = 0 $ for $ p \neq n = \mathop{\rm dim} X $. In this case the homology sequence of the pair $ ( X , A ) $ with coefficients in $ G $ coincides with the cohomology of the pair $ ( X , X \setminus A ) $ with coefficients in the sheaf $ {\mathcal H} _ {n} $( Poincaré–Lefschetz duality). The similar facts for the local cohomology of locally compact spaces do not hold.

References

[1] E.G. Sklyarenko, "On the theory of generalized manifolds" Math. USSR Izv. , 5 : 4 (1971) pp. 845–858 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 831–843
[2] A.E. [A.E. Kharlap] Harlap, "Local homology and cohomology, homology dimension and generalized manifolds" Math. USSR Sb. , 25 : 3 (1975) pp. 323–349 Mat. Sb. , 96 (1975) pp. 347–373

Comments

References

[a1] A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Sect. IV.3
How to Cite This Entry:
Local homology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_homology&oldid=47681
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article