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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649001.png" /> with a unique non-trivial reduced homology group:
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<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/m/m064/m064900/m0649002.png" /></td> </tr></table>
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649003.png" /> is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of the group of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649005.png" /> is the Moore space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649006.png" />, then
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==Homology==
 +
A topological space $  M $
 +
with a unique non-trivial reduced 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/m/m064/m064900/m0649007.png" /></td> </tr></table>
+
$$
 +
\widetilde{H}  _ {k} ( M)  = G ; \ \
 +
\widetilde{H}  _ {i} ( M)  = 0 ,\ \
 +
i \neq k .
 +
$$
  
that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649008.png" /> is the spectrum of the cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m0649009.png" />. This allows one to extend the idea of cohomology with arbitrary coefficients to a generalized cohomology theory. For any spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490010.png" />, the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490011.png" /> defines a cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490012.png" />, called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490014.png" />-cohomology theory with coefficient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490015.png" />. For the definition of generalized homology theories with coefficients in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490016.png" />, the so-called co-Moore space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490017.png" /> is used, which is characterized by
+
If  $  K ( \mathbf Z , n ) $
 +
is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of the group of integers  $  \mathbf Z $
 +
and  $  M _ {k} ( G) $
 +
is the Moore space with  $  \widetilde{H}  _ {k} ( M _ {k} ( G) ) = G $,  
 +
then
  
<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/m/m064/m064900/m06490018.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {N \rightarrow \infty } \
 +
\left [
 +
\Sigma  ^ {N+} k X , K ( \mathbf Z , N + n ) \wedge M _ {k} ( G) \right ]
 +
\cong  H  ^ {n} ( X , G ) ,
 +
$$
  
For example, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490019.png" /> is called the homotopy group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490020.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490021.png" />. However, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490022.png" /> does not exist for all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490024.png" /> is a finitely-generated group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490025.png" /> does exist.
+
that is,  $  \{ K ( \mathbf Z , n ) \wedge M _ {k} ( G) \} $
 +
is the spectrum of the cohomology theory  $  H  ^ {*} (  , G ) $.
 +
This allows one to extend the idea of cohomology with arbitrary coefficients to a generalized cohomology theory. For any spectrum  $  E $,
 +
the spectrum  $  E \wedge M _ {k} ( G) $
 +
defines a cohomology theory  $  ( E \wedge M _ {k} ( G) )  ^ {*} $,  
 +
called the  $  E  ^ {*} $-
 +
cohomology theory with coefficient group  $  G $.
 +
For the definition of generalized homology theories with coefficients in a group $  G $,
 +
the so-called co-Moore space  $  M  ^ {k} ( G) $
 +
is used, which is characterized by
 +
 
 +
$$
 +
\widetilde{H}  {}  ^ {k} ( M  ^ {k} ( G) )  = G ,\ \
 +
\widetilde{H}  {}  ^ {i} ( M  ^ {k} ( G) )  = 0 ,\ \
 +
i \neq k .
 +
$$
 +
 
 +
For example, the group  $  \pi _ {i} ( X , G ) = [ M  ^ {k} ( G) , X ] $
 +
is called the homotopy group of the space $  X $
 +
with coefficients in $  G $.  
 +
However, the space $  M  ^ {k} ( G) $
 +
does not exist for all pairs $  ( G , k ) $.  
 +
If $  G $
 +
is a finitely-generated group, then $  M  ^ {k} ( G) $
 +
does exist.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.C. Moore,  "On homotopy groups of spaces with a single non-vanishing homotopy group"  ''Ann. of Math.'' , '''59''' :  3  (1954)  pp. 549–557</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.C. Moore,  "On homotopy groups of spaces with a single non-vanishing homotopy group"  ''Ann. of Math.'' , '''59''' :  3  (1954)  pp. 549–557</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For a construction of a Moore space as a CW-complex with one zero cell and further only cells in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490027.png" />, cf. [[#References|[a1]]]. The Eilenberg–MacLane space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490028.png" /> can be obtained from the Moore space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490029.png" /> by killing the higher homotopy groups.
+
For a construction of a Moore space as a CW-complex with one zero cell and further only cells in dimensions $  n $
 +
and $  n + 1 $,  
 +
cf. [[#References|[a1]]]. The Eilenberg–MacLane space $  K ( G , n ) $
 +
can be obtained from the Moore space $  M ( G , n ) $
 +
by killing the higher homotopy groups.
  
In general topology, a Moore space is a [[Regular space|regular space]] with a development. (A development is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490030.png" /> of open coverings such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490031.png" /> and every open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490032.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490033.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490034.png" /> such that
+
====References====
 
+
<table>
<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/m/m064/m064900/m06490035.png" /></td> </tr></table>
+
<TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Gray,  "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. §17</TD></TR>
 +
</table>
  
(in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490036.png" /> is a [[neighbourhood base]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064900/m06490037.png" />.)
+
==General topology==
 +
In general topology, a Moore space is a [[regular space]] with a [[Refinement|development]]: a sequence  $  \{ {\mathcal U} _ {n} \} _ {n} $
 +
of open coverings such that for every  $  x $
 +
and every open set  $  O $
 +
containing  $  x $
 +
there is an  $  n $
 +
such that
 +
$$
 +
\mathop{\rm St} ( x , {\mathcal U} _ {n} )  =  \cup
 +
\{ {U \in {\mathcal U} _ {n} } : {x \in U } \}
 +
\subseteq  O ;
 +
$$
 +
(in other words, $  \{  \mathop{\rm St} ( x , {\mathcal U} _ {x} ) \} _ {n} $
 +
is a [[neighbourhood base]] at $  x $.)
  
The idea of a development can be found in [[#References|[a4]]] (Axiom 1). Moore spaces are generalizations of metric spaces and one can show that collectionwise normal Moore spaces are metrizable [[#References|[a2]]]. The question whether every normal Moore space is metrizable generated lots of research; its solution is described in [[#References|[a3]]].
+
The idea of a development can be found in [[#References|[a4]]] (Axiom 1). Moore spaces are generalizations of [[metric space]]s and one can show that [[Collection-wise normal space|collectionwise normal]] Moore spaces are metrizable [[#References|[a2]]]. The question whether every normal Moore space is metrizable generated lots of research; its solution is described in [[#References|[a3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Gray,  "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. §17</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.H. Bing,  "Metrization of topological spaces"  ''Canad. J. Math.'' , '''3'''  (1951)  pp. 175–186</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.G. Fleissner,  "The normal Moore space conjecture and large cardinals"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set-Theoretic Topology'' , North-Holland  (1984)  pp. 733–760</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.L. Moore,  "Foundations of point set theory" , Amer. Math. Soc.  (1962)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.H. Bing,  "Metrization of topological spaces"  ''Canad. J. Math.'' , '''3'''  (1951)  pp. 175–186</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  W.G. Fleissner,  "The normal Moore space conjecture and large cardinals"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set-Theoretic Topology'' , North-Holland  (1984)  pp. 733–760</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.L. Moore,  "Foundations of point set theory" , Amer. Math. Soc.  (1962)</TD></TR>
 +
</table>

Latest revision as of 19:33, 20 January 2021


Homology

A topological space $ M $ with a unique non-trivial reduced homology group:

$$ \widetilde{H} _ {k} ( M) = G ; \ \ \widetilde{H} _ {i} ( M) = 0 ,\ \ i \neq k . $$

If $ K ( \mathbf Z , n ) $ is the Eilenberg–MacLane space of the group of integers $ \mathbf Z $ and $ M _ {k} ( G) $ is the Moore space with $ \widetilde{H} _ {k} ( M _ {k} ( G) ) = G $, then

$$ \lim\limits _ {N \rightarrow \infty } \ \left [ \Sigma ^ {N+} k X , K ( \mathbf Z , N + n ) \wedge M _ {k} ( G) \right ] \cong H ^ {n} ( X , G ) , $$

that is, $ \{ K ( \mathbf Z , n ) \wedge M _ {k} ( G) \} $ is the spectrum of the cohomology theory $ H ^ {*} ( , G ) $. This allows one to extend the idea of cohomology with arbitrary coefficients to a generalized cohomology theory. For any spectrum $ E $, the spectrum $ E \wedge M _ {k} ( G) $ defines a cohomology theory $ ( E \wedge M _ {k} ( G) ) ^ {*} $, called the $ E ^ {*} $- cohomology theory with coefficient group $ G $. For the definition of generalized homology theories with coefficients in a group $ G $, the so-called co-Moore space $ M ^ {k} ( G) $ is used, which is characterized by

$$ \widetilde{H} {} ^ {k} ( M ^ {k} ( G) ) = G ,\ \ \widetilde{H} {} ^ {i} ( M ^ {k} ( G) ) = 0 ,\ \ i \neq k . $$

For example, the group $ \pi _ {i} ( X , G ) = [ M ^ {k} ( G) , X ] $ is called the homotopy group of the space $ X $ with coefficients in $ G $. However, the space $ M ^ {k} ( G) $ does not exist for all pairs $ ( G , k ) $. If $ G $ is a finitely-generated group, then $ M ^ {k} ( G) $ does exist.

References

[1] J.C. Moore, "On homotopy groups of spaces with a single non-vanishing homotopy group" Ann. of Math. , 59 : 3 (1954) pp. 549–557

Comments

For a construction of a Moore space as a CW-complex with one zero cell and further only cells in dimensions $ n $ and $ n + 1 $, cf. [a1]. The Eilenberg–MacLane space $ K ( G , n ) $ can be obtained from the Moore space $ M ( G , n ) $ by killing the higher homotopy groups.

References

[a1] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. §17

General topology

In general topology, a Moore space is a regular space with a development: a sequence $ \{ {\mathcal U} _ {n} \} _ {n} $ of open coverings such that for every $ x $ and every open set $ O $ containing $ x $ there is an $ n $ such that $$ \mathop{\rm St} ( x , {\mathcal U} _ {n} ) = \cup \{ {U \in {\mathcal U} _ {n} } : {x \in U } \} \subseteq O ; $$ (in other words, $ \{ \mathop{\rm St} ( x , {\mathcal U} _ {x} ) \} _ {n} $ is a neighbourhood base at $ x $.)

The idea of a development can be found in [a4] (Axiom 1). Moore spaces are generalizations of metric spaces and one can show that collectionwise normal Moore spaces are metrizable [a2]. The question whether every normal Moore space is metrizable generated lots of research; its solution is described in [a3].

References

[a2] R.H. Bing, "Metrization of topological spaces" Canad. J. Math. , 3 (1951) pp. 175–186
[a3] W.G. Fleissner, "The normal Moore space conjecture and large cardinals" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 733–760
[a4] R.L. Moore, "Foundations of point set theory" , Amer. Math. Soc. (1962)
How to Cite This Entry:
Moore space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moore_space&oldid=41968
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article