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Difference between revisions of "Eilenberg-MacLane space"

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A space, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352001.png" />, representing the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352003.png" /> is a non-negative number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352004.png" /> is a group which is commutative for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352006.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352007.png" />-dimensional cohomology group of a cellular space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352008.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e0352009.png" />. It exists for any such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520011.png" />.
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A space, denoted by $K(\pi,n)$, representing the functor $X\to H^n(X;\pi)$, where $n$ is a non-negative number, $\pi$ is a group which is commutative for $n>1$ and $H^n(X;\pi)$ is the $n$-dimensional cohomology group of a cellular space $X$ with coefficients in $\pi$. It exists for any such $n$ and $\pi$.
  
The Eilenberg–MacLane space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520012.png" /> can also be characterized by the condition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520017.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520018.png" />-th [[Homotopy group|homotopy group]]. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520019.png" /> is uniquely defined up to a weak homotopy equivalence. An arbitrary topological space can, up to a weak homotopy equivalence, be decomposed into a twisted product of Eilenberg–MacLane spaces (see [[Postnikov system|Postnikov system]]). The cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520020.png" /> coincide with those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035200/e03520021.png" />. Eilenberg–MacLane spaces were introduced by S. Eilenberg and S. MacLane .
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The Eilenberg–MacLane space $K(\pi,n)$ can also be characterized by the condition: $\pi_i(K(\pi,n))=\pi$ for $i=n$ and $\pi_i(K(\pi,n))=0$ for $i\neq n$, where $\pi_i$ is the $i$-th [[Homotopy group|homotopy group]]. Thus, $K(\pi,n)$ is uniquely defined up to a weak homotopy equivalence. An arbitrary topological space can, up to a weak homotopy equivalence, be decomposed into a twisted product of Eilenberg–MacLane spaces (see [[Postnikov system|Postnikov system]]). The cohomology groups of $K(\pi,1)$ coincide with those of $\pi$. Eilenberg–MacLane spaces were introduced by S. Eilenberg and S. MacLane .
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  S. Eilenberg,  S. MacLane,  "Relations between homology and homotopy groups of spaces"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 480–509</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  S. Eilenberg,  S. MacLane,  "Relations between homology and homotopy groups of spaces. II"  ''Ann. of Math.'' , '''51'''  (1950)  pp. 514–533</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  S. Eilenberg,  S. MacLane,  "Relations between homology and homotopy groups of spaces"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 480–509</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  S. Eilenberg,  S. MacLane,  "Relations between homology and homotopy groups of spaces. II"  ''Ann. of Math.'' , '''51'''  (1950)  pp. 514–533</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>

Latest revision as of 23:52, 7 July 2014

A space, denoted by $K(\pi,n)$, representing the functor $X\to H^n(X;\pi)$, where $n$ is a non-negative number, $\pi$ is a group which is commutative for $n>1$ and $H^n(X;\pi)$ is the $n$-dimensional cohomology group of a cellular space $X$ with coefficients in $\pi$. It exists for any such $n$ and $\pi$.

The Eilenberg–MacLane space $K(\pi,n)$ can also be characterized by the condition: $\pi_i(K(\pi,n))=\pi$ for $i=n$ and $\pi_i(K(\pi,n))=0$ for $i\neq n$, where $\pi_i$ is the $i$-th homotopy group. Thus, $K(\pi,n)$ is uniquely defined up to a weak homotopy equivalence. An arbitrary topological space can, up to a weak homotopy equivalence, be decomposed into a twisted product of Eilenberg–MacLane spaces (see Postnikov system). The cohomology groups of $K(\pi,1)$ coincide with those of $\pi$. Eilenberg–MacLane spaces were introduced by S. Eilenberg and S. MacLane .

References

[1a] S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces" Ann. of Math. , 46 (1945) pp. 480–509
[1b] S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces. II" Ann. of Math. , 51 (1950) pp. 514–533
[2] R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[4] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
How to Cite This Entry:
Eilenberg-MacLane space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Eilenberg-MacLane_space&oldid=22371
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article