# Difference between revisions of "Eilenberg-MacLane space"

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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 $\pi$. | 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|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 . | + | 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 & 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 & 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 13:05, 24 December 2020

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://encyclopediaofmath.org/index.php?title=Eilenberg-MacLane_space&oldid=51062