Eilenberg-MacLane room

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In topology , a branch of mathematics , Eilenberg-MacLane spaces are an important class of topological spaces , which on the one hand serve as building blocks in homotopy theory to assemble any CW complexes by means of fibers , and on the other hand the aspheric manifolds, which is important in differential geometry include.


For a group and a natural number , a connected CW-complex is an Eilenberg-MacLane space , if for its homotopy groups

for all


Existence and uniqueness

If and is an Abelian group or if and is anything, then there is a CW complex that is a.

This CW-complex is clearly defined except for homotopy equivalence , in the homotopy theory these CW-complexes are therefore simply referred to as "the" .


- Spaces are aspherical spaces in particular ; they occur in numerous applications in mathematics.

  • The infinite-dimensional real-projective space is a .
  • The circle is a .
  • The bouquet of circles is one for the free group .
  • The closed, orientable area of the gender is one for the area group .
  • A closed, orientable, prime 3-manifold is a .
  • Every manifold of non-positive intersection curvature (and more generally every metric space whose universal superposition is a CAT (0) space ) is a . This includes locally symmetric spaces of non-compact type , especially hyperbolic manifolds .

- Spaces for any play an important role in numerous applications of algebraic topology .

  • The infinite-dimensional complex-projective space is a .
  • The product of one and one is one .

Postnikov decomposition

Each CW complex can be dismantled as a Postnikov tower , i. H. as iterated fibers , the fibers of which are Eilenberg-MacLane spaces. Using spectral sequences , one can then try to calculate the homotopy groups of the CW complex from the known homotopy groups of the Eilenberg-MacLane spaces.

Singular cohomology

Eilenberg-MacLane areas represent the singular cohomology is: for any topological space , each and every abelian group applies

where the square brackets denote the set of homotopy classes of continuous mappings.

Group homology and group cohomology

The group homology of a group (with coefficients ) is by definition the singular homologue of the Eilenberg-MacLane space :

accordingly for group cohomology :


  • S. Eilenberg, S. MacLane: Relations between homology and homotopy groups of spaces Ann. of Math. 46 (1945) pp. 480-509
  • S. Eilenberg, S. MacLane: Relations between homology and homotopy groups of spaces. II Ann. of Math. 51 (1950) pp. 514-533
  • Chapter 8.1 in: Edwin H. Spanier, Algebraic topology. Corrected reprint. Springer-Verlag, New York-Berlin, 1981. ISBN 0-387-90646-0

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