# Eilenberg-MacLane room

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.

## definition

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

- and
- for all

applies.

## 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" .

## Examples

- 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 :

## literature

- 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