# 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${\ displaystyle G}$${\ displaystyle n}$ ${\ displaystyle X}$ ${\ displaystyle K (G, n)}$

${\ displaystyle \ pi _ {n} (X) = G}$ and
${\ displaystyle \ pi _ {k} (X) = 0}$ for all ${\ displaystyle k \ neq 0, n}$

applies.

## Existence and uniqueness

If and is an Abelian group or if and is anything, then there is a CW complex that is a. ${\ displaystyle n \ geq 2}$${\ displaystyle G}$${\ displaystyle n = 1}$${\ displaystyle G}$ ${\ displaystyle X}$${\ displaystyle K (G, n)}$

This CW-complex is clearly defined except for homotopy equivalence , in the homotopy theory these CW-complexes are therefore simply referred to as "the" . ${\ displaystyle K (G, n)}$

## Examples

${\ displaystyle K (G, 1)}$- Spaces are aspherical spaces in particular ; they occur in numerous applications in mathematics.

• The infinite-dimensional real-projective space is a .${\ displaystyle \ mathbb {R} P ^ {\ infty}}$${\ displaystyle K (\ mathbb {Z} / 2 \ mathbb {Z}, 1)}$
• The circle is a .${\ displaystyle S ^ {1}}$${\ displaystyle K (\ mathbb {Z}, 1)}$
• The bouquet of circles is one for the free group .${\ displaystyle n}$${\ displaystyle K (F_ {n}, 1)}$ ${\ displaystyle F_ {n}}$
• The closed, orientable area of the gender is one for the area group .${\ displaystyle S_ {g}}$ ${\ displaystyle g \ geq 1}$${\ displaystyle K (\ pi _ {1} S_ {g}, 1)}$ ${\ displaystyle \ pi _ {1} S_ {g}}$
• A closed, orientable, prime 3-manifold is a .${\ displaystyle M \ not = S ^ {2} \ times S ^ {1}}$${\ displaystyle K (\ pi _ {1} M, 1)}$
• 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 .${\ displaystyle K (G, 1)}$

${\ displaystyle K (G, n)}$- Spaces for any play an important role in numerous applications of algebraic topology . ${\ displaystyle n}$

• The infinite-dimensional complex-projective space is a .${\ displaystyle \ mathbb {C} P ^ {\ infty}}$${\ displaystyle K (\ mathbb {Z}, 2)}$
• The product of one and one is one .${\ displaystyle K (G_ {1}, n)}$${\ displaystyle K (G_ {2}, n)}$${\ displaystyle K (G_ {1} \ times G_ {2}, n)}$

## 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 ${\ displaystyle X}$${\ displaystyle n}$${\ displaystyle G}$

${\ displaystyle H ^ {n} (X, G) = \ left [X, K (G, n) \ right],}$

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 : ${\ displaystyle G}$${\ displaystyle A}$${\ displaystyle K (G, 1)}$

${\ displaystyle H _ {*} (G; A): = H _ {*} (K (G, 1); A),}$

accordingly for group cohomology :

${\ displaystyle H ^ {*} (G; A): = H ^ {*} (K (G, 1); A).}$

## 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