Eilenberg-MacLane space

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

Definition

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

Applies.

Existence and uniqueness

Falls and an Abelian group or if and is arbitrary, then there is a CW - complex, which is a.

This CW - complex is uniquely determined up to weak homotopy equivalence in the homotopy theory of these CW complexes are therefore referred to simply as "the".

Examples

Areas are also referred to as an aspherical spaces, they come in a variety of applications in mathematics before.

  • The infinite-dimensional real projective space is a.
  • The circle is a.
  • The bouquet of circles is for the free group.
  • The closed orientable surface of genus is a group for the area.
  • A closed, orientable, prime 3-manifold is a.
  • Every CAT (0 ) space is a. This includes locally - symmetric spaces of nichtkompaktem type, in particular hyperbolic manifolds.

Rooms for any desired play an important role in many applications of algebraic topology.

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

Postnikov decomposition

Every CW complex can be decomposed as a Postnikov tower, ie as iterated fibration whose fibers are Eilenberg - MacLane spaces. Spectral sequences means you can then try to compute the homotopy groups of the CW - complex of the known homotopy groups of Eilenberg - MacLane spaces.

Singular homology

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

Where the square brackets the amount of homotopy classes of continuous maps, respectively.

Group homology

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

Mutandis to the cohomology of groups.

Product topology Group cohomology
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