as referring to a weak homotopy equivalence class of spaces.
Moreover, it is common to assume that this space is a CW-complex (which is always possible via CW approximation).
The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.
As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system.
These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology.
Note that if G has a torsion element, then every CW-complex of type K(G,1) has to be infinite-dimensional.
There are multiple techniques for constructing higher Eilenberg-Maclane spaces.
Another useful technique is to use the geometric realization of simplicial abelian groups.
[4] This gives an explicit presentation of simplicial abelian groups which represent Eilenberg-Maclane spaces.
Another simplicial construction, in terms of classifying spaces and universal bundles, is given in J. Peter May's book.
This was exploited by Jean-Pierre Serre while he studied the homotopy groups of spheres using the Postnikov system and spectral sequences.
is in natural bijection with the n-th singular cohomology group
are representing spaces for singular cohomology with coefficients in G. Since there is a distinguished element
A constructive proof of this theorem can be found here,[6] another making use of the relation between omega-spectra and generalized reduced cohomology theories can be found here,[7] and the main idea is sketched later as well.
the structure of an abelian group, where the operation is the concatenation of loops.
Also this property implies that Eilenberg–MacLane spaces with various n form an omega-spectrum, called an "Eilenberg–MacLane spectrum".
In a more general context, Brown representability says that every reduced cohomology theory on based CW-complexes comes from an omega-spectrum.
Taking the direct limit over these maps, one can verify that this defines a reduced homology theory on CW complexes.
It follows from the universal coefficient theorem for cohomology that the Eilenberg MacLane space is a quasi-functor of the group; that is, for each positive integer
is any homomorphism of abelian groups, then there is a non-empty set satisfying
possesses a Postnikov tower, that is an inverse system of spaces: such that for every
: Dually there exists a Whitehead tower, which is a sequence of CW-complexes: such that for every
: With help of Serre spectral sequences computations of higher homotopy groups of spheres can be made.
[9] For fixed natural numbers m,n and abelian groups G,H exists a bijection between the set of all cohomology operations
Some interesting examples for cohomology operations are Steenrod Squares and Powers, when
becomes apparent quickly;[10] some extensive tabeles of those groups can be found here.
The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence with
lies in the fact that there are the homotopy equivalences for the classifying space
can be used to start a short exact sequence that kills the homotopy group
The Cartan seminar contains many fundamental results about Eilenberg-Maclane spaces including their homology and cohomology, and applications for calculating the homotopy groups of spheres.