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Topological space with only one nontrivial homotopy group
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type , if it has n th homotopy group isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that , Eilenberg–MacLane spaces of type always exist, and are all weak homotopy equivalent. Thus, one may consider as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a " or as "a model of ". 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. A generalised Eilenberg–Maclane space is a space which has the homotopy type of a product of Eilenberg–Maclane spaces
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The unit circle is a The infinite dimensional complex projective space is a model of The infinite dimensional real projective space is a The wedge sum of k unit circles is a , where is the free group on k generators. The complement to any connected knot or graph in a 3 dimensional sphere is of type ; this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos. Any compact, connected, non positively curved manifold M is a , where is the fundamental group of M. This is a consequence of the Cartan–Hadamard theorem. An infinite lens space given by the quotient of by the free action for is a This can be shown using covering space theory and the fact that the infinite dimensional sphere is contractible. Note this includes as a The configuration space of points in the plane is a , where is the pure braid group on strands. Correspondingly, the th unordered configuration space of is a , where denotes the strand braid group. The infinite symmetric product of a n sphere is a More generally is a for all Moore spaces
Some further elementary examples can be constructed from these by using the fact that the product is For instance the dimensional Torus is a .

An important property of 's is that for any abelian group G, and any based CW complex X, the set of based homotopy classes of based maps from X to is in natural bijection with the n th singular cohomology group of the space X. Thus one says that the are representing spaces for singular cohomology with coefficients in G. Since

there is a distinguished element corresponding to the identity. The above bijection is given by the pullback of that element This is similar to the Yoneda lemma of category theory. A constructive proof of this theorem can be found here, another making use of the relation between omega spectra and generalized reduced cohomology theories can be found here and the main idea is sketched later as well.

The loop space of an Eilenberg–MacLane space is again an Eilenberg–MacLane space: Further there is an adjoint relation between the loop space and the reduced suspension: , which gives the structure of an abelian group, where the operation is the concatenation of loops. This makes the bijection mentioned above a group isomorphism. Also this property implies that Eilenberg–MacLane spaces with various n form an omega spectrum, called an "Eilenberg–MacLane spectrum". This spectrum defines via a reduced cohomology theory on based CW complexes and for any reduced cohomology theory on CW complexes with for there is a natural isomorphism , where denotes reduced singular cohomology. Therefore these two cohomology theories coincide. In a more general context, Brown representability says that every reduced cohomology theory on based CW complexes comes from an omega spectrum.

One can define the group (co)homology of G with coefficients in the group A as the singular (co)homology of the Eilenberg MacLane space with coefficients in A.

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. http://www. numdam. org/volume/SHC 1954 1955 7/