the -dimensional Torus is a . Remark on constructing Eilenberg–MacLane spaces For and an arbitrary group the construction of is identical to that of the classifying space of the group . 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. One of which is to construct a Moore space for an abelian group : Take the wedge of n-spheres, one for each generator of the group A and realise the relations between these generators by attaching (n+1)-cells via corresponding maps in of said wedge sum. Note that the lower homotopy groups are already trivial by construction. Now iteratively kill all higher homotopy groups by successively attaching cells of dimension greater than , and define as direct limit under inclusion of this iteration. Another useful technique is to use the geometric realization of simplicial abelian groups. 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. Since taking the loop space lowers the homotopy groups by one slot, we have a canonical homotopy equivalence , hence there is a fibration sequence . Note that this is not a cofibration sequence ― the space is not the homotopy cofiber of . This fibration sequence can be used to study the cohomology of from using the Leray spectral sequence. This was exploited by Jean-Pierre Serre while he studied the homotopy groups of spheres using the Postnikov system and spectral sequences. Properties of Eilenberg–MacLane spaces Bijection between homotopy classes of maps and cohomology 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