sketched later as well. Loop spaces / Omega spectra 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. Relation with Homology For a fixed abelian group there are maps on the stable homotopy groups induced by the map . Taking the direct limit over these maps, one can verify that this defines a reduced homology theory on CW complexes. Since vanishes for , agrees with reduced singular homology with coefficients in G on CW-complexes. Functoriality 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 if is any homomorphism of abelian groups, then there is a non-empty set satisfying where denotes the homotopy class of a continuous map and Relation with Postnikov/Whitehead tower Every connected CW-complex possesses a Postnikov tower, that is an inverse system of spaces: such that for every : there are commuting maps , which induce isomorphism on for , for , the maps are fibrations with fiber . Dually there exists a Whitehead tower, which is a sequence of CW-complexes: such that for every : the maps induce isomorphism on for , is n-connected, the maps are fibrations with fiber With help of Serre spectral sequences computations of higher homotopy groups of spheres can be made. For instance and using a Whitehead tower of can be found here, more generally those of using a Postnikov systems can be found here. Cohomology operations For fixed natural numbers m,n and abelian groups G,H exists a bijection between the set of all cohomology operations and defined by , where is a fundamental class. As a result, cohomology operations cannot decrease the degree of the cohomology