discrete topology. The real projective space is a classifying space for G. Let the field of two elements. Then a polynomial k-algebra on a single generator, since this is the cellular cohomology ring of Künneth formula If, M = k is a field, then H*(G; k) is a graded k-algebra and the cohomology of a product of groups is related to the ones of the individual groups by a Künneth formula: For example, if G is an elementary abelian 2-group of rank r, and then the Künneth formula shows that the cohomology of G is a polynomial k-algebra generated by r classes in H1(G; k)., Homology vs. cohomology As for other cohomology theories, such as singular cohomology, group cohomology and homology are related to one another by means of a short exact sequence where A is endowed with the trivial G-action and the term at the left is the first Ext group. Amalgamated products Given a group A which is the subgroup of two groups G1 and G2, the homology of the amalgamated product (with integer coefficients) lies in a long exact sequence The homology of can be computed using this: This exact sequence can also be applied to show that the homology of the and the special linear group agree for an infinite field k. Change of group The Hochschild–Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro-)finite groups G). From it, one gets the inflation-restriction exact sequence. Cohomology of the classifying space Group cohomology is closely related to topological cohomology theories such as sheaf cohomology, by means of an isomorphism The expression at the left is a classifying space for . It is an Eilenberg–MacLane space , i.e., a space whose fundamental group is and whose higher homotopy groups vanish). Classifying spaces for and are the 1-sphere S1, infinite real projective space and lens spaces, respectively. In general, can be constructed as the quotient , where is a contractible space on which acts freely. However, does not usually have an easily amenable geometric description. More generally, one can attach to any -module a local coefficient system on and the above isomorphism generalizes to an isomorphism Further examples Semi-direct products of groups There is a way to compute the semi-direct product of groups using the topology of fibrations and properties of Eilenberg-Maclane