Kan-Thurston theorem

In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group formula_1 to every path connected topological space formula_2 in such a way that the group cohomology of formula_1 is the same as the cohomology of the space formula_2. The group formula_1 might then be regarded as a good approximation to the space formula_2, and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory.
More precisely, the theorem
states that every path connected topological space is homology-equivalent to the classifying space formula_7 of a discrete group formula_1, where homology-equivalent means there is a map formula_9 inducing an isomorphism on homology.
The theorem is attributed to Daniel Kan and William Thurston who published their result in 1976.
Statement of the Kan-Thurston theorem.
Let formula_2 be a path connected topological space. Then, naturally associated to formula_2, there is a Serre fibration formula_12 where formula_13 is an aspherical space. Furthermore,