Kan-Thurston theorem

In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group $$G$$ to every path-connected topological space $$X$$ in such a way that the group cohomology of $$G$$ is the same as the cohomology of the space $$X$$. The group $$G$$ might then be regarded as a good approximation to the space $$X$$, 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 $$K(G,1)$$ of a discrete group $$G$$, where homology-equivalent means there is a map $$K(G,1) \rightarrow X$$ 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 $$X$$ be a path-connected topological space. Then, naturally associated to $$X$$, there is a Serre fibration $$t_x \colon T_X \to X$$ where $$T_X$$ is an aspherical space. Furthermore,
 * the induced map $$\pi_1(T_X) \to \pi_1(X)$$ is surjective, and
 * for every local coefficient system $$A$$ on $$X$$, the maps $$H_*(TX;A) \to H_*(X;A)$$ and $$H^*(TX;A) \to H^*(X;A)$$ induced by $$t_x$$ are isomorphisms.