Kan–Thurston theorem

In mathematics, particularly algebraic topology, the Kan–Thurston theorem associates a discrete group to every path-connected topological space in such a way that the group cohomology of is the same as the cohomology of the space . The group might then be regarded as a good approximation to the space , 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 of a discrete group , where homology-equivalent means there is a map 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 be a path-connected topological space. Then, naturally associated to , there is a Serre fibration <Math>t_x \colon T_X \to X</Math> where is an aspherical space. Furthermore,

• the induced map <Math>\pi_1(T_X) \to \pi_1(X)</math> is surjective, and
• for every local coefficient system on , the maps and induced by are isomorphisms.

Notes

References