### 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 {\displaystyle t_{x}} t_{x} are isomorphisms.

Notes

Kan-Thurston theorem in nLab

References
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