In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group ${\displaystyle G}$ to every path-connected topological space ${\displaystyle X}$ in such a way that the group cohomology of ${\displaystyle G}$ is the same as the cohomology of the space ${\displaystyle X}$. The group ${\displaystyle G}$ might then be regarded as a good approximation to the space ${\displaystyle X}$, and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory.

More precisely,^[1] the theorem states that every path-connected topological space is homology-equivalent to the classifying space ${\displaystyle K(G,1)}$ of a discrete group ${\displaystyle G}$, where homology-equivalent means there is a map ${\displaystyle K(G,1)\rightarrow X}$ inducing an isomorphism on homology.

The theorem is attributed to Daniel Kan and William Thurston who published their result in 1976.

Let ${\displaystyle X}$ be a path-connected topological space. Then, naturally associated to ${\displaystyle X}$, there is a Serre fibration ${\displaystyle t_{x}\colon T_{X}\to X}$ where ${\displaystyle T_{X}}$ is an aspherical space. Furthermore,

• the induced map ${\displaystyle \pi _{1}(T_{X})\to \pi _{1}(X)}$ is surjective, and
• for every local coefficient system ${\displaystyle A}$ on ${\displaystyle X}$, the maps ${\displaystyle H_{*}(TX;A)\to H_{*}(X;A)}$ and ${\displaystyle H^{*}(TX;A)\to H^{*}(X;A)}$ induced by ${\displaystyle t_{x}}$ are isomorphisms.