nLab Kan-Thurston theorem

Idea

The Kan-Thurston theorem says that every path-connected topological space is homology-equivalent to the classifying space $K(G,1)$ of a discrete group $G$.

Statement

For every path connected space $X$ with base point there exists a (Serre-)-fibration

$$t X \colon T X \rightarrow X$$

which is natural with respect to $X$ and has the following properties.

(i) The map $t X$ induces an isomorphism on (singular-)homology and cohomology with local coefficients

$$H_*(T X;A)\cong H_*(X;A),\;\; H^*(T X;A)\cong H^*(X;A)$$

for every local coefficient system $A$ on $X$, and

(ii) $\pi_i\;(T X)$ is trivial for $i\neq 1$ and $\pi_1\;t X$ is onto.

Furthermore the homotopy type of $X$ is completely determined by the pair of groups $(G_X, P_X)$ where $G_X = \pi_1T X,\;\; P_X = ker\;\pi_1t X$ and $P_X$ is a perfect normal subgroup of $G_X$.

More precisely, $X$ can be recovered, up to homotopy, by applying the plus construction to $K(G_X,1)$ with respect to $P_X$.

More generally, the Kan-Thurston theorem gives an embedding of the homotopy category of topological spaces into the category of presheaves on the category whose objects are groups and whose morphisms are conjugacy classes of group homomorphisms. Under this embedding a topological space $X$ is sent to the presheaf $G \mapsto [B G, X]$.

References

• Daniel Kan and William Thurston, Every connected space has the homology of a $K(\pi,1)$, Topology Vol. 15. pp. 253–258, 1976.