Kan-Thurston theorem


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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Paths and cylinders

Homotopy groups

Basic facts




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


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

tX:TXX t X \colon T X \rightarrow X

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

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

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

for every local coefficient system AA on XX, and

(ii) π i(TX)\pi_i\;(T X) is trivial for i1i\neq 1 and π 1tX\pi_1\;t X is onto.

Furthermore the homotopy type of XX is completely determined by the pair of groups (G X,P X)(G_X, P_X) where G X=π 1TX,P X=kerπ 1tXG_X = \pi_1T X,\;\; P_X = ker\;\pi_1t X and P XP_X is a perfect normal subgroup of G XG_X.

More precisely, XX can be recovered, up to homotopy, by applying the plus construction to K(G X,1)K(G_X,1) with respect to P XP_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 XX is sent to the presheaf G[BG,X]G \mapsto [B G, X].


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

Last revised on June 16, 2013 at 20:21:43. See the history of this page for a list of all contributions to it.