Kan-Thurston theorem

In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group [math]\displaystyle{ G }[/math] to every path-connected topological space [math]\displaystyle{ X }[/math] in such a way that the group cohomology of [math]\displaystyle{ G }[/math] is the same as the cohomology of the space [math]\displaystyle{ X }[/math]. The group [math]\displaystyle{ G }[/math] might then be regarded as a good approximation to the space [math]\displaystyle{ X }[/math], 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 [math]\displaystyle{ K(G,1) }[/math] of a discrete group [math]\displaystyle{ G }[/math], where homology-equivalent means there is a map [math]\displaystyle{ K(G,1) \rightarrow X }[/math] 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 [math]\displaystyle{ X }[/math] be a path-connected topological space. Then, naturally associated to [math]\displaystyle{ X }[/math], there is a Serre fibration [math]\displaystyle{ t_x \colon T_X \to X }[/math] where [math]\displaystyle{ T_X }[/math] is an aspherical space. Furthermore,

• the induced map [math]\displaystyle{ \pi_1(T_X) \to \pi_1(X) }[/math] is surjective, and
• for every local coefficient system [math]\displaystyle{ A }[/math] on [math]\displaystyle{ X }[/math], the maps [math]\displaystyle{ H_*(TX;A) \to H_*(X;A) }[/math] and [math]\displaystyle{ H^*(TX;A) \to H^*(X;A) }[/math] induced by [math]\displaystyle{ t_x }[/math] are isomorphisms.

References

• Kan, Daniel M.; Thurston, William P. (1976). "Every connected space has the homology of a K(π,1)". Topology 15 (3): 253–258. doi:10.1016/0040-9383(76)90040-9. ISSN 0040-9383. 
• McDuff, Dusa (1979). "On the classifying spaces of discrete monoids". Topology 18 (4): 313–320. doi:10.1016/0040-9383(79)90022-3. ISSN 0040-9383. 
• Maunder, Charles Richard Francis (1981). "A short proof of a theorem of Kan and Thurston". The Bulletin of the London Mathematical Society 13 (4): 325–327. doi:10.1112/blms/13.4.325. ISSN 0024-6093. 
• Hausmann, Jean-Claude (1986). "Every finite complex has the homology of a duality group". Mathematische Annalen 275 (2): 327–336. doi:10.1007/BF01458466. ISSN 0025-5831. 
• Leary, Ian J. (2013). "A metric Kan-Thurston theorem". Journal of Topology 6 (1): 251–284. doi:10.1112/jtopol/jts035. ISSN 1753-8416. 
• Kim, Raeyong (2015). "Every finite complex has the homology of some CAT(0) cubical duality group". Geometriae Dedicata 176: 1–9. doi:10.1007/s10711-014-9956-4. ISSN 0046-5755.