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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[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 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.
Statement of the Kan-Thurston theorem

Let X be a path connected topological space. Then, naturally associated to 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 A on X, the maps \( {\displaystyle H_{*}(TX;A)\to H_{*}(X;A)} \) and \( {\displaystyle H^{*}(TX;A)\to H^{*}(X;A)} \) induced by t x {\displaystyle t_{x}} t_{x} are isomorphisms.

Notes

Kan-Thurston theorem in nLab

References
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. MR 1439159.
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. MR 0551013.
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. MR 0620046.
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. MR 0854015.
Leary, Ian J. (2013). "A metric Kan-Thurston theorem". Journal of Topology. 6 (1): 251–284. arXiv:1009.1540. doi:10.1112/jtopol/jts035. ISSN 1753-8416. MR 3029427.
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. MR 3347570.

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