Kan-Thurston theorem

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 $K(G,1)$ of a discrete group $G$ , where homology-equivalent means there is a map $K(G,1)\rightarrow X$ 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 $t_{x}\colon T_{X}\to X$ where $T_{X}$ is an aspherical space. Furthermore,

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

Notes

^ Kan-Thurston theorem at the 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. S2CID 119913298.
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. S2CID 119162788.
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. S2CID 119644662.

[1] Kan-Thurston theorem at the nLab

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