Kan-Thurston theorem

In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group ${\displaystyle G}$ to every path-connected topological space ${\displaystyle X}$ in such a way that the group cohomology of ${\displaystyle G}$ is the same as the cohomology of the space ${\displaystyle X}$. The group ${\displaystyle G}$ might then be regarded as a good approximation to the space ${\displaystyle X}$, 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 ${\displaystyle G}$, 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 ${\displaystyle X}$ be a path-connected topological space. Then, naturally associated to ${\displaystyle X}$, 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 ${\displaystyle A}$ on ${\displaystyle X}$, the maps ${\displaystyle H_{*}(TX;A)\to H_{*}(X;A)}$ and ${\displaystyle H^{*}(TX;A)\to H^{*}(X;A)}$ induced by ${\displaystyle t_{x}}$ are isomorphisms.

Notes

1. 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.