Kan-Thurston theorem
In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group to every path-connected topological space in such a way that the group cohomology of is the same as the cohomology of the space . The group might then be regarded as a good approximation to the space , 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 of a discrete group , where homology-equivalent means there is a map 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 be a path-connected topological space. Then, naturally associated to , there is a Serre fibration where is an aspherical space. Furthermore,
- the induced map is surjective, and
- for every local coefficient system on , the maps and induced by 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.