Kan-Thurston theorem explained

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.K(G, 1 ) of a discrete group G , where homology-equivalent means there is a map K(G, 1 ) → 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 $t\_x\; \backslash colon\; T\_X\; \backslash to\; X$ where T _{X} is an aspherical space. Furthermore,

• the induced map $\backslash pi\_1(T\_X)\; \backslash to\; \backslash pi\_1(X)$ 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.

References

• Kan . Daniel M. . Daniel_Kan . Thurston . William P. . William_Thurston . 1439159 . Every connected space has the homology of a K(π,1) . Topology . 15 . 3 . 253–258 . 1976 . 0040-9383 . 10.1016/0040-9383(76)90040-9 .
• McDuff . Dusa . Dusa_McDuff . 551013 . On the classifying spaces of discrete monoids . Topology . 18 . 4 . 313–320 . 1979 . 0040-9383 . 10.1016/0040-9383(79)90022-3 . free .
• Maunder . Charles Richard Francis . 620046 . A short proof of a theorem of Kan and Thurston . The Bulletin of the London Mathematical Society . 13 . 4 . 325–327 . 1981 . 0024-6093 . 10.1112/blms/13.4.325 .
• Hausmann . Jean-Claude . 854015 . Every finite complex has the homology of a duality group . Mathematische Annalen . 275 . 2 . 327–336 . 1986 . 0025-5831 . 10.1007/BF01458466 . 119913298 .
• Leary . Ian J. . 3029427 . A metric Kan-Thurston theorem . Journal of Topology . 6 . 1 . 251–284 . 2013 . 1753-8416 . 10.1112/jtopol/jts035 . 1009.1540 . 119162788 .
• Kim . Raeyong . 3347570 . Every finite complex has the homology of some CAT(0) cubical duality group . Geometriae Dedicata . 176 . 1–9 . 2015 . 0046-5755 . 10.1007/s10711-014-9956-4 . 119644662 .