• Source: Kan-Thurston theorem

In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group



G


{\displaystyle G}

to every path-connected topological space



X


{\displaystyle X}

in such a way that the group cohomology of



G


{\displaystyle G}

is the same as the cohomology of the space



X


{\displaystyle X}

. The group



G


{\displaystyle G}

might then be regarded as a good approximation to the space



X


{\displaystyle X}

, and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory.
More precisely, the theorem states that every path-connected topological space is homology-equivalent to the classifying space



K
(
G
,
1
)


{\displaystyle K(G,1)}

of a discrete group



G


{\displaystyle G}

, where homology-equivalent means there is a map



K
(
G
,
1
)
→
X


{\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


{\displaystyle X}

be a path-connected topological space. Then, naturally associated to



X


{\displaystyle X}

, there is a Serre fibration




t

x


:

T

X


→
X


{\displaystyle t_{x}\colon T_{X}\to X}

where




T

X




{\displaystyle T_{X}}

is an aspherical space. Furthermore,

the induced map




π

1


(

T

X


)
→

π

1


(
X
)


{\displaystyle \pi _{1}(T_{X})\to \pi _{1}(X)}

is surjective, and
for every local coefficient system



A


{\displaystyle A}

on



X


{\displaystyle X}

, the maps




H

∗


(
T
X
;
A
)
→

H

∗


(
X
;
A
)


{\displaystyle H_{*}(TX;A)\to H_{*}(X;A)}

and




H

∗


(
T
X
;
A
)
→

H

∗


(
X
;
A
)


{\displaystyle H^{*}(TX;A)\to H^{*}(X;A)}

induced by




t

x




{\displaystyle t_{x}}

are isomorphisms.




Notes






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.

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