Kan and Thurston's theorem

The set of Kan and Thurston is a theorem from the mathematical field of algebraic topology .

It was proved in 1976 by Daniel Marinus Kan and Bill Thurston and says that for every path-connected topological space there is a discrete group such that the Eilenberg-MacLane space is a good approximation . More precisely, there is a "homology isomorphism" ; H. a continuous mapping that induces an isomorphism in singular homology . ${\ displaystyle X}$ ${\ displaystyle \ pi}$ ${\ displaystyle K (\ pi, 1)}$ ${\ displaystyle X}$ ${\ displaystyle K (\ pi, 1) \ rightarrow X}$

Occasionally it is interpreted to mean that homotopy theory can be viewed as part of group theory .

Kan and Thurston's theorem

For each connected room there is an Eilenberg-MacLane room (for a group ) and a Serre fiber ${\ displaystyle X}$ ${\ displaystyle K (G, 1)}$ ${\ displaystyle G}$

{\ displaystyle \ iota \ colon K (G, 1) \ to X}

so that:

${\ displaystyle \ iota _ {*} \ colon \ pi _ {1} (K (G, 1)) = G \ to \ pi _ {1} (X)}$ is surjective ,
${\ displaystyle \ iota _ {*} \ colon H _ {*} (K (G, 1), A) = H _ {*} (G, A) \ to H _ {*} (X, A)}$ is an isomorphism from the group homology of to the homology of for each local coefficient system . ${\ displaystyle G}$ ${\ displaystyle X}$ ${\ displaystyle A}$

In particular, it is homotopy equivalent to the space that is obtained by applying Quillen's Plus construction to the perfect normal divider . ${\ displaystyle X}$ ${\ displaystyle K (G, 1)}$ ${\ displaystyle N: = ker (\ iota _ {*} \ colon G \ to \ pi _ {1} (X))}$

Quote

It is a long-standing joke (and for things like Mal'cev completion rather more than a joke) that group theory is contained in algebraic topology as the homotopy theory of Eilenberg-MacLane spaces K (G, 1). The paper under review has the effect of reversing the joke, showing that, in a sense, the homotopy theory of connected spaces is contained in the homotopy theory of K (G, 1) 's and thus in group theory. ( J. Peter May in discussion of Kan and Thurston's work in Mathematical Reviews )

literature

DM Kan, WP Thurston: Every connected space has the homology of a K (π, 1). In: Topology. 15, No. 3, 1976, pp. 253-258. (online ; pdf)
Dusa McDuff : On the classifying spaces of discrete monoids. In: Topology. 18, No. 4, 1979, pp. 313-320.
CRF Maunder: A short proof of a theorem of Kan and Thurston. In: Bull. London Math. Soc. 13, No. 4, 1981, pp. 325-327.
J.-C. Hausmann: Every finite complex has the homology of a duality group. In: Math. Ann. 275, No. 2, 1986, pp. 327-336.
I. Leary: A metric Kan – Thurston theorem. In: J. Topol. 6, No. 1, 2013, pp. 251–284.
Raeyong Kim: Every finite complex has the homology of some CAT (0) cubical duality group. In: Geom. Dedicata. 176, 2015, pp. 1-9.

Web links

Kan-Thurston Theorem (nLab)