Group completion theorem

in algebraic topology

Let $ M $ be a topological monoid and $ BM $ its classifying space. Let $ M \rightarrow \Omega BM $ be the canonical mapping. Then $ H _ {*} ( M ) \rightarrow H _ {*} ( \Omega BM ) $ induces an isomorphism

$$ H _ {*} ( M ) [ \pi _ {0} ( M ) ^ {- 1 } ] \rightarrow H _ {*} ( \Omega BM ) . $$

This theorem plays an important role in $ K $- theory.

References

[a1]	D. McDuff, G. Segal, "Homology fibrations and the "group completion" theorem" Invent. Math. , 31 (1976) pp. 279–287
[a2]	J.F. Jardine, "The homotopical foundations of algebraic -theory" , Algebraic -Theory and Algebraic Number Theory , Contemp. Math. , 83 , Amer. Math. Soc. (1989) pp. 57–82 MR991976
[a3]	J.P. May, "Classifying spaces and fibrations" , Memoirs , 155 , Amer. Math. Soc. (1975) MR0370579 Zbl 0321.55033
[a4]	M.B. Barrat, S.B. Priddy, "On the homology of non-connected monoids and their associated groups" Comm. Math. Helvetici , 47 (1972) pp. 1–14 MR314940
[a5]	I. Moerdijk, "Bisimplicial sets and the group-completion theorem" , Algebraic -Theory: Connections with Geometry and Topology , Kluwer Acad. Publ. (1989) pp. 225–240 MR1045852 Zbl 0708.18008