# 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
How to Cite This Entry:
Group completion theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_completion_theorem&oldid=47143
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article