Group completion theorem

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in algebraic topology

Let be a topological monoid and its classifying space. Let be the canonical mapping. Then induces an isomorphism

This theorem plays an important role in -theory.


[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:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article