View source for Group completion theorem

''in algebraic topology''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102601.png" /> be a topological [[Monoid|monoid]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102602.png" /> its classifying space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102603.png" /> be the canonical mapping. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102604.png" /> induces an isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102605.png" /></td> </tr></table>

This theorem plays an important role in [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102606.png" />-theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. McDuff, G. Segal, "Homology fibrations and the "group completion" theorem" ''Invent. Math.'' , '''31''' (1976) pp. 279–287</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.F. Jardine, "The homotopical foundations of algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102607.png" />-theory" , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102608.png" />-Theory and Algebraic Number Theory'' , ''Contemp. Math.'' , '''83''' , Amer. Math. Soc. (1989) pp. 57–82 {{MR|991976}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.P. May, "Classifying spaces and fibrations" , ''Memoirs'' , '''155''' , Amer. Math. Soc. (1975) {{MR|0370579}} {{ZBL|0321.55033}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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 {{MR|314940}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> I. Moerdijk, "Bisimplicial sets and the group-completion theorem" , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110260/g1102609.png" />-Theory: Connections with Geometry and Topology'' , Kluwer Acad. Publ. (1989) pp. 225–240 {{MR|1045852}} {{ZBL|0708.18008}} </TD></TR></table>