Group completion

Given a topological monoid $ M$, the group completion is the space $ \Omega BM$, where $ BM$ is the classifying space of $ M$ (thinking of $ M$ as a topological category). If $ M$ is already a topological group, then this operation does not change $ M$ up to homotopy equivalence. Under some assumptions, we have the following description of the homology of the group completion. If we treat the monoid $ \pi_{0}(M)$ as a directed system (with maps given by the monoid operation), then

$\displaystyle \lim_{\alpha \in \pi_{0}(M)} H_{\ast}(M_{\alpha}) = H_{\ast}((\Omega

in situations where the direct limit on the left-hand side is well-defined. In particular, if we consider the monoid $ \amalg_g
B\Gamma_{g,2}$, the homology of the group completion is precisely the stable homology. The plus construction can often be used to give an alternative construction of the group completion.

Jeffrey Herschel Giansiracusa 2005-06-27