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

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

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.