at.algebraic topology - Group completion theorem

Well, if $pi_0=pi_0(M)$ is already a group, then $H_*(M)approx H_*(M)[pi_0^{-1}]$. So $M$ and $Omega B M$ have the same homology in this case. This isn't quite enough on its own, but if you can produce a map $Mto Omega BM$ which induces this homology isomorphism, then the result follows using the Hurewicz theorem.

What McDuff-Segal actually do is show that if $M$ is a topological monoid which acts on a space $X$, in such a way that every $min M$ induces a homology equivalence $xmapsto mxcolon Xto X$, then you can produce a "homology fibration" $f:X_Mto BM$ with fiber $X$. "Homology fibration" means that the fibers of $f$ are homology-equivalent to the homotopy fibers of $f$.

If $pi_0M$ is an abelian group, you can find an $X$ such that $X_M$ is contractible, and the fiber of $f:X_Mto BM$ is $X$. This gives the homology equivalence you want, since the homotopy fibers of $f$ look like $Omega BM$.

Take a look at McDuff and Segal's paper, it's nice. There is a also a treatment in terms of simplicial sets in Goerss-Jardine, *Simplicial Homotopy Theory".

Added: The functor $Mmapsto Omega BM$ is the "total derived functor of group completion". The only convincing explanation of why this is so (that I'm aware of) is in Dwyer-Kan, Simplicial Localizations of Categories, JPAA (17) 267-283. Though they work simplicially, and work more generally (with categories in place of monoids), they show that $M$ is a cofibrant simplicial monoid, then the simplicial monoid $M[M^{-1}]$ is weakly equivalent to the space $Omega |BM|$.

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