TITLE: Applications of the group completion theorem
QUESTION [9 upvotes]: I've been reading Hatcher's exposition of the Madsen-Weiss theorem here.  One of the key results needed is the group completion theorem.  Hatcher gives several applications of the group completion theorem (the Barratt-Priddy theorem relating the stable cohomology of the symmetric group to the stable stems, the description of the loop space associated to the stable braid group, the Madsen-Weiss theorem, and Galatius's theorem on the stable cohomology of automorphism groups of free groups).  Looking around the internet, I have not been able to locate other applications.  The above theorems are all proven in kind of the same way (using "scanning").  Can people provide other interesting applications of the group completion theorem?
This should probably be community wiki, but I can't seem to find the box to check to make it so.

REPLY [3 votes]: I use the group completion theorem quite a lot.  
For example, I used it (along with Yang-Mills theory and work of Tyler Lawson) to study the spaces Hom$(\pi_1 S, U)$ and Hom$(\pi_1 S, U)/U$ when $S$ is an aspherical surface (or a product of aspherical surfaces) and $U =$ colim $U(n)$ is the infinite unitary group.  These spaces turn out to be the homotopy group completions of the monoids $\coprod_n$ Hom$(\pi_1 S, U(n))$ and $\coprod_n$ Hom$(\pi_1 S, U(n))/U(n)$, respectively.  Here the monoid structure comes from block sum of unitary matrices.  This means, in particular, that both of these representation spaces are infinite loop spaces (because block sum is commutative up to coherent isomorphisms).  For orientable surfaces, the picture is quite nice: 

Hom$(\pi_1 S, U)/U \simeq (S^1)^{2g} \times \mathbb{C} P^\infty$.

The first factor can be seen entirely via the determinants of representations ($g$ is the genus).  The second factor gives a (non-canonical) 2-dimensional cohomology class (or line bundle) over the moduli space of representations.  This should be a reflection of Goldman's symplectic form, but I don't have any idea how to prove that.
The group completion story in this case is spelled out in quite a bit of detail in my paper "Excision for deformation K-theory..." (AGT, 2007).  It goes back to the basic ideas of McDuff and Segal.  The applications to surface groups show up in "The stable moduli space..." (Trans. AMS, 2011).  You also can find these papers on my webpage or on the arXiv.