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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.

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If you click on the "edit" button you'll see the Community Wiki checkbox near the bottom-right of the text window where the body of your (modifiable) question appears. –  Ryan Budney Mar 23 '11 at 19:00
    
Done. Thanks Ryan! –  Jessica L Mar 23 '11 at 19:07
    
Two small comments: (1) These notes don't cover Galatius' theorem on the stable homology of automorphism groups of free groups. I wish they did, but that's a harder story to tell. (2) The version of the notes linked to is a revision posted a couple days ago on March 19, so anyone who happens to have an earlier version should replace it with this new one. –  Allen Hatcher Mar 24 '11 at 0:42
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4 Answers

Bruno Harris has a neat proof of Bott periodicity that uses the group completion theorem.

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To begin with, my personal opinion and experience is that the name ''group completion theorem'' is quite bombastic and mainly a source of confusion. This is not to say that one cannot \emph{interpret} the results as ''group completions'' in a useful way. I understood the story when I read the last chapter of Galatius, Madsen, Tillmann, Weiss ''The homotopy type of the cobordism category''. It is a superb exposition, not least because the geometric and algebraic-topological arguments are cleanly separated.

In all the results you quote, the ''scanning'' (or Pontrjagin-Thom, or h-principle)arguments and the ''group completion'' arguments belong to really different parts of the story.

To explain these two different parts, let us go through the classical example of configuration spaces of points in $\mathbb{R}^n$.

The ''scanning'' part of the argument: Let $C^{n,k}$ be the space of configurations in $R^n$, contained in $I^k \times R^{n-k}$, with the topology that Hatcher describes. There are several maps that could be called scanning maps in this context. One is $C^{n,k} \to \Omega C^{n,k-1}$ and it is a homotopy equivalence if $k \leq n-1$. Another one is the homotopy equivalence $C^{n,0} \to S^n$. Both together give homotopy equivalences $C^{n,k} \to \Omega^k S^n$; both can be derived also using Gromov's general h-principle. There are no homology, no monoids or categories, let alone ''group completion'' in that part.

''Group completion'' comes in when you consider the scanning map $ C^{n,n} \to \Omega C^{n,n-1}$; it is not a homotopy equivalence. It can be stabilized and the stabilized map is a map $Z \times Conf^{\infty}(R^n) \to \Omega C^{n,n-1}$, which is a homology isomorphism.

But you asked for other examples; a classical one is Quillen's algebraic $K$-theory; and there is no ''scanning map'', in absence of a geometric model. Let $R$ be a ring and let $Proj(R)$ be a set of representatives for the isomorphism classes of finitely generated projective $R$-modules and $M:= \coprod_{P \in Proj(R)} B Aut(P)$. The statement is that there is a homology isomorphism

$$K_0 (R)\times B GL_{\infty}(R) \to \Omega BM.$$

This is the analogue of the statement on $Z \times Conf^{\infty}(R^n) \to \Omega C^{n,n-1}$, but here there is no way to interpret it as a scanning map. The statement $ \Omega C^{n,n-1} \simeq \Omega^n S^n$ does not have an analogue; basically because there is no real geometric model for $B GL_n (R)$ (the $K$-theory spectrum is no Thom spectrum!).

The proof begins with turning $M$ into something that you can take $B$ of, say a monoid under direct sum. Without a geometric model, this is not a complete triviality. Then there is a self-map $M \to M$, given by adding a trivial rank one $R$-module and form $M_{\infty} = colim (M \to M \to \ldots ) \simeq K_0 (R)\times B GL_{\infty}(R) $.

$M$ ''acts'' on $M_{\infty}$ and you can form $EM \times_M M_{\infty}$ (after you made sense out of this). There is a map $f:EM \times_M M_{\infty} \to BM$. The source of $f$ is contractible, and hence the homotopy fibre of $f$ is $\Omega BM$. The actual fibre of $f$ is $M_{\infty}$ and these observations yield a map $M_{\infty} \to \Omega BM$, which we claim is a homology isomorphism.

The translation maps $M_{\infty} \to M_{\infty}$ given by multiplication by elements of $M$ are homology equivalences. This step is easy in the situation of modules; but for the Madsen-Weiss or the Galatius theorem, the analogous argument requires homological stablity. Because the translations are homology equivalences, $f$ is a homology fibration. By MacDuff-Segal, the natural map from the actual fibre of $f$ to the homotopy fibre is a homology isomorphism, which completes the proof. I like to think about this last step as a local-to-global principle, similar in spirit to, say, the result that a fibre bundle (local condition) is a fibration (global condition).

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Ah, the algebraic K-Theory example is exactly the kind of thing I was looking for (examples of group completion more than examples of scanning -- I guess my question was a little unclear). Thanks! –  Jessica L Mar 23 '11 at 21:32
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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.

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This isn't an application of the group-completion theorem, but the theorem provides context for a problem I'm quite interested in.

Let $\mathcal K$ be the space of $C^\infty$-smooth embeddings of $\mathbb R$ into $\mathbb R^3$ which agree with the map $t \longmapsto (t,0,0)$ outside of the interval $[-1,1]$. This is the space of "long knots" in $\mathbb R^3$. There is a homotopy-equivalence between $\mathcal K \times_{SO_2} SO_4$ and the space of smooth embeddings of $S^1$ in $S^3$, which is usually thought of as "the space of classical knots".

There is a homotopy-associative pairing $\mathcal K^2 \to \mathcal K$ which in a suitable setting (up to a homotopy-equivalence of $\mathcal K$ with some other space) can be made into a strictly assosiative pairing. This pairing is the "connect sum operation".

So you can talk about the "group completion" of $\mathcal K$ with respect to connect-sum. In particular, it's homology is closely related to the homology of the space $\mathcal K$.

It turns-out that $\Omega B\mathcal K$ has the homotopy-type of

$$\Omega^2 \Sigma^2 (\mathcal P \sqcup \{*\})$$

where $\mathcal P \subset \mathcal K$ is the subspace consisting of knots that can't be expressed as connect-sums of non-trivial knots (space of all knots which happen to be prime with respect to connect-sum).

Some structure theorems for $\mathcal K$ tell us that this space contains (as a retract) spaces such as:

$$\Omega^2 \left( \bigvee_{\infty} S^2 \vee S^3 \vee S^4 \vee \cdots \right)$$

i.e. the double loop space on an infinite wedge of spaces -- containing countably-many spheres of every dimension ($S^k$ occuring a countable-infinite number of times for all $k \geq 2$).

The space $\Omega B \mathcal K$ is bigger than that, but it's a striking curiosity.

I've often thought this should fit in well with the Embedding calculus and the Vassiliev approach to knots since ultimately those are comparisons between knot spaces and things derived from configuration spaces, which when you loop, give you spaces that are loop-spaces on products of wedges of spheres.

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