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 pairings. 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.

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.