There have been a couple of posts and questions on MathOverflow about the proofs of the following two facts:
Fact 1: if $X$ is a topological space, then $\pi_k(X,x)$ is abelian for $k\ge 2$.
Fact 2: if $G$ is a topological group, then $\pi_1(G,e)$ is abelian.
Both facts can be proven using the Eckmann-Hilton argument, which is a cool algebraisation of two more topological proofs that one can get by staring for sufficiently long to a couple of pictures* (after drawing them): while, up until two hours ago, I thought that the two proofs were actually distinct, Ryan Budney proved me wrong and showed me the connection (see comments below)
Now, to the question:
Is there any other example of an incursion of the Eckmann-Hilton argument into the realm of topology? Is there any other such application outside category theory/algebra?
I would like to see some results for which no proof is known that doesn't make use of the E-H argument, or such that any proof avoiding the argument (even somehow disguised) is significantly longer/harder.
EDIT: Thanks to Ryan Budney for pointing out how the two kind of proofs are actually the same proof, and to Tom Goodwillie for making me realise that the question was a bit too rough in an earlier version.
* One such picture is linked above. The other one is just a square with the diagonal, representing the map $(s,t)\mapsto (f(s),g(t))$: this picture gives homotopies between $f\cdot g$ and $f*g$, so they recover as much information as the E-H argument produces, see the question linked above.
Best Answer
I can't resist pointing out that while the EH-argument shows that a group object in the category of groups is an abelian group, this does not apply to a group object in the category of groupoids, which is equivalent instead to a crossed module, which represents a pointed, connected homotopy 2-type.
Higher groupoids are in some sense ``more nonabelian'' than groups, and provide a route to some nonabelian calculations in higher homotopy theory.
So circumstances in which the EH-argument fails (for example a group object in the category of semigroups) are maybe more interesting.