There have been a couple of [posts](https://mathoverflow.net/questions/8846/proofs-without-words/8883#8883) and [questions](https://mathoverflow.net/questions/35868/fundamental-group-of-lie-groups) 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](http://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_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.




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\* 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.



Applications of Eckmann-Hilton argument to topology
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. 


