Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?:

Have the homotopy groups of spheres ever been applied to

anything, including in algebraic topology itself?

It started to get some answers in those comments, but comments are a lousy place to record answers to a question like this so I'm reposting it as a question.

In order to add some more value to the question (and justify *my* reposting it), let me say that I can foresee answers coming in several different flavours and I'd like the answers to explicitly say which flavour they use.

Firstly, there is the distinction between *stable* and *unstable* homotopy groups. Briefly, there is a natural map $\pi_k(S^n) \to \pi_{k+1}(S^{n+1})$ and eventually (you will see the phrase, "in the stable range") this becomes an isomorphism. Once it is an isomorphism, we refer to them as the *stable* homotopy groups. So there are more unstable homotopy groups than stable ones, but to balance that, the stable ones are better behaved.

Secondly, there is the point that I was trying to make in the aforementioned question: the fact that the homotopy groups are so complicated is correlated with their usefulness. So there may be some uses of the homotopy groups of spheres that explicitly rely on their complexity: if they weren't so complicated, they wouldn't be able to detect *X*.

Thirdly, and partly in converse to the above, we do know some of the homotopy groups of spheres. So a use might be: because we know $\pi_7(S^{16})$ then we know *X*.

So in your answer, please indicate which of the above best fits (or if none do, try to classify it in some way). Also, please note that this is a question about the homotopy groups of **spheres**, not homotopy theory in general, and that although I'm an algebraic topologist (some of the time), answers outside algebraic topology will be more useful in "selling" our subject!

This question is a fairly obvious one for community wiki: it wasn't originally my question (though I hope that I've expanded it a little to add extra value) and I appear to be asking for a "big list". However, I suspect that the really good answers will involve some work to explain to a non-expert the key idea of why the homotopy groups of spheres are so important - merely linking to a paper will not be very satisfactory because it is likely that that paper is written for algebraic topologists rather than a general audience, and I would like to reward such efforts with the only coinage MO has. If the only answers I get are "see this paper" then I will gladly hit the "community wiki" button (indeed, if that was all I got, I'd consider closing the question).

Proposition.If $n \ge 3$, then $\sqrt[n]{2}$ is irrational.Proof:suppose $\sqrt[n]{2} = z/y$ for $z$ and $y$ positive integers. Taking $n$th powers yields that $2 = z^n/y^n$, so $y^n + y^n = z^n$. By FLT, there is a contradiction. $\Box$ See rjlipton.wordpress.com/2010/03/31/april-fool $\endgroup$ – Steve Huntsman Apr 28 '10 at 12:10