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The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is a 'stable range' of indices which one understands better than the unstable part.

I think that there is an analogy (have to be careful with that word) to the distribution of primes: It seems that there exists a general pattern but no one has found it yet. It is a construction producing an infinite list of numbers (or groups) but no numbers were put into it. Such a thing always fascinates me.

The largely unknown prime pattern leads to applications in cryptography for example. Are there similar applications of the knowledge (or not-knowledge) of the homotopy groups of spheres? Are there applications to real natural sciences or does one study the homotopy groups of spheres only for their inherent beauty?

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    $\begingroup$ I don't think there's an analogy between homotopy groups of spheres and the distribution of primes in any way. Yes, they both yield some numbers which are not very well understood -- but you can't say that any two poorly understood sequences are analogous, just because they're both poorly understood! So I think these questions are not that well-motivated, at least by the middle paragraph above. On the other hand, I too wonder about questions like "what are homotopy groups of spheres good for, outside of the easy and obviously useful cases?" $\endgroup$
    – Marty
    Commented Feb 26, 2010 at 22:56

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A few comments on applications that aren't covered by the above Wikipedia article.

I don't know any applications to cryptography. Most cryptosystems require some kind of one-way lossless function and it's not clear how to do that with the complexity of the homotopy groups of spheres. Moreover, the homotopy-groups of spheres have a lot of redundancy, there are many patterns.

There's work by Fred Cohen, Jie Wu and John Berrick's where they relate Brunnian braid groups to the homotopy-groups of the 2-sphere. It's not clear if that has any cryptosystem potential but it's an interesting aspect of how the homotopy-groups of a sphere appear in a natural way in what might otherwise appear to be a completely disjoint subject.

Homotopy groups of spheres and orthogonal groups appear in a natural way in Haefliger's work on the group structure (group operation given by connect sum) on the isotopy-classes of smooth embeddings $S^j \to S^n$. I suppose that shouldn't be seen as a surprise though. Moreover, it's not clear to me that this is always the most efficient way of computing these groups. But I think all techniques that I know of ultimately would require some input in the form of computations of some relatively simple homotopy groups of spheres.

I think one of the most natural applications of homotopy groups of spheres, Stiefel manifolds and orthogonal groups would be obstruction-theoretic constructions. Things like Whitney classes, Stiefel-Whitney classes and general obstructions to sections of bundles. Not so much the construction of the individual classes, more just the understanding of the general method.

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Wikipedia already gives a list of some examples.

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