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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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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?" – Marty Feb 26 '10 at 22:56
up vote 6 down vote accepted

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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In many physical problems S^n symmetry grouo is important internal or even for smal n physical symmetry of tre system. For such system it is convinient to check for topological invariants which rise to conserved quantities during evolution. Article from wikipedia says on solitons:

A topological soliton, or topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution." Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise almost always because the differential equations must obey a set of boundary conditions, and the boundary has a non-trivial homotopy group, preserved by the differential equations.

If internal symmetry is symmetry of sphere then number of windings is one of examples of such property ( and in fact is the simplest one). You may find such creatures ( topological solitons) not only in string theory ( too speculative for physics interesting and great for mathematics) but also in liquid crystal physics, solid state theory ( in Ising models for example) etc. You may even made by Yourself model of such physical system at home by gluing matches or sticks to a thread and twisting such "chain". This model is used in demonstration for physicist, and it is related to Sine-Gordon equation. which appears in theory of Josephson junction fro example.

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I don't see much connection to the question, besides some word-association. The question asks specifically about the homotopy groups of spheres (beyond $\pi_1$, of course!). I don't see how this response could be useful. – Marty Feb 26 '10 at 12:18
I do not understand. Could You point me where in the question I answer is any requirement that i>1? veit79 asks for application of homotopy groups which I gave. Do You not agree that winding number is topological invariant and appears in sine-Gordon equation which may be applied in Josephson junction ( and liquid crystals also) theory? Fro sine-Gordon equation configuration space for field is $S^1$ for liquid crystals it is in some cases $S^3$. There are also models in Field Theory where You using $S^n$ directly and even in limit $n-> \infinity$ but this is pure theoretical. – kakaz Feb 26 '10 at 13:19
Why the question does not make the $i>1$ requirement, it is generally understood that "homotopy groups of spheres" refers to the higher homotopy groups, the ones that are hard to compute and therefore those for which one would love to have applications which justfy the effort! The computation of $\pi_1$ of the spheres can be done one or two classes after having defined homotopy in a topology course, so, while the knowledge of $\pi_1(S^1)$ is surely a nice example and a very useful example, it is not the kind of example the question as in mind (in all likelihood...) – Mariano Suárez-Alvarez Feb 26 '10 at 15:09
Conversely, that the higher homotopy groups of $S^1$ are trivial, this has more applications than I can count. – Ryan Budney Feb 26 '10 at 15:13

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