What are the uses of the homotopy groups of spheres?

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).

-
@Pete: I agree with your second sentence completely. However, if you had turned it round, I would have been unhappy. My contention (and I feel as though it's getting more exposure than it's worth!) is that for something to convey lots of information, it must be complicated in some way. (I don't like the word "deep", but that's just a personal dislike.) As for your emphasised sentence, again I'm in complete agreement. But nonetheless, I'd like to know if they have had significant applications! If nothing else, I could sell my subject a bit better than I do at the moment! –  Andrew Stacey Apr 28 '10 at 12:07
Here is a (silly) application of FLT due to Schultz that is on Dick Lipton's blog. 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 –  Steve Huntsman Apr 28 '10 at 12:10
This question seems related: mathoverflow.net/questions/16495/… –  j.c. Apr 28 '10 at 14:35
Well, I'd say more than related. –  Andrea Ferretti Apr 28 '10 at 16:15

I used to think that the entire theory was intellectual masturbation, but two examples in particular completely changed my mind.

The first is the Pontryagin-Thom construction, which exhibits an isomorphism between the $k$th stable homotopy group $\pi_{n+k}(S^n)$ and the framed cobordism group of smooth $k$-manifolds. This is even interesting (though more elementary) in the case $k = 0$, where it recovers the basic degree theory that you learn in your first course on topology. This was originally developed by Pontryagin to compute homotopy groups of spheres, but now it is regarded as a tool in manifold theory. These matters are discussed in Chapter 3 of Luck's book on Surgery theory, for example.

The second application is to physics. Unfortunately I don't understand this story very well at all, so I'll begin with what I more or less DO understand (which may or may not be well-known). The basic idea begins with the problem of situating electromagnetism in a quantum mechanical framework. Dirac began this process by imagining a "magnetic monopole", i.e. a particle that would play the role for magnetic fields that the electron plays for electric fields. The physical laws for a charged particle sitting in the field determined by a magnetic monopole turn out to depend on a choice of vector potential for the field (the choice is necessarily local), and Dirac found that changing the vector potential corresponds to multiplying the wave function $\psi$ for the particle by a complex number of modulus 1 (i.e. an element of U(1)). If we think of the magnetic monopole as sitting at the origin, then these phases can naturally be regarded as elements of a principal $U(1)$-bundle over $M = \mathbb{R}^3 - \{0\}$. But $M$ is homotopy equivalent to $S^2$, and principal $U(1)$-bundles over $S^2$ are classified by $\pi_1(U(1)) = \mathbb{Z}$. Proof: think about the Hopf fibration. The appearance of the integers here corresponds exactly to the observation of Dirac (the Dirac quantization condition) that the existence of a magnetic monopole implies the quantization of electric charge. It is remarkable to note that Hopf's paper on the Hopf fibration and Dirac's paper on magnetic monopoles were published in the same year, though neither had any clue that the two ideas were related!

The story goes on. The so-called "Yang-Mills Instantons" correspond in a similar way to principal $SU(2)$ bundles over $S^4$, which are classified by $\pi_3(SU(2)) = \mathbb{Z}$. Again, the integers have important physical significance. So these two classical examples motivate the computation of $\pi_1(S^1)$ and $\pi_3(S^3)$, but as is always the case this is just the tip of an iceberg. I am not familiar with anything deeper than the tip, but I have it on good authority that physicists have become interested in homotopy groups of other spheres as well, presumably to classify other principal bundles (it seems like a bit of a coincidence that the groups which came up in these examples are spheres, but maybe one reduces homotopy theory for other spaces to homotopy theory for spheres). People who know more about physics and/or the classification of principal bundles should feel free to chime in.

A great reference for the mathematician who wants to learn something about the physics that I discussed here is the book "Topology, Geometry, and Gauge Fields: Foundations" by Naber.

-
I would be amazed (and delighted) to learn of any applications in physics of nontrivial homotopy groups of spheres π_m(S^k) where m or k get very large. –  j.c. Apr 28 '10 at 15:08
@jc: Hopkins hints at an invariant of manifolds that is not known or constructed. I am not sure how exactly this works, but the reference i am thinking of is his talk at santa barbara online.kitp.ucsb.edu/online/mp03/hopkins4 He is working by analogy in that other invariants of manifolds correspond to certain elements in the homotopy of other spectra, such as ko. I dont really understand this story very well, but I would like to. –  Sean Tilson Apr 28 '10 at 22:58
My friend's question that inspired the original question was inspired by the applications to physics. –  Andrew L Apr 29 '10 at 3:26
There's also a great introduction in Frankel's book. –  Andrew L Apr 29 '10 at 3:28
I'm not sure if this ends up having a connection to the homotopy groups on spheres, but part of the AdS/CFT correspondence is the idea (due to t'Hooft) that String Theories can be viewed as "large-N" Gauge Theories with gauge group $\mathsf{SU}(N)$ [0]. As mentioned in reference [0], the idea is to turn YM fields into a set of $\mathsf{SU}(N)$ fields and thehn take a "limit" (probably ill-defined) of $N\uparrow \infty$. In the paper they show that the propagator of theory depends on 'large-N sphere topology of spheres.' [0] Section 1.2 of hep-th:9905111 –  Tarun Chitra Mar 23 '13 at 4:02
show 1 more comment

Here are four applications, but some of them are cheating. I'd like to give more examples involving homotopy groups of more general spaces, but will try not to.

(Tagged cw - feel free to add.)

• The fact that $\pi_k(S^n) = 0$ for $k < n$ is used implicitly all the time; this high connectivity means there are few obstructions to embedding small complexes $X$ into $S^n$, and hence having tools like Alexander duality available to study $X$.
• In another question the classification of simply-connected 4-manifolds up to homotopy type is mentioned. This uses $\pi_3$ of a wedge of 2-spheres, so technically it's not what the original question asked.
• The existence of any kind of unital multiplication $S^n \times S^n \to S^n$, such as one that would be obtained by a division algebra structure on $\mathbb{R}^{n+1}$, is equivalent to a certain element existing in stable homotopy groups of spheres (a class of Hopf invariant one). Adams proved that this element is not realized in a homotopy group unless n is 0, 1, 3, or 7.
• The more serious computations of algebraic K-theory groups using topological cyclic homology, such as the K-groups of the integers, often require serious input knowledge from stable homotopy theory. For example, when Ausoni-Rognes computed the V(1)-homology groups of $K(\ell)$, they used knowledge about the stable homotopy groups of spheres to identify certain classes as necessarily survivors of several of the many spectral sequences involved. This is not the deepest part of their work by any means, but is supposed to indicate that this kind of knowledge is "jacks or better to open".
-

In a very influential paper called An SU(2) anomaly, Edward Witten employed $\pi_4(S^3) \cong \mathbb{Z}/2\mathbb{Z}$ to derive a constraint on the field content of a mathematically consistent (read, anomaly-free) $\mathrm{SU}(2)$ gauge theory.

-

Other posters have alluded to the Kervaire-Milnor theory (from "Groups of homotopy spheres. I") which shows how, via Pontryagin-Thom, our knowledge and ignorance about the stable homotopy groups of spheres is reflected in knowledge and ignorance about classification of manifolds. Maybe it's worth telling this (really beautiful!) story.

In each dimension $n$, one has a group $\Theta^n$ of smooth $n$-manifolds that are homotopy $n$-spheres, up to h-cobordism, under connected sum. This has a subgroup $bP^{n+1}$ of boundaries of parallelizable $n+1$-manifolds. Assume $n>4$, so h-cobordism classes are diffeomorphism classes.

Every homotopy $n$-sphere $S$ can be shown to have a stable framing. Hence (by P-T) $S$ is a regular fibre of a map $S^{n+k}\to S^k$ for $k\gg 0$ whose class in $\pi_{n+k}(S^k)$ is the obstruction to $S$ (with chosen stable framing) being a framed boundary. Changing the stable framing amounts to adding something in the image of the J-homomorphism $J: \pi_n(SO(k))\to \pi_{n+k}(S^k)$. So we get an injective homomorphism $\Theta^n/bP^{n+1}\to coker(J)$ (which is onto e.g. for $n$ odd).

We don't know $coker(J)$ in high dimensions, so we don't know the order of $\Theta^n/bP^{n+1}$. But Serre's finiteness theorem for the stable stems tells us that $\Theta^n/bP^{n+1}$ is finite!

The subgroup $bP^{n+1}$ is analyzed via surgery theory and the h-cobordism theorem. There's a nice summary in Lück's Basic introduction to surgery theory.

We have $bP^{odd}=0$. There's a formula for $bP^{4p}$ involving Bernoulli numerators; this comes from a known (thanks to Adams) part of the stable stems, namely, $im (J)$.

Finally, $bP^{4p+2}$ is at most $\mathbb{Z}/2$. Here $S$ bounds a parallelizable manifold $P$. We'd like to make this contractible. By framed surgery, we can kill the homotopy groups of $P$ below the middle dimension but the Arf invariant of the pairing on middle-dimensional homology obstructs the final step, that of killing $\pi_{p+1}$. Say this is non-zero. Can we do better by starting with a different $P$? Yes, if and only if there's a closed, framed $2p+2$-manifold of Kervaire invariant one.

Browder showd that the Kervaire invariant can be one only when $4p+2=2^l-2$ for some $l$, and Hill-Hopkins-Ravenel have shown that $l\leq 7$. Conclusion: $bP^{4p+2}$ is $\mathbb{Z}/2$ except in dimensions 6, 14, 30, 62, and possibly 126, where it's zero.

-

If I may try my hand at an answer:

I don't think anyone with an iota of familiarity with algebraic topology has any serious doubts about the importance of the computation of the homotopy groups of spheres. What little is known about them is connected to very deep algebraic and topological information. For instance, as someone else commented on the related question, if one understood even the stable homotopy groups of spheres very well, one would therefore have a near complete understanding of the group (I assume that $n \neq 4$) of differentiable structures on the $n$-sphere: see e.g.

http://en.wikipedia.org/wiki/Exotic_sphere

The tightness of the relationship between homotopy groups and differentiable structures recently took a big step forward via the work of Hill-Hopkins-Ravenel. (I was fortunate enough to hear a very nice expository talk on this by Prof. Michael Ching of UGA.)

I am already at the limits of my meager knowledge in this area, but I surmise that there would be other dramatic consequences of a systematic computation of homotopy groups of spheres.

The homotopy groups of spheres are also extremely challenging to compute. My understanding is that ever since Serre's work circa 1950, every generation of algebraic topologists has had a small number of luminaries who push the theory further -- modestly, but intriguingly. Therefore the combination of certified applicability to other results and intrinsic difficulty makes it easy to see why this has remained an irreresistible problem for algebraic topologists. In a roughly similar way, the Taniyama-Shimura conjecture was made irresistible by Ribet's proof in the late 80's that it implies Fermat's Last Theorem.

What is not clear to me (and again, I am an outsider in these matters to say the least) is to what extent anyone has ever been able to say, "Aha, I have computed sufficiently many homotopy groups of spheres in order to deduce the following striking consequence in [say] geometric topology". So far as I know, it may well be the case that the information flows in the other direction: one finds some link between homotopy groups of spheres and some fascinating and slightly more tractable other structure...and uses this to push a little further on the computation of homotopy groups of spheres. I.e., I wonder whether in practice, the homotopy groups of spheres are the end, not the means.

In a perhaps too brief summary: if the homotopy groups of spheres were less complicated, they would probably be more directly useful but also less interesting.

-
"I wonder whether in practice, the homotopy groups of spheres are the application, not the tool." I would say this is fairly on the mark, though I wouldn't use the word "application." Homotopy groups of spheres in part serve as a measuring stick for our understanding of homotopy theory. For example, one way to measure the depth of the construction of elliptic cohomology was the consequence of its existence for homotopy groups of spheres. I once heard a number theorest talk about special values of L-functions similarly, as a good place to measure progress. –  Dev Sinha Apr 28 '10 at 15:35
@Dev: Thanks for your comment; I changed the language slightly to something which I hope is more apt. By the way, I agree with you that special values of L-functions are a reasonable number-theoretic analogue of homotopy groups of spheres. –  Pete L. Clark Apr 28 '10 at 16:20
"For instance, as someone else commented on the related question, if one understood even the stable homotopy groups of spheres very well, one would therefore have a near complete understanding of the group (I assume that n=4 ) of differentiable structures on the n -sphere". Wasn't the set of differentiable structures on a given $n-$sphere just a cyclic group under connected sum? What is there to be understood about the structure of that group? –  Qfwfq Apr 28 '10 at 17:33
As you can see in the very nice Wikipedia article that Pete linked to, the group of differentiable structures on the n-sphere has a cyclic subgroup, but the cokernel injects into the quotient of the n'th stable stem by the image of the J-homomorphism (it is usually isomorphic). For example, the group of differentiable structures on S^9 has Z/2 x Z/2 as a quotient. –  Tyler Lawson Apr 28 '10 at 17:52
I think that TMF and TAF may provide a relationship between homotopy groups of spheres and special values of L-functions. –  Spice the Bird Nov 14 '12 at 5:11

My favourite application of the stable homotopy of spheres is the Rokhlin theorem that the signature of a compact smooth spin 4-manifold is divisible by 16. Rokhlin proved this as a corollary of πS3 the third stable homotopy group of spheres being cyclic of order 24. There are other nice proofs now (using the Hirzebruch signature theorem for example), however.
This theorem leads to the Rokhlin invariant, which is quite an important invariant in 4-dimensional topology.

-
Isn't this also a consequence of the index theorem for the Dirac operator? –  José Figueroa-O'Farrill Apr 29 '10 at 14:00
Yes, but you can calculate the third stable stem pretty easily using the Serre spectral sequence, which seems to me to be a heck of a lot more elementary than the index theorem. But that probably says more about me than about the index theorem... –  Andy Putman May 6 '10 at 0:44

Knowing homotopy groups of spheres would help tremendously in classifying manifolds (up to diffeomorphism) in a given homotopy type. Indeed, one of the big stumbling blocks in the classification problem is poor knowledge of the structure of $F/O$, the homotopy fiber of the $J$-homomorphism $BO\to BF$, where $BF$ is the classifying space for stable spherical fibrations. The space $F/O$ appears in the smooth surgery exact sequence. The homotopy groups of $F$ are the stable homotopy groups of spheres, and getting a firm grip on $F/O$ is hard precisely because it is hard to understand homotopy groups of spheres.

-
Igor, I guess the easy(ish) way out is to prove classification theorems up to connect sum with a homotopy-sphere. But I wonder how much we lose this way. Does connect summing with an exotic sphere always changes the manifold? –  Tim Perutz Apr 30 '10 at 3:16

Sorry for posting this as an answer. This should actually be a comment (to the Paul's and Jose''s answers).

I doubt that there are (and will be in the nearest future at least) any applications to physics of higher homotophy groups of spheres, which are not Lie groups. Of course, something may come up accidentally, but this will hardly be a conceptual application.

So, my point is that the applications to physics mentioned above are misleading. Spheres which appear there admit lie group structure, and applications mentioned illustrate only the fact that Lie groups are tremendously important in physics. There is no single hope for spheres $S^n$ with $n \geq 10$ to have such sorts of applications in physics.

Of course, someone may cook up a "homotopical quantum field theory", where the key role will be played by spheres and their homotopy groups, but such a theory will hardly agree with experiment. (At least, at the current state of knowledge.)

In fact, I don't quite understand why someone could worry about "why the knowledge of homotopy groups of spheres is useful". I think the definition of homotopy groups is very natural (much more natural than many other definitions in topology which I have seen, at least to my (rather limited) understanding), and spheres are one of the simplest geometrical species. So, it is reasonable to try to compute those groups for the spheres.

The fact that we can't do this easily is already very important. It signifies that our computational power is rather (probably, shamely) limited. I don't think that this fact per se says anything about the importance of homotopy groups of spheres. They are important because they are natural things to ask and to compute. They also provide us with a challenge. There are many other things in math which are also not known.

Let me also mention, that as it is being repeated often, (here I refer to what is commonly said, am not a historian) the whole modern algebraic number theory appeared as a result of people's work attempting to solve Fermat Last Theorem. Could someone even ask "what are the applications of FLT in number theory"? Without FLT, there would be no algebraic number theory at all. (at least in the form we know it today)

-
"There is no single hope..." The history of Mathematical Physics has proved many such statements wrong -- what Wigner called the "unreasonable effectiveness of Mathematics in the Natural Sciences." –  José Figueroa-O'Farrill Apr 29 '10 at 14:04
Well, I just stressed the confidence that today nothing indicates a possibility for such applications in Physics. Who knows what can happen, and what will be applied in future? –  rat Apr 29 '10 at 15:05
i am curious what you would think of the audio i reference on my comment to paul's answer. have you had a chance to listen to it? I will interpret your main point as follows: The applications people are mentioning are for the most part related to Stable homotopy groups of spheres whereas the unstable ones are more interesting to physicists for obvious reasons and further that the applications mentioned earlier are more physical because they are really about lie groups. So should homotopy theorists be spending more time thinking about lie groups? ... –  Sean Tilson Jun 12 '10 at 22:26
...or is the nature of homotopical results (lack of certain "physical" rigidity) going to be a hinderance in application? –  Sean Tilson Jun 12 '10 at 22:26