What are the uses of the homotopy groups of spheres?

Question

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!

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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).

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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".

Answer 5

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.

Answer 6

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 π^S₃ 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.

Answer 7

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.

Answer 8

If you believe that CW-complexes are nice spaces, then two-cell complexes are among the nicest spaces of all. These are spaces of the form $X = S^n \cup_\alpha e^{m+1}$, where $\alpha: S^m\to S^n$ is a map of spheres (technically there are three cells, but never mind).

The cohomology of a two-cell complex is usually not very interesting as an algebra for dimensional reasons, but it may support non-trivial cohomology operations which allow you to detect non-triviality of $\alpha$.

On the other hand, particular choices of $\alpha$ may produce two-cell complexes with very interesting homotopy properties. A famous case of this is due to Iwase, who exhibited two-cell complexes $X$ which give counterexamples to the Ganea conjecture on LS-category. The conjecture, which was open for around 30 years, asked whether $\operatorname{cat}(X\times S^k) = \operatorname{cat}(X)+\operatorname{cat}(S^k)$ for all finite complexes $X$ and $k\ge 1$.

The point is, without an intimate knowledge of the homotopy groups of spheres, Iwase wouldn't have known where to start looking for such spaces, and LS-category theorists would probably still be none the wiser about Ganea's question (which on the face of it has nothing to do with homotopy groups of spheres).

Answer 9

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)

Answer 10

I am just reporting something I am not familiar with, but apparently the existence of the Breen-Deligne resolution for Abelian groups requires finite generation of stable homotopy groups of spheres.

The result is the following

Theorem (Breen–Deligne). For an abelian group $A$, there exists a resolution of the form $$ \dots → \bigoplus_{j = 0}^{n_i} ℤ[A^{r_{i,j}}] → \dots → ℤ[A^2] → ℤ[A] → A → 0 $$ that is functorial in $A$.

The proof appears for instance in Appendix to Lecture IV in Scholze's lectures on condensed mathematics and I learned about this example from this answer.

In particular, the fact that the proof (that I have not read) requires finite generation of stable homotopy groups of spheres is a comment by @ReidBarton