In Infinite Commutativity: Part I, I described infinite commutativity in general terms with some basic examples, including real infinite series. In this post, I’ll discuss how the homotopy groups , are more than just abelian in the ordinary sense. Their infinite product operations behave like absolutely convergent series of real numbers in the sense that they are invariant under infinite permutation. I’m not going to put you through the technical proofs of the most general results. I just really love it so I’m going to try and make it as accessible and understandable as possible. The natural infinitary operations turn out to be commutative in a very strong, infinite sense. The approach I like also highlights the existence of a natural extension of the little n-cubes operad to a space consisting of configurations of infinitely many disjoint -cubes.
History and Recent Work
The idea that higher homotopy groups are infinitely commutative is only 20 years old (due to Eda-Kawamura) because the proof is fairly intricate and it could easily seem overwhelming if you don’t already know how it goes. You can find it implicitly built into the main proof in  but even though I understand the fundamental ideas of the proof, beware that the details there are difficult to read. This idea was also used by Kawamura in  to study the higher dimensional earrings (sometimes called Barratt-Milnor Spheres) Following  and , infinite commutativity in has become something of “folklore” among experts.
In the effort to formalize the infinite product framework for pushing the homotopy theory of locally complicated spaces, I’ve worked both on infinite commutativity in fundamental groups with a former student  and also higher homotopy groups in . Part of my recent work in  and  is to make this business all more practical for computations and future applications. So I’ve been trying to formalize, simplify, and strengthen the set of available tools. This kind of math is really fun (in my opinion) and has potential to open up a lot of new avenues for research. The truth is that, right now, there are many fundamental and important open questions related to wild higher homotopy groups that are likely to become more accessible with the development of new tools/foundations, so lately I’ve been thinking a lot about these ideas.
Infinite Products in Higher Homotopy Groups
Let’s start with ordinary infinite products indexed by the natural numbers at the n-th loop space level. Since we’re talking about higher homotopy, will always be an integer .
For a based space , let denote the space of relative maps with the compact open topology. I’ll refer to the elements of as n-loops. In Part I, I defined the usual binary concatenation operation , . Let’s see if we can build up infinitary operations on n-loops. Since the domain .
Definition: Let be a countable indexing set. A collection of maps is a null if for every neighborhood of , all but finitely many of the maps have image in .
If is the natural numbers, then we call a null-sequence. Such a sequence is exactly a sequence in that converges to the constant map at . Let be the space of null-sequences in topologized as a subspace of the infinite direct product .
Let’s start with the obvious first choice: given a null-sequence of maps , we may for the infinite concatenation , which is defined as on and . This is continuous at the points of precisely because is a null-sequence.
By taking the homotopy class of , we get an operation . This operation is technically a function .
Warning/Red Alert!! We may want to jump quickly to writing down a partially-defined infinite sum operation of homotopy classes by setting . The operation is “partially-defined” because the concatenation is continuous if and only if is a null-sequence. Moreover, such an operation would only be well defined on homotopy classes if homotopic factors result in homotopic products, i.e. whenever for all . This happens a lot of the time, but not always. Higher-dimensional analogues of the harmonic archipelago are counterexamples.
Here’s how infinite commutativity works for these particular kinds of infinite products:
Lemma: If is a null-sequence of n-loops and is any bijection, then and are homotopic by a homotopy with image in .
Idea of the Proof. Let’s not do a technical proof here. Instead, let’s just see how this is done in pictures in dimension with an example. Suppose is a null sequence and that is a specific bijection where , , , , , and so on….. To prove the lemma in this case, we’ll need to show that
is homotopic to
Our pictures will look a lot like the Eckmann-Hilton gif from Part I but with infinitely many rectangles sliding around instead of just two. I’m afraid that I wasn’t ambitious enough to animate this so all you’ll get is a sequence of still images.
Throughout the following sequence of steps, the blue squares will represent the domain of the maps with the appropriate affine correction to the domain. If we start with the infinite product where the domains are rectangles, of height , we may shrinking these rectangles within themselves to squares of side-length so that the horizontal projects of the squares have disjoint interiors.
Now that we’ve done a vertical “reparameterization” we want to switch to a horizontal slide/stretch. This is where comes into play. Since , we want to move the new square domain for horizontally to where used to be. As we move it, we should stretch it to width . Similarly, since , we should move the square domain for horizontally to where use to be. As we move it, we should stretch it to width . Do this for all already! Since we had initially adjusted the heights of the domains to be disjoint we can accomplish this in one infinite simultaneous shifting process.
Now that we’ve taken care of things horizontally, it’s now a matter of some more vertical stretching to ensure that what we end up with is precisely an infinite product.
Finally, we end up with where the -th factor is because of our careful horizontal shifting in the middle step. Putting this all together gives a continuous homotopy . Let’s see if we can visualize the entire homotopy all at once…Remember the second animation from Part I (see figure on the left)?
Now imagine the same animation in this situation. There will be infinitely many cylinders with rectangle cross-sections twisting around each other in just the right way! This visualization now might have you worried that shifting around infinitely many rectangles is going to make this thing discontinuous (a valid concern!) but remember that the domain of the homotopy here is the cube . The homotopy is definitely continuous when restricted to each twisted cylinder because its restriction to each cross-section is just . Also, maps to the basepoint . So the only “hard” case is checking continuity of at a boundary point of some twisted cylinder . But maps to and if is a neighborhood of , then for all but finitely many (since is a null-sequence). This means for all but finitely many cylinders . Combined with the earlier observations about where already restricts to a continuous map, it’s possible to find an open neighborhood of with (filling in the details is a good exercise for those newer to topology). This means is continuous. .
Hooray! We conclude that homotopy classes of infinite products in higher homotopy groups don’t change if you start shuffling around factors. So when infinite sums are well defined and the notation for null-sequences makes sense, these sums behave very much like absolutely convergent series of real numbers in ! But not all real infinite series are absolutely convergent. In this sense, infinite sums in wild homotopy groups are a fair bit nicer than infinite sums of real numbers. If you want to get a little fancier with your terminology, another way to phrase the above lemma is: homotopy classes of infinite products are invariant under the natural action of the infinite symmetric group on !
Now you can imagine how this type of argument might be generalized and altered.
Definition: An n-domain is a (possibly infinite) collection of -cubes (subsets of the form ) in the unit n-cube with pairwise-disjoint interiors, i.e. if .
The next step is to generalize the idea of infinite product to make sense for a general n-domain.
Definition: If is an n-domain and is a -sequence of n-loops, which is null if is infinite, then the -product of this sequence is the map defined as on and which maps to .
Be warned that n-domains are arbitrary.
So the main question now is….
Question: Given any two n-domains and and a null -sequence are the products and always homotopic?
I can’t help but spoiling it… The answer is YES! This is a Theorem in . Here’s why you should be even more startled that the answer is YES…The approach we took (in the lemma) for simultaneously sliding around individual squares/cubes fails miserably in general. It fails when you run into an n-domain that look like this:
Yikes! There’s no way to make individual moves and hope to shuffle them (after some choice of ordering) back into the simple n-domain for infinite products. If you try to move individual squares (all at once or even finitely many at a time), you will end up breaking continuity. Remember our original n-domain for defining ordinary infinite products (on the right)? How do you continuously shuffle the squares in the complicated one above to align with these ones, all of which have height 1? Aren’t we turning arbitrarily small things into big things???
It seems like there’s just too much in the way and like sequences of squares that converge to the boundary are going to be a problem. Despite all this potential concern, there is a way to do it. Once you realize the main tricks, you have a ton of freedom to move things around in the domain while still holding a continuous homotopy in your hand at the end of the day. When I was writing the paper , I worked really hard to find the simplest geometric procedure I could possibly imagine to get the job done. The whole time, I was imagining these infinite swarms of rectangular cylinders twisting around each other in time just like the above gif with two cylinders. I had a lot of fun working it out and I am excited to see how this formalized interpretation of “infinite commutativity” will be applied.
So… not only are higher homotopy groups commutative for infinite products, they are commutative in this super-duper-geometric-infinite way.
A comment for operad fans
One cool thing here is the extension of the little -cubes operad. The -th space of the little -cubes operad is the space of ordered collections of -many -dimensional cubes (not geometric cubes but products of closed intervals) in with disjoint interiors. There are a few standard ways to topologize this kind of space. I recommend Peter May’s book The Geometry of Iterated Loop Spaces.
For us infinitary-minded folks, we can take be the space of all enumerated infinite -domains, that is, all infinite collections of -cubes in with disjoint interiors. Our YES answer to infinite commutativity proves exactly that is always path connected. It’s less clear if is always -connected like its finite counterparts but this seems pretty likely to me and could be an interesting future project.
Just as acts on the -fold product , the space acts on ! Also, if you think about the structure maps of the little n-cubes operad, you can throw in obvious structure maps for to extend it to something we could call an “infinitary operad.” Fun stuff that is ripe for exploration!
 J. Brazas, P. Gillespie, Infinitary commutativity and abelianization in fundamental groups. (2020). To appear in the Journal of the Australian Math. Society.
 K. Eda, K. Kawamura, Homotopy and Homology Groups of the n-dimensional Hawaiian earring, Fund. Math. 165 (2000) 17-28.
 K. Kawamura, Low dimensional homotopy groups of suspensions of the Hawaiian earring, Colloq. Math. 96 (2003) no. 1 27-39.