Infinite Commutativity (Part II)

Posted on August 3, 2020 by Jeremy Brazas

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 \pi_n(X,x_0), n\geq 2 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 C_n(1),C_n(2),C_n(3),... to a space C_n(\aleph_0) consisting of configurations of infinitely many disjoint n-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 [3] 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 [4] to study the higher dimensional earrings (sometimes called Barratt-Milnor Spheres) Following [3] and [4], infinite commutativity in \pi_n 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 [2] and also higher homotopy groups in [1]. Part of my recent work in [1] and [2] 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, n will always be an integer \geq 2.

For a based space (X,x_0), let \Omega^n(X,x_0) denote the space of relative maps (I^n,\partial I^n)\to (X,x_0) with the compact open topology. I’ll refer to the elements of \Omega^n(X,x_0) as n-loops. In Part I, I defined the usual binary concatenation operation \Omega^n(X,x_0)\times \Omega^n(X,x_0)\to \Omega^n(X,x_0), (\alpha,\beta)\mapsto \alpha\cdot\beta. Let’s see if we can build up infinitary operations on n-loops. Since the domain I^n.

Definition: Let K be a countable indexing set. A collection of maps \{f_k\mid k\in K\}\subseteq\Omega^{n}(X,x_0) is a null if for every neighborhood U of x_0, all but finitely many of the maps f_k have image in U.

If K is the natural numbers, then we call \{f_k\}_{k\in K} a null-sequence. Such a sequence is exactly a sequence in \Omega^{n}(X,x_0) that converges to the constant map at x_0. Let Null_n(X,x_0) be the space of null-sequences in \Omega^n(X,x_0) topologized as a subspace of the infinite direct product \prod_{k\in \mathbb{N}}\Omega^n(X,x_0).

Let’s start with the obvious first choice: given a null-sequence of maps f_k:(I^n,\partial I^n)\to (X,x_0), we may for the infinite concatenation \prod_{k=1}^{\infty}f_k=f_1\cdot f_2\cdot f_3\cdots, which is defined as f_k on \left[1-\frac{1}{2^{k-1}},1-\frac{1}{2^{k}}\right]\times I^{n-1} and f_{\infty}(\{1\}\times I^{n-1})=x_0. This is continuous at the points of \{1\}\times I^{n-1} precisely because f_k is a null-sequence.

Infinite product of 2-loops

By taking the homotopy class of \prod_{k=1}^{\infty}f_k, we get an operation \{f_k\}_{k=1}^{\infty}\mapsto\left[\prod_{k=1}^{\infty}f_k\right]. This operation is technically a function Null_n(X,x_0)\to \pi_n(X,x_0).

Warning/Red Alert!! We may want to jump quickly to writing down a partially-defined infinite sum operation of homotopy classes by setting \sum_{k=1}^{\infty}[f_k]=\left[\prod_{k=1}^{\infty}f_k\right]. The operation \{[f_k]\}_{k=1}^{\infty}\mapsto\left[\prod_{k=1}^{\infty}f_k\right] is “partially-defined” because the concatenation is continuous if and only if \{f_k\}_{k=1}^{\infty} 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. \prod_{k=1}^{\infty}f_k\simeq\prod_{k=1}^{\infty}g_k whenever f_k\simeq g_k for all k. 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 \{f_k\}_{k=1}^{\infty} is a null-sequence of n-loops and \phi:\mathbb{N}\to\mathbb{N} is any bijection, then \prod_{k=1}^{\infty}f_k=f_1\cdot f_2\cdot f_3\cdots and \prod_{k=1}^{\infty}f_{\phi(k)}=f_{\phi(1)}\cdot f_{\phi(2)}\cdot f_{\phi(3)}\cdots are homotopic by a homotopy with image in \bigcup_{k=1}^{\infty}Im(f_k).

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 n=2 with an example. Suppose \{f_k\}_{k=1}^{\infty} is a null sequence and that \phi is a specific bijection where \phi(1)=3, \phi(2)=5, \phi(3)=1, \phi(4)=2, \phi(5)=4, and so on….. To prove the lemma in this case, we’ll need to show that

\prod_{k=1}^{\infty}f_k=f_1\cdot f_2\cdot f_3\cdot f_4\cdot f_5\cdots

is homotopic to

\prod_{k=1}^{\infty}f_{\phi(k)}=f_{3}\cdot f_{5}\cdot f_{1}\cdot f_2\cdot f_4\cdots.

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 f_1,f_2,f_3,... with the appropriate affine correction to the domain. If we start with the infinite product f_1\cdot f_2\cdot f_3\cdots where the domains are rectangles, of height 1, we may shrinking these rectangles within themselves to squares of side-length \frac{1}{2^k} 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 \phi comes into play. Since \phi(1)=3, we want to move the new square domain for f_3 horizontally to where f_1 used to be. As we move it, we should stretch it to width 1/2. Similarly, since \phi(2)=5, we should move the square domain for f_5 horizontally to where f_2 use to be. As we move it, we should stretch it to width 1/4. Do this for all k 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 f_3\cdot f_5\cdot f_1\cdot f_2\cdot f_4\cdots where the k-th factor is f_{\phi(k)} because of our careful horizontal shifting in the middle step. Putting this all together gives a continuous homotopy H:[0,1]^3\to X. 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)?

Brazas

Eckmann Hilton Homotopy

Now imagine the same animation in this situation. There will be infinitely many cylinders S_1,S_2,S_3,...\subseteq [0,1]^3 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 [0,1]^3. The homotopy H is definitely continuous when restricted to each twisted cylinder S_k because its restriction to each cross-section is just f_k. Also, H maps [0,1]^3\backslash \bigcup_{k=1}^{\infty}int(S_k) to the basepoint x_0. So the only “hard” case is checking continuity of H at a boundary point a\in \partial S_{m} of some twisted cylinder S_m. But H maps a to x_0 and if U is a neighborhood of x_0, then Im(f_k)\subseteq U for all but finitely many k (since f_k is a null-sequence). This means S_k\subseteq f^{-1}(U) for all but finitely many cylinders S_k. Combined with the earlier observations about where H already restricts to a continuous map, it’s possible to find an open neighborhood V of a with f(V)\subseteq U (filling in the details is a good exercise for those newer to topology). This means H is continuous. \square.

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 \sum_{k=1}^{\infty}[f_k] for null-sequences makes sense, these sums behave very much like absolutely convergent series \sum_{k=1}^{\infty}a_k of real numbers in \mathbb{R}! 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 \Sigma_{\infty} on Null_n(X,x_0)!

Now you can imagine how this type of argument might be generalized and altered.

Definition: An n-domain is a (possibly infinite) collection \mathscr{R} of n-cubes (subsets of the form \prod_{m=1}^{n}[a_m,b_m]) in the unit n-cube [0,1]^n with pairwise-disjoint interiors, i.e. int(R)\cap int(S)=\emptyset if R\neq S.

The next step is to generalize the idea of infinite product to make sense for a general n-domain.

Definition: If \mathscr{R}=\{R_k\mid k\in K\} is an n-domain and \{f_k\}_{k\in K} is a K-sequence of n-loops, which is null if K is infinite, then the \mathscr{R}-product of this sequence is the map \prod_{R_k\in\mathscr{R}}f_k defined as f_k on R_k and which maps [0,1]^n\backslash \bigcup_{k\in K}int(R_k) to x_0.

Be warned that n-domains are arbitrary.

Some examples of 2-domains. Each rectangular region with a blue boundary is an element of the intended n-domain.

So the main question now is….

Question: Given any two n-domains \mathscr{R}=\{R_k\mid k\in K\} and \mathscr{S}=\{S_k\mid k\in K\} and a null K-sequence \{f_k\}_{k\in K} are the products \prod_{\mathscr{R}}f_k and \prod_{\mathscr{S}}f_k always homotopic?

I can’t help but spoiling it… The answer is YES! This is a Theorem in [1]. 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 [1], 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 n-cubes operad. The k-th space of the little n-cubes operad is the space C_n(k) of ordered collections of k-many n-dimensional cubes (not geometric cubes but products of closed intervals) in [0,1]^n 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 C_n(\aleph_0) be the space of all enumerated infinite n-domains, that is, all infinite collections of n-cubes in [0,1]^n with disjoint interiors. Our YES answer to infinite commutativity proves exactly that C_n(\aleph_0) is always path connected. It’s less clear if C_n(\aleph_0) is always (n-2)-connected like its finite counterparts but this seems pretty likely to me and could be an interesting future project.

Just as C_n(k) acts on the k-fold product \Omega^n(X,x_0)^k, the space C_n(\aleph_0) acts on Null_n(X,x_0)! Also, if you think about the structure maps of the little n-cubes operad, you can throw in obvious structure maps for C_n(\aleph_0) to extend it to something we could call an “infinitary operad.” Fun stuff that is ripe for exploration!

References:

[1] J. Brazas, The infinitary n-cube shuffle. Topology Appl. 287 (2020) 107446. arXiv:2006.08738

[2] J. Brazas, P. Gillespie, Infinitary commutativity and abelianization in fundamental groups. (2020). To appear in the Journal of the Australian Math. Society.

[3] K. Eda, K. Kawamura, Homotopy and Homology Groups of the n-dimensional Hawaiian earring, Fund. Math. 165 (2000) 17-28.

[4] K. Kawamura, Low dimensional homotopy groups of suspensions of the Hawaiian earring, Colloq. Math. 96 (2003) no. 1 27-39.

Posted in Algebraic Topology, Higher Homotopy groups, Homotopy theory, Infinite Group Theory, Infinite products | 3 Comments

Infinite Commutativity (Part I)

Posted on August 2, 2020 by Jeremy Brazas

The Eckmann-Hilton Principle is a classical argument in algebraic topology/algebra. This argument allows you to conclude that an operation which may be expressed in two different ways (imagine that it may be applied both horizontally and vertically when written) is always commutative. This all has a very topological flavor that is important in homotopy theory. It is helpful to think of the two ways of expressing such an operation as having two-dimensions to move around in. This principle is fun to learn and revisit but it’s infinitary extension has become increasingly relevant in infinitary/wild topology. I’m spending a lot of time on the “elementary” Eckmann-Hilton homotopy in this post, because the infinite one in Part II is a fair bit more complicated and it will make a lot more sense if you remember how to “see” it.

The Algebraic Eckmann-Hilton Principle

Here’s the set-up of the Eckmann-Hilton Principle: suppose you have a set M with two unital operations a\cdot b and a\ast b satisfying the distributive rule: (a\ast b)\cdot (c\ast d)=(a\cdot c)\ast (b\cdot d). Using only this set-up, we have the following theorem.

Eckmann-Hilton Theorem: The operations a\cdot b and a\ast b are equal. Moreover, they are commutative and associative.

Proof. Let  e_{\cdot} and e_{\ast} be the identity elements for the operations \cdot and \ast respectively. We check the desired equalities in sequence. Each argument uses a previous one.

Identities are equal:

e_{\cdot}=e_{\cdot}\cdot e_{\cdot}=(e_{\cdot}\ast e_{\ast})\cdot (e_{\ast} \ast e_{\cdot})=(e_{\cdot}\cdot e_{\ast})\ast (e_{\ast}\cdot e_{\cdot})=e_{\ast}\ast e_{\ast}=e_{\ast}

The operations agree:

a\cdot b=(a \ast e_{\ast})\cdot (e_{\ast}\ast b)=(a\cdot e_{\ast})\ast (e_{\ast}\cdot b)=(a\cdot e_{\cdot})\ast (e_{\cdot}\cdot b)=a\ast b

The operations are commutative:

a\cdot b=(e_{\ast}\ast a)\cdot (b\ast e_{\ast})=( e_{\ast}\cdot b)\ast (a\cdot e_{\ast})=( e_{\cdot}\cdot b)\ast (a\cdot e_{\cdot})=b\ast a=b\cdot a

Since the operation \cdot is commutative and agrees with \ast, the later is also commutative.

The operations are associative:

a\cdot (b\cdot c)=(a\cdot e_{\cdot})\cdot (b\cdot c)=(a\cdot e_{\cdot})\ast (b\cdot c)=(a \ast b)\cdot (e_{\cdot} \ast c)=(a\cdot b)\cdot (e_{\ast} \ast c)=(a\cdot b)\cdot c. \square

The proof can be translated nicely into pictures if we transition to algebraic topology. For a based topological space (X,x_0), let \Omega^n(X,x_0) be the set of all relative maps (I^n,\partial I^n)\to (X,x_0), which we call n-loops.

The Homotopical Eckmann-Hilton Principle

Given two n-loops \alpha,\beta\in \Omega^n(X,x_0), we can define the concatenation n-loop \alpha\cdot\beta\in \Omega^n(X,x_0) by the formula:

\alpha\cdot\beta(t_1,t_2,t_3,\dots,t_n)=\begin{cases} \alpha(2t_1,t_2,t_3,\dots,t_n), & 0\leq t_1\leq 1/2 \\ \beta(2t_1-1,t_2,t_3,\dots,t_n), & 1/2\leq t_1\leq 1 \end{cases} .

Then [\alpha]+[\beta]:=[\alpha\cdot\beta] defines the group operation of \pi_n(X,x_0). Let’s see why we can always commute [\alpha]+[\beta]=[\beta]+[\alpha]. The following gif is basically a topological picture of the Eckmann-Hilton principle applied in dimension n=2.

Slices of the commuting homotopy in dimension 2

In the animation, the the black boundaries and the gray shaded region are mapped to the basepoint x_0\in X. The blue rectangle is the domain of \alpha and the red rectangle is the domain of \beta (with suitable scaling applied). The animation basically tells us how to define a homotopy from \alpha\cdot\beta to \beta\cdot\alpha as a map on a solid cube by showing how to define it on each vertical slice. Think of the start (height t=0) as mapping the bottom of a cube [0,1]^3. It’s precisely \alpha\cdot\beta. As time goes along, this animation realizes the how we want to map the cube into X on higher slices [0,1]^2\times\{t\}. Finally, the end (t=1) is how we map the top of the cube as \beta\cdot\alpha. Overall, we get a map on the cube, that is, a homotopy H:[0,1]^2\times [0,1]\to X from \alpha\cdot\beta to \beta\cdot\alpha. Hence, [\alpha]+[\beta]=[\alpha\cdot\beta]=[\beta\cdot\alpha]=[\beta]+[\alpha].

It’s helpful for me to imagine what shape the red and blue squares will trace out in the cube [0,1]^3. They look like cylinders with rectangle cross-sections that twist around each other in 3-space.

Brazas

Commuting homotopy in dimension 2

This construction is one of the first things you learn when you start studying homotopy groups. It’s worth pointing out that this homotopy \alpha\cdot\beta\simeq \beta\cdot\alpha has image in Im(\alpha)\cup Im(\beta) and, in fact, has constant image for all “time” of the homotopy. In particular, you can commute small loops with small homotopies.

What is Infinite Commutativity?

Something quite remarkable is that as soon as we move up to the higher homotopy groups, things aren’t just commutative in the usual sense. The product operation on \pi_n(X,x_0), n\geq 2 turns out to be “infinitely commutative.” In Part II of this post, I’ll describe exactly what this means for homotopy groups. So, in the rest of the current post, I’m just going to try and give a simple answer to the question: what is an infinitary operation and what does it mean for it to be infinitely commutative?

If you’ve learned some abstract algebra, you know about binary operations and what it means to have a commutative binary operation. Typically, you’d end up with a commutative semigroup, monoid, group, ring, etc.

Definition: An infinitary operation on a set X is a partially defined operation \{x_n\}_{n=1}^{\infty}\mapsto \prod_{n=1}^{\infty}x_n, which assigns an output (represented here using product notation) to certain infinite sequences in X. It is also possible to index these products by sets other than the natural numbers 1,2,3,\dots. For example if I is another indexing set (typically with an ordering or some other structure on it), then an infinitary I-operation on X is a partially defined operation \{x_i\}_{i\in I}\mapsto \prod_{i\in I}x_i.

Of course, these kinds of operation can have a unit and you can impose axioms on them to express exactly how you’d like for them to be associative. The infinite sum/product operations I’m talking about here are not formal infinite sums. I’m interested in operations that are induced by topological limits and which extend familiar binary operations. Perhaps the most familiar one is the infinite sum operation on the real/complex numbers that you might learn about in Calculus or Analysis \{x_n\}_{n=1}^{\infty}\mapsto \sum_{n=1}^{\infty}x_n. Chances are that if you’re reading this blog, you’ve seen these before.

Infinitary operations are all over the place, sometimes hiding in plain sight. Prominent examples include infinite sums, products related topological fields and rings of continuous functions on topological fields. Maybe a little less well-known are infinite compositions f_1\circ f_2\circ f_3\circ\cdots and \cdots \circ f_3\circ f_2\circ f_1 which have applications to fixed point theory and number theory. There are analogous infinitary sum/product operations for matrices.

I hear that there are some folks who don’t enjoy working with actual topological spaces but like category theory. Well, I hate to break it to these folks but there is something implicitly topological about limits and colimits of infinite diagrams. Maybe in a future post I will clarify exactly how this is the case but I kind of describe how this goes in the introduction to this paper (published version [1]). Anyway, if X_1\to X_2\to X_3\to\cdots is a directed system in a category and X is the colimit of this diagram, then the canonical map X_1\to X is very much the infinite composition \cdots\circ f_3\circ f_2\circ f_1 where f_n:X_n\to X_{n+1} are the bonding maps. More generally, the canonical map X_n\to X is the infinite composition \cdots f_{n+2}\circ f_{n+1}\circ f_n. The dual situation works for inverse limits and you can replace the naturals \omega with any well-ordered indexing set. Sometimes this kind of thing is called transfinite composition. The unavoidable topology that creeps in is hiding in the fact that the ordered set X_1,X_2,X_3,\dots, X of objects (including the colimit) is indexed by a non-discrete compact ordered set, namely \omega+1. It is not a coincidence that \omega+1 is order isomorphic to the set of cuts of \omega.

Definition: An infinitary operation \{x_n\}_{n=1}^{\infty}\mapsto \prod_{n=1}^{\infty}x_n on a set X is infinitely commutative if for every bijection \phi:\mathbb{N}\to\mathbb{N}, we have \prod_{n=1}^{\infty}x_n=\prod_{n=1}^{\infty}x_{\phi(n)}.

For a more general I-indexed operation, you would consider bijections \phi:I\to I and demand that \prod_{i\in I}x_i=\prod_{i\in I}x_{\phi(i)} always holds.

In short: Infinite commutativity means that you can permute the terms in the product in any way you like and the product will still be defined and its value will not change.

In fancy: Infinite commutativity means the natural action of the symmetric group S_I of the indexing set on the I-sequence \{x_i\}_{i\in I} is invariant under the infinitary operation \{x_i\}_{i\in I}\mapsto \prod_{i\in I}x_i.

Example: Even for infinite sums from Calculus, there are some subtleties involved. For instance, it is not enough to know the terms a_1,a_2,a_3,... shrink to 0 to ensure that \sum_{n=1}^{\infty}a_n is well-defined, e.g. 1+\frac{1}{2}+\frac{1}{3}+\cdots diverges. Which sequences have a well-defined sum has more to do with the existence of a shrinking sequence of tails t_n=a_n+a_{n+1}+a_{n+1}+\cdots as I described for infinite words. Also, there is a dichotomy of real infinite series: absolutely convergent series and conditionally convergent series. A series \sum_{n=1}^{\infty}a_n is absolutely convergent if \sum_{n=1}^{\infty}|a_n| converges and there is a Rearrangement Theorem stating that if \phi:\mathbb{N}\to\mathbb{N} is any bijection then \sum_{n=1}^{\infty}a_n= \sum_{n=1}^{\infty}a_{\phi(n)}. A series is conditionally convergent if it is not absolutely convergent and the Rearrangement Theorem states that if \sum_{n=1}^{\infty}a_n is conditionally convergent and L is any real number, then there exists a bijection \phi such that \sum_{n=1}^{\infty}a_{\phi(n)}=L. For instance, the alternating harmonic series \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots is conditionally convergent and its terms can be rearranged so the new sum converges to 42 or \pi or whatever number you want.

Takeaway: The infinite series operation in the real line is finitely commutative but is NOT infinitely commutative.

There are infinitary operations out there (like composition) which are not even finitely commutative. Here is one that is actually infinitely commutative.

Example: The Baer-Specker group \mathbb{Z}^{\mathbb{N}} is the infinite direct product of discrete groups \mathbb{Z}\times \mathbb{Z}\times\mathbb{Z}\times \cdots and consists of all infinite sequences of integers \mathbf{a}=(a_1,a_2,a_3,\dots). We give \mathbb{Z}^{\mathbb{N}} the product topology so that a sequence of \mathbf{a}_k=(a_{k,1},a_{k,2},a_{k,3},\dots) of sequences converges to \mathbf{b}=(b_1,b_2,b_3,\dots) if and only if there initial coordinates in the sequence \mathbf{a}_k stabilize to the terms of \mathbf{b}, that is if for every n\geq 1, there exists an K\geq 1 such that a_{k,n}=b_n for all k\geq K. So if you keep going through the sequence \mathbf{a}_k, the first coordinate will eventually stabilize, then the second coordinate will eventually stabilize, and so on.

Now, given a sequence \{\mathbf{a}_k\}_{k=1}^{\infty}\to (0,0,0,\dots) that converges to the identity, we can define a sum \sum_{k=1}^{\infty}\mathbf{a}_k as the sequence

\left(\sum_{k=1}^{\infty}a_{k,1},\sum_{k=1}^{\infty}a_{k,2},\sum_{k=1}^{\infty}a_{k,3},\dots \right)

The infinite sum in each sequence is really just a finite sum since for given coordinate n, the sequence \{a_{k,n}\}_{n=1}^{\infty} is eventually 0.

Let’s check infinite commutativity. Suppose that \phi:\mathbb{N}\to\mathbb{N} is a bijection. Then

\sum_{k=1}^{\infty}\mathbf{a}_{\phi(k)}= \left(\sum_{k=1}^{\infty}a_{\phi(k),1},\sum_{k=1}^{\infty}a_{\phi(k),2},\sum_{k=1}^{\infty}a_{\phi(k),3},\dots \right)

But each coordinate is just an ordinary finite sum. Hence, all that \phi is really doing is commuting the finitely many non-zero terms in each coordinate. We can conclude that \sum_{k=1}^{\infty}\mathbf{a}_k=\sum_{k=1}^{\infty}\mathbf{a}_{\phi(k)}, which means that the natural infinite sum operation on the Specker group is infinitely commutative. So, in an infinitary algebra sense, the Specker group is much simpler that the infinite series operation on the real line.

In Part II, we’ll explore why the natural infinitary operations on all higher homotopy groups are infinitely commutative!

[1] J. Brazas, Transfinite Product Reduction in Fundamental Groupoids. To Appear in European Journal of Mathematics. (2020). https://doi.org/10.1007/s40879-020-00413-0  arXiv version.

Posted in Baer-Specker group, Homotopy theory, Infinite Group Theory, Infinite products | Tagged , , , , , , | 1 Comment

Spanier Groups: a modern take on vintage covering space theory

Posted on December 15, 2019 by Jeremy Brazas

Edwin H. Spanier’s 1966 Algebraic Topology book is a true classic. Well-written and precise, I still find myself referring to it regularly even though it is really “old.” Spanier takes a unique approach to covering space theory that I haven’t seen anywhere else. I’ve found his approach to covering space theory so much more intuitive and general than modern books, that I use it when I teach covering space theory to my students. In particular, Spanier defined subgroups \pi(\mathscr{U},x_0) of a fundamental group \pi_1(X,x_0) to “detect” when a covering map p:Y\to X exists that corresponds to a given subgroup H\leq \pi_1(X,x_0). In particular,

Modern Interpretation of Spanier’s Classification Theoreom: A path-connected, locally path-connected space X has a covering space classifying (unique up to equivalence) a subgroup H\leq \pi_1(X,x_0) if and only if H is open in the Spanier topology on \pi_1(X,x_0).

Notice that X doesn’t have to be semilocally simply connected (SLSC) for this to work. The term “Spanier group(s)” was coined in [1]; it immediately stuck and has become fairly standard.

Spanier groups with respect to open covers

Notation: \alpha\cdot\beta denotes path-concatenation and \alpha^{-} denotes the reverse path of \alpha. We’ll work in the fundamental groupoid and write [\alpha][\beta]=[\alpha\cdot\beta] for the operation on path-homotopy classes.

Definition: Let \mathscr{U} be an open cover of a space X with basepoint x_0\in X. The Spanier group of (X,x_0) with respect to \mathscr{U} is the subgroup of \pi_1(X,x_0) generated by all homotopy classes of loops based at x_0 of the “lasso” form \alpha\cdot\gamma\cdot\alpha^{-} where Im(\gamma)\subseteq U for some U\in\mathscr{U}. In short, it’s

\pi(\mathscr{U},x_0)=\langle [\alpha\cdot\gamma\cdot\alpha^{-}]\in \pi_1(X,x_0)\mid \exists U\in\mathscr{U}\,\,Im(\gamma)\subseteq U\rangle

spanier

This means that a generic element of \pi(\mathscr{U},x_0) has the form \prod_{i=1}^{n}[\alpha_i][\gamma_i][\alpha_{i}^{-}] where \alpha_i:([0,1],0)\to (X,x_0) are paths and each \gamma_i is a loop with image in some member of \mathscr{U}.

Let’s first make some basic observations:

  1. Normality: The Spanier group \pi(\mathscr{U},x_0) is always a normal subgroup of \pi_1(X,x_0) since the conjugate [\beta][\alpha][\gamma][\alpha^{-}][\beta]^{-}=[\beta\cdot\alpha][\gamma][(\beta\cdot\alpha)^{-}] of a generator [\alpha][\gamma][\alpha^{-}] still has the form of a generator of \pi(\mathscr{U},x_0).
  2. Invariant under basepoint-change: if x_1\in X is another point and \beta:[0,1]\to X is a path from x_1 to x_0 and \varphi_{\beta}:\pi_1(X,x_0)\to\pi_1(X,x_1), \varphi_{\beta}([\eta])=[\beta\cdot\eta\cdot\beta^{-}] is the basepoint-change isomorphism, then \varphi_{\beta}(\pi(\mathscr{U},x_0))=\pi(\mathscr{U},x_1).
  3. Refinement: If an open cover \mathscr{V} refines \mathscr{U} (every V\in \mathscr{V} is contained in some U\in\mathscr{U}), then \pi(\mathscr{V},x_0)\leq \pi(\mathscr{U},x_0).
  4. They generate a topology on \pi_1: The cosets of Spanier groups form a basis for a topology on \pi_1(X,x_0) sometimes called the Spanier topology. In particular, a set W\subseteq \pi_1(X,x_0) is open if for every g\in W, there exists an open cover \mathscr{U} of X such that g\pi(\mathscr{U},x_0)\subseteq W. There are many topologies one can put on \pi_1. This one is closely related to covering space theory as we’ll see below.
  5. Continuity: If f:(Y,y_0)\to (X,x_0) is a continuous function, then f^{-1}\mathscr{U}=\{f^{-1}(U)\mid U\in\mathscr{U}\} is an open cover of Y such that the induced homomorphism f_{\#}:\pi_1(Y,y_0)\to\pi_1(X,x_0) satisfies f_{\#}(\pi(f^{-1}\mathscr{U},y_0))\leq\pi(\mathscr{U},x_0). I use the term continuity here because this is equivalent to saying that the induced homomorphisms f_{\#} are continuous with respect to the Spanier topology. Hence, the Spanier topology gives us one example of a fundamental group functor \pi_1:\mathbf{Top}_{\ast}\to \mathbf{TopGrp} to the category of topological groups.

Some intuition may be lost in the definition of the Spanier group \pi(\mathscr{U},x_0) in terms of it’s generators. I’ll give here an example of a more complicated element of \pi(\mathscr{U},x_0):

bcs

Let \Delta_2 be a 2-simplex with basepoint vertex d_0 and (bd_{k}\Delta_2)_1 be the 1-skeleton of some barycentric subdivision of \Delta_2 (the union of the black edges in the image on the right). Let f:((bd_{k}\Delta_2)_1,d_0)\to (X,x_0) be a map such that for every 2-simplex \tau of bd_{k}\Delta_2, we have f(\partial \tau)\subseteq U for some U\in\mathscr{U}. In other words, each small black triangle gets mapped into an element of the cover. With some careful choices of conjugating paths, the inclusion loop \partial \Delta_2\to (bd_{k}\Delta_2)_1 of the outermost triangle factors as a huge product of “lasso” loops where the hoops of the lasso are the boundaries of the 2-simplices in the subdivision bd_{k}\Delta_2. Since lassos map to lassos, the “large” loop f|_{\partial \Delta_2}:\partial \Delta_2\to X factors as a huge product of Spanier group generators and therefore represents an element \pi(\mathscr{U},x_0).

In my view, this construction really helps to clarify which homotopy classes end up in the Spanier groups and which ones do not.

Two definitions of “semilocally simply connected”

Spanier basically used these groups to provide an alternative definition of the “semilocally simply connected” property. But one oversight in Spanier’s book is that there are two competing notions that are not always the same if your spaces are not necessarily locally path connected.

Red Alert: There are two non-equivalent definitions of the “semilocally simply connected” property that differ by the addition of a single quantifier.

  1. A space X is based semilocally simply connected at x\in X if there exists an open neighborhood U of X such that the induced homomorphism \pi_1(U,x)\to \pi_1(X,x) is trivial.
  2. A space X is unbased semilocally simply connected at x\in X if there exists an open neighborhood U of X such that for all y\in U, the induced homomorphism \pi_1(U,y)\to \pi_1(X,y) is trivial.

Certainly the unbased property implies the based property and these are equivalent for locally path-connected spaces. However, these two definitions are not equivalent in general. Here’s a counterexample:

lassospace

A one-dimensional compact space that is based SLSC but not unbased SLSC.

In particular, any small neighborhood of the right endpoint of the horizontal line contains circles but contains no path from that point to those circles.

The following lemma is why the unbased SLSC property is typically used – it is a necessary condition for having a simply connected covering space.

Lemma 1: If X admits a simply connected covering space, then X is unbased SLSC.

Proof. This lemma is a special case of Theorem 4 below (where p_{\#}(\pi_1(Y,y_0))=1 since the covering space Y is simply connected and p_{\#} is inejctive) but this is a nice exercise to work out on it’s own. \square

So…even though the space pictured above is based SLSC, it can’t possibly have a simply connected covering space.

Lemma 2: Suppose X is locally path-connected. Then X is unbased SLSC if and only if there exists an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)=1.

Proof. Suppose X is (unbased) SLSC. For each x\in X, let U_x be an open neighborhood of x such that for every y\in U_x, the inclusion induces the trivial homomorphism \pi_1(U_x,x)\to \pi_1(X,x), i.e. every loop in U_x based at x is null-homotopic in X. Since X is locally path-connected, we may replace each U_x with a smaller path-connected neighborhood that necessarily has the same property. Now \mathscr{U}=\{U_x\mid x\in X\} is an open cover of X. Consider a generator [\alpha\cdot\gamma\cdot\alpha^{-}] of \pi(\mathscr{U},x_0) where \gamma has image in some U_x. Since U_x is path-connected, there is a path \beta:[0,1]\to U_x from x to \gamma(0). Now [\alpha\cdot\gamma\cdot\alpha^{-}]=[\alpha\cdot\beta][\beta\cdot\gamma\cdot\beta^{-}][\alpha\cdot\beta]^{-1} where \beta\cdot\gamma\cdot\beta^{-} is a loop based a x and thus is null-homotopic in X. Thus [\alpha\cdot\gamma\cdot\alpha^{-}]=[\alpha\cdot\beta][\beta\cdot\gamma\cdot\beta^{-}][\alpha\cdot\beta]^{-1}=[\alpha\cdot\beta][\alpha\cdot\beta]^{-1}=1. Since all the generators of \pi(\mathscr{U},x_0) are trivial, this subgroup is the trivial subgroup.

Conversely, suppose there exists an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)=1. For each x\in X, let V_x be a path-connected neighborhood of x such that V_x\subseteq U for some U\in\mathscr{U}. Since \mathscr{V}=\{V_x\mid x\in X\} refines \mathscr{U}, we have \pi(\mathscr{V},x_0)\leq \pi(\mathscr{U},x_0) and thus \pi(\mathscr{V},x_0)=1. We check that the inclusion (V_x,x)\to (X,x) induces the trivial homomorphism on fundamental groups. If \gamma:([0,1],\{0,1\})\to (V_x,x) is a loop, consider any path \alpha:[0,1]\to X from x_0 to x. Then [\alpha\cdot\gamma\cdot\alpha] is generator of \pi(\mathscr{V},x_0) and must therefore, represent the identity element in \pi_1(X,x_0). Since [\alpha\cdot\gamma\cdot\alpha]=1 in \pi_1(X,x_0), path conjugation in the fundamental groupoid gives [\gamma]=1 in \pi_1(X,x). Thus \gamma is null-homotopic in X, completing the proof. \square

Notice that local path-connectivity is necessary for both directions of the above proof. If you want to do away with this, you’ll need to use “based Spanier groups” which are also defined in [1]. Or, if you want to put on your categorical fancy pants you can use the locally path-connected coreflection lpc(X).

Theorem 3: lpc(X) is unbased SLSC if and only if there exists an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)=1.

Lemma 2 also says something about the Spanier topology.

Corollary 4: If X is path-connected and locally path-connected, then X is SLSC if and only if \pi_1(X,x_0) is discrete with the Spanier topology.

Spanier Groups and Covering Spaces

The following theorem has no hypotheses in the spaces involved except for path connectivity.

Theorem 5: If p:Y\to X is a covering map of path-connected spaces, then there exists an open cover \mathscr{U} of X such that for any choice of basepoints p(y_0)=x_0, we have \pi(\mathscr{U},x_0)\leq p_{\#}(\pi_1(Y,y_0)).

Proof. Let \mathscr{U} be the open cover of X by neighborhoods that are evenly covered by p. Let y_0\in Y and p(y_0)=x_0. Consider a generator [\alpha\cdot\gamma\cdot\alpha^{-}] of the Spanier group \pi(\mathscr{U},x_0) where \gamma is a loop with image in U\in\mathscr{U}. Let y=\widetilde{\alpha}(1) be the end point of the lift \widetilde{\alpha}:([0,1],0\to (Y,y_0) of \alpha and let V be an open subset of p^{-1}(U) containing y that is mapped homeomorphically onto U by p. Since \gamma has image in U, the lift \widetilde{\gamma} of \gamma starting at y is a loop in V based at y. Thus \widetilde{\alpha}\cdot\widetilde{\gamma}\cdot \widetilde{\alpha}^{-} is a loop in Y based at y_0. We have [\alpha\cdot\gamma\cdot\alpha^{-}]=p_{\#}([\widetilde{\alpha}\cdot\widetilde{\gamma}\cdot \widetilde{\alpha}^{-}])\in p_{\#}(\pi_1(Y,y_0)). Since p_{\#}(\pi_1(Y,y_0)) contains all generators of \pi(\mathscr{U},x_0), we have \pi(\mathscr{U},x_0)\leq p_{\#}(\pi_1(Y,y_0)). \square

The following classification of coverings blows many “standard” approaches out of the water because it includes the entire lattice of subgroups classified by covering maps even if the space in question is not SLSC. Even though spaces like the Hawaiian earring or Menger cube don’t have simply connected coverings, they do have lots and lots of intermediate covering spaces and I often find myself in need of these. Weaker classifications, e.g. in Munkres and Hatcher assume SLSC and say nothing about these intermediate coverings that Spanier’s approach includes.

Spanier’s Covering Space Classification Theorem: Suppose X is path-connected and locally path-connected and H\leq \pi_1(X,x_0) is a subgroup. Then there exists a covering map p:(Y,y_0)\to (X,x_0) such that p_{\#}(\pi_1(Y,y_0))=H if and only if there exists an open covering \mathscr{U} such that \pi(\mathscr{U},x_0)\leq H.

Proof. Theorem 5 above is the only if direction that holds generally. For the converse, suppose the hypotheses on X and that there exists an open covering \mathscr{U} such that \pi(\mathscr{U},x_0)\leq H. I’m not going to show all of the nitty gritty details here, but I’ll give all the ingredients for how to build a corresponding covering map.

Let \widetilde{X}_H be the set of “groupoid cosets” H[\alpha]=\{h[\alpha]\mid h\in H\} for paths \alpha:([0,1],0)\to (X,x_0) starting at a fixed basepoint x_0\in X. Notice that H[\alpha]=H[\beta] if and only if \alpha(1)=\beta(1) and [\alpha\cdot\beta^{-}]\in H. Give \widetilde{X}_H the topology generated by the sets B(H[\alpha],U)=\{H[\alpha\cdot\epsilon]\mid Im(\epsilon)\subseteq U\} where U is an open neighborhood of \alpha(1) in X. Let p_H:\widetilde{X}_H\to X, p_H(H[\alpha])=\alpha(1) be the endpoint projection map. Since X is path-connected, p_H is onto and if V is a path-connected open set in X, then p_H(B(H[\alpha],V))=V. Hence, p_H is an open map since X is locally path-connected.

By assumption, we may find an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)\leq H. By refining \mathscr{U}, we may assume each element of U is path-connected. Given x\in X pick U\in\mathscr{U} containing x. It’s easy to see that p_{H}^{-1}(H)=\bigcup_{\alpha(1)=x}B(H[\alpha],U). Also, some routine arguments (using the path-connectivity of U) show that for any two paths \alpha,\beta:([0,1],0,1)\to (X,x_0,x), the open sets B(H[\alpha],U) and B(H[\beta],U) are either disjoint are equal. Since we already know that p_H is an open map, it suffices to show that p_H is injective on B(H[\alpha],U). Suppose p_{H}(H[\alpha\cdot\epsilon])=\alpha\cdot\epsilon(1)=\alpha\cdot\delta(1)=p_{H}(H[\alpha\cdot\delta]) for paths \epsilon,\delta in U. Now \alpha\cdot(\epsilon\cdot\delta^{-})\cdot\alpha^{-} is a well-defined “lasso” loop where \epsilon\cdot\delta^{-} has image in U\in\mathscr{U}. Therefore, [\alpha\cdot(\epsilon\cdot\delta^{-})\cdot\alpha^{-}]\in \pi(\mathscr{U},x_0)\leq H, which implies H[\alpha\cdot\epsilon]=H[\alpha\cdot\delta], proving injectivity. \square

Notice that the subgroup condition \pi(\mathscr{U},x_0)\leq H in the statement of the classification is precisely what is needed to verify the locally injective part of the definition of a covering map. The typical arguments for uniqueness don’t require SLSC so equivalent based coverings still correspond to conjugate subgroups of \pi_1(X,x_0) and so on and so forth.

The classification theorem at the start of the post is a slick restatement of this theorem.

Corollary 6: Suppose X is path connected and locally path connected. The lattice of subgroups corresponding to covering maps over X is upward closed and closed under finite intersection.

Now, if X is also SLSC, then, as we proved above, \pi_1(X,x_0) is discrete with the Spanier topology, which means all subgroups are open, which means all subgroups are classified by covering spaces! This case gives you back the specific classification theorem you might be used to for SLSC spaces.

References

[1] H. Fischer, D. Repovs, Z. Virk, A. Zastrow, On semilocally simply-connected spaces, Topology Appl. 158 (2011) 397-408.

[2] E.H. Spanier, Algebraic Topology, McGraw-Hill, 1966.

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