One of my posts where I did some substantial hand-waving is my original post on the fundamental group of the Hawaiian earring. I wrote about how to understand and work with this group, but I never gave a proof of the key fact that the Hawaiian earring group naturally injects into an inverse limit of free groups . This is one of the two primary viewpoints that researchers take to study and apply the beautiful algebra of this group (and more generally fundamental groups of one-dimensional spaces). Seriously, I’m using this machinery like 1.) it’s going out of style and 2.) I understand fashion. The second approach is to identify the Hawaiian earring group as a group of reduced countable, linear words over a countable alphabet. They’re logically equivalent, but sometimes one is more convenient than the other.

To be honest, I hesitated about writing this post. I say with confidence that there is no completely elementary proof. While I’ve read and understood many different proofs of shape injectivity, most of them are either super technical or they gloss over details by applying continuum and dimension theory. Some inquisitive and kind readers have given me the motivation to do it.

When trying to write this post, I dug deep into the literature trying to weasel out an almost entirely self-contained proof that a grad student would believe. After some reading, I worked things out and these posts are the results of the effort. This first post will mostly be used to set up the technical tools about arcs in inverse limits that we’ll need to prove shape injectivity.

Let be the circle of radius centered at . Then is the Hawaiian earring with basepoint . We need to set up a little more notation:

Let be the bouquet of the first n-circles with free fundamental group .

be the bouquet of the first n-circles with free fundamental group . Let be the retraction, which collapses to and is the identity elsewhere.

be the retraction, which collapses to and is the identity elsewhere. Let be the retraction, which collapses the smaller copy of the Hawaiian earring to and is the identity elsewhere.

This gives an inverse system of retracts:

The closed mapping theorem should help convince you that the Hawaiian earring is homeomorphic to the inverse limit of this system of bouquets. Now apply to this inverse system: the maps induce homomorphisms , which together induce a canonical homomorphism defined by .

The inverse limit of free groups, which we can abbreviate as is precisely the first shape (or Cech) homotopy group .

First, let’s notice that is homeomorphic to . Basically, we just need to believe that is precisely the infinite wedge viewed as a subspace of the infinite torus with the product topology. It’s then very tempting to think that is an isomorphism. However, doesn’t always preserve inverse limits!

Nevertheless, we can still understand and work with if we identify it as a subgroup of . Hence, the motivation for showing that is injective.

Shape Injectivity Theorem: is injective.

Basically, this theorem says that a loop in is null-homotopic if and only if every projection is null-homotopic in the wedge of circles . The contrapositive says that is not null-homotopic in iff there exists some such that represents a non-trivial word in .

Why this is not so obvious

Let be the loop going once around counterclockwise and let be its reverse loop. Given a loop , we may assume that for each component of , the restriction of to is one of the paths or . Obviously, for each , the loops and can show up as subloops at most finitely many times or we would violate the uniform continuity of .

Suppose we know is null-homotopic in . The primary difficulty is that the null-homotopies for might have nothing to do with each other. We just know one exists for each approximation level. There is no guarantee that we can “fix em up right” so that they agree with the bonding maps, i.e. satisfy and thus induce a null-homotopy of in .

Here is a more algebraic way to look at it. It doesn’t hurt to think each projection loop as an unreduced finite word in the letters . Then means a concatenation of length and means a concatenation of length . For instance, suppose is the loop described by the following projections.







and so on where between any two letters in the previous projection you insert an inverse pair of the form .

Notice that deleting the ‘s from gives , deleting the ‘s from gives , and so on. Moreover, each letter is only used finitely many times. We conclude that this does indeed give a loop in . Notice that even though these finite projection words are getting pretty long, the homotopy class will cancel to the trivial word in the free group . After all, we just inserted inverse pairs that cancel!

But the infinite limit loop in is a “transfinite word” of dense order. It is null-homotopic (due to the main result of the post) but it’s much harder to come up with an explicit contraction because there is no finite reduction scheme that can do the job. Between any two of the inverse pairs that you wanted to cancel in the n-th projection, you actually had letters up in the next level. You can’t just cancel in the -th level, forget about what you just did, and then move on up to the -st level.

The trouble is that the process of cancellation requires choice. Ok, there’s not much choice in cancelling the first word. But look at the second one.

You could cancel all the ‘s first and then the remaining ‘s. Or you could cancel the middle ‘s first, then the ‘s, and then the remaining ‘s. It may not seem like a big deal but these are different homotopies! The only thing we have going for us it that we have some contracting reduction for each . This leaves us with an infinite sequence of reductions for the projection words – one for each level. We have no idea if these reductions match up or can be chosen so that the projection of a reduction of to the next level down is exactly the reduction for . Sure, once I choose a reduction/null-homotopy for , I can project it down to make sure the reductions on lower levels match up, but then you have to start over and worry about . If you project down and fix all the lower levels and continue this process all the way up, you’re going to end up with a rearranging the reduction choice at each level infinitely many times. There is no guarantee this can be done “continuously.”

History

The first attempt to identify was by H.B. Griffiths [3]. However, there was a critical error in Griffiths’ proof of the injectivity. The error was observed and a correct proof finally given (30 years later!) by Morgan and Morrison [4]. Many years back when I read the original proof for the first time, I was a bit unsatisfied with how specific and technical it all was. Later on, I read the proof given by Eda and Kawamura in [2], which felt more intuitive because all I had to do was understand inverse limits and believe a little continuum theory. Bonus: It applies to all spaces with Lebesgue covering dimension 1, not just . The key idea is originally due to the work in [1] by Curtis and Fort from the 1950’s.

Trees and Inverse Limits

An important theme in wild topology is the idea of a space being “uniquely arc-wise connected.” Here an “arc” in a space refers to a subspace of homeomorphic to . The image of and in are the endpoints of the arc. A “simple closed curve” in is a homeomorphic copy of the unit circle .

Definition: A space is uniquely arc-wise connected if for all distinct points , there is a unique arc in whose endpoints are and .

The next proposition gives another useful way to describe uniquely arc-wise connected spaces.

Proposition: If is uniquely arc-wise connected, then is path connected and contains no simple closed curves. The converse holds if is weakly Hausdorff.

Proof. Since is not uniquely arc-wise connected, one direction is obvious. Now suppose is weakly Hausdorff and not uniquely arc-wise connected. Then there are distinct arcs sharing the same endpoints. Since , without loss of generality, we may suppose there is a point . Note that since is weakly Hausdorff, and are closed. It follows that is non-empty and closed in and thus is open in . Choosing a homeomorphism , let be the component of containing . Now is a subarc of with endpoints . If is the subarc of with endpoints and , then we have . Now it’s clear that is a homeomorphic image of a circle, i.e. a simple closed curve.

The uniquely arc-wise connected spaces you’re most likely to already be familiar with are trees.

Definition: A simplicial tree is a one-dimensional simplicial complex without any cycles. A (topological) tree is a space, which is the geometric realization of a simplicial tree.

Basic algebraic topology tells us that trees are contractible and uniquely arc-wise connected. Since a tree is simply connected, between any two points there is a single homotopy (rel. endpoints) class of paths from to . This means of the unique arc from to is a reduced representative of the single homotopy class of paths from to in the sense that it has no null-homotopic subloops. This reduced representative is unique up to reparameterization. A non-reduced path in a tree would have some null-homotopic zig-zags that we could “delete” by a homotopy to obtain a reduced representative. Of course, there could be infinitely many zig-zags but since trees are semilocally simply connected, this is not much of an obstacle to overcome.

Now what about an inverse limit of trees ? Informally, such an inverse limit “glues” together the trees according to their bonding maps. The result should be one-dimensional and if maps to the same point of , then will send the unique arc connecting and to a finite topological subtree of . So there should be no way for a simple closed curve to magically appear in the gluing process. We’ll prove exactly this using the simplest proof I could come up with.

Recall that an inverse limit is topologized as a subspace of . If are the projection maps, then a point is represented by the sequence . A basic open neighborhood latex of is of the form where is an open neighborhood of and there is an such that for . Since the functions are continuous and , we may replace with where . Terminating at , we can inductively replace with where . In this way, we may take a basic open neighborhood of to satisfy for and for .

Lemma: Suppose is an inverse limit of Hausdorff spaces and are the projection maps. If are disjoint compact subsets of , then there exists an such that .

Proof. Suppose, to obtain a contradiction, that for every there exists and such that . Notice that since the coordinates of each and must agree with the bonding maps , this means for all . Since and are compact, we may find a subsequences and that converge to and respectively. We’re going to prove that also converges to . Consider a basic open neighborhood of . Let be the minimal such that . Since , there exists a such that for all . We can choose large enough so that . Now pick any . Since , we have for all . Since , this implies that for . Hence (for ) and we conclude that . However, this means the converges to both of the points and ; an impossibility in a Hausdorff space.

Remark: Notice that if , then it must also be the case that for . So for given and , we can choose to be as large as we want.

Theorem: An inverse limit of trees contains no simple closed curves.

Proof. Since topological trees are always Hausdorff, is Hausdorff. Suppose, to obtain a contradiction, that is an embedding. For , let be the intersection of and -th quadrant of the plane (include the bounding axes). Now are four (compact) arcs in the meet at endpoints to form the simple closed curve . Let and . Notice that and . Let be the paths that trace the arcs with the orientation shown below so that and are injective paths from to .

According to the previous Lemma (and the following Remark), if we denote the projection maps by , then we can find an such that and . Note that and are distinct points in the tree (as and are disjoint) and are thus connected in by a unique arc. Let trace out this arc from to . Now is the reduced representative of both and .

Considering the reduction of the path to , we see that an initial segment has image in and the terminal segment has image in .

to , we see that an initial segment has image in and the terminal segment has image in . Considering the reduction of the path to , we see that an initial segment has image in and the terminal segment has image in .

If , then . If , then . If , then lies in every .

In any of these cases, even the degenerate ones where one of or is or , we arrive at a contradiction.

Corollary: Every path-component of the limit of an inverse system of trees is uniquely arc-wise connected.

In Part II, the fact that an inverse limit of trees contains no simple closed curves will be a critical part of proving the Shape Injectivity Theorem.

References.

[1] M.L. Curtis and M.K. Fort, The fundamental group of one-dimensional spaces, Proc. Amer. Math. Soc. 10 (1959) 140-148.

[2] K. Eda and K. Kawamura, The fundamental groups of one-dimensional spaces, Topology and its Applications 87 (1998) 163-172.

[3] H.B. Griffiths, Infinite products of semigroups and local connectivity, Proc. London Math. Soc. (3), 6 (1956), 455-485.

[4] J.W. Morgan and I. Morrison, A van Kampen theorem for weak joins, Proceedings of the London Mathematical Society 53 (1986) 562-576.