Higher Dimensional Earrings

2d Hawaiian Earring 03

The (1-dimensional) earring space \mathbb{E} is a 1-dimensional Peano continuum (connected, locally path-connected, compact metric space) constructed by adjoining a shrinking sequence of circles at a single point.

Hawaiian earring space

The earring space

The importance of \mathbb{E} stems from the fact that this is the prototypical space that has infinite products in it’s fundamental group. This makes it a useful test-space for detecting the existence of non-trivial infinite products (or algebraic wildness) in other fundamental groups. The earring group is not a free group but it is basically an infinitary (i.e. having infinite products) analogue of a free group. Moreover, \mathbb{E} is aspherical, i.e. \pi_k(\mathbb{E},b_0)=0 for k\geq 2. In this post, I’m going to give a summary about their higher dimensional analogues \mathbb{E}_n\subseteq \mathbb{R}^{n+1}, the n-dimensional earring.

Constructing \mathbb{E}_n

Definition: let n\geq 0 and C_{k}^{n}=\{\mathbf{x}\in\mathbb{R}^{n+1}\mid \|\mathbf{x}-(1/k,0,0,\dots ,0)\|=1/k\} be the n-sphere of radius 1/k centered at (1/k,0,0,\dots,0). The n-dimensional earring is the set

\mathbb{E}_n=\bigcup_{k\in\mathbb{N}}C_{k}^{n}

with the subspace topology inherited from \mathbb{R}^{n+1}. Let b_0=(0,0,\dots,0) denote the origin, which we typically take as the basepoint.

2dim

Two views of \mathbb{E}_2, the 2-dimensional earring

The 0-dimensional earring \mathbb{E}_0=\{0,\dots,1/3,2/5,1/2,2/3,1,2\} is simply a convergent sequence and \mathbb{E}=\mathbb{E}_1 is the usual earring. We can build \mathbb{E}_n using a variety of constructions:

  1. One-point compactification: \mathbb{E}_n is homeomorphic to the one-point compactification of a countable disjoint union of open n-cells \coprod_{k\in\mathbb{N}}(0,1)^n, i.e. \mathbb{E}_n\cong \coprod_{k\in\mathbb{N}}[0,1]^n/\coprod_{k\in\mathbb{N}}\partial [0,1]^n.
  2. Subspace of a product: \mathbb{E}_n is homeomorphic to the wedge sum \bigvee_{k\in\mathbb{N}}S^n viewed naturally as a subspace of \prod_{k\in\mathbb{N}}S^n with the product topology. To make this identification, the m-th wedge summand is identified with the set \prod_{k\in\mathbb{N}}A_k where A_k=\{b_0\} if k\neq m and A_m=S^n.
  3. Inverse Limit: Let X_k=\bigcup_{i=1}^{k}C^{n}_{i} be the union of the first k n-spheres and r_{k+1,k}:X_{k+1}\to X_k be the retraction that collapses C_{k+1}^{n} to b_0 and is the identity elsewhere. Then the natural retractions r_k:\mathbb{E}_n\to X_k collapsing \bigcup_{i>k}C_{i}^{n} to b_0 induce a homeomorphism \mathbb{E}\cong\varprojlim_{k}(X_k,r_{k+1,k}).
  4. Reduced Suspension: For n\geq 1, the n-dimensional earring \mathbb{E}_{n} is homeomorphic to the reduced suspension \Sigma \mathbb{E}_{n-1} of the (n-1)-dimensional earring. By iteration, we obtain a formula similar to that for ordinary spheres: \mathbb{E}_{n+m}\cong\Sigma^m\mathbb{E}_n for all m,n\geq 0. It is important here that we use the reduced suspension and not the unreduced suspension. The undreduced suspension S\mathbb{E}_{n-1} is not homotopy equivalent to \Sigma \mathbb{E}_{n-1}. This means that your typical tricks, including Mayer Vietoris Sequences, won’t be helpful for computing the singular homology groups of \mathbb{E}_{n}.

The 2-dimensional earring \mathbb{E}_2, was shown to have some interesting properties in the famous paper [1] by Barratt and Milnor. To put their observation into context, recall that if we consider an ordinary wedge of 2-spheres, elementary computations show that

\displaystyle\widetilde{H}_k\left(\bigvee_{\mathbb{N}}S^2\right)=\begin{cases} \bigoplus_{\mathbb{N}}\mathbb{Z} & k=2 \\ 0 & k\neq 2 \end{cases}.

It might seem that \mathbb{E}_2 should be similar. In particular, \mathbb{E}_2 is simply connected and locally simply connected. However, Barratt and Milnor showed that there are infinitely many k>2 such that H_k(\mathbb{E}_2) is uncountable! This seems really strange and “anomalous” until one reads the paper to see how it’s done.

I’ve said it once and I’ll say it again: the “wildness” in wild algebraic topology is really just about infinite products. The reason why higher homology groups of \mathbb{E}_2 are often non-trivial is precisely because one can consider infinite sums \sum_{m=1}^{\infty}[f_m] of elements of [f_m]\in \pi_k(\mathbb{E}_2), e.g. given by shrinking sequences of non-trival Whitehead products. Even though [f_m] is trivial in homology of any finite wedge of the spheres C_{k}^{2}, the k-cycle represented by the infinite \sum_{m=1}^{\infty}[f_m] (apply the Hurewicz homomorphism) is not the boundary of a finite sum of n+1-chains and therefore represents a non-trivial homology class.

The homotopy groups \pi_k(\mathbb{E}_n), k\leq n

Recall that a space (X,x_0) is n-connected if \pi_m(X,x_0) is trivial for 0\leq m\leq n.

Theorem (Eda-Kawamura): \mathbb{E}_n is (n-1)-connected and locally (n-1)-connected. Moreover, the embedding \varphi:\mathbb{E}_n\to \prod_{k=1}^{\infty}C_{k}^{n} induces an isomorphism on \pi_n.

This means that

\pi_n(\mathbb{E}_n,b_0)\cong \prod_{k=1}^{\infty}\mathbb{Z}

is the Baer-Specker group.

Easy part of the proof: \varphi_{\#}:\pi_n(\mathbb{E}_n,b_0)\to\prod_{k=1}^{\infty}\mathbb{Z} is onto.

Identify \pi_n(C_{k}^{n},b_0) with \mathbb{Z}. Given a sequence of integers (a_1,a_2,a_3,\dots)\in\prod_{k=1}^{\infty}\mathbb{Z}, let f_k:(I^n,\partial I^n)\to (C_{k}^{n},b_0) be a map such that [f_k]=a_k\in \pi_n(C_{k}^{n},b_0). Define f=\prod_{k=1}^{\infty}f_k as the infinite concatenation of n-loops. Explicitly, f:(I^n,\partial I^n)\to (\mathbb{E}_n,b_0) is defined so that the restriction f:\left[\frac{k-1}{k},\frac{k}{k+1}\right]\times I^{n-1}\to\mathbb{E}_n is f(t_1,t_2,\dots, t_n)=f_k((1 + k) (1 + k (t_1-1)),t_2,t_3,\dots,t_n), for all k\geq 1 and f(1,t_2,t_3,\dots,t_n)=b_0.

higherinfproduct

The map f as an infinite concatenation of 2-loops

It’s now easy to see that \varphi_{\#}([f])=(a_1,a_2,a_3,\dots ), proving that \varphi_{\#} is surjective.

Hard part of the proof: \varphi_{\#} is injective.

The Eda-Kawamura proof in [2] is inspired but is written in a very technical fashion and has scared off many-a-reader. Essentially, the key idea of the proof is to simultaneously perform simplicial/cellular approximation on an infinite sequence of disjoint polyhedral domains in I^n. Then you work to perform an infinite Eckmann-Hilton type argument, to show that every map (I^n,\partial I^n)\to (\mathbb{E}_n,b_0) is homotopic to one of the form f=\prod_{k=1}^{\infty}f_k as described above, i.e. where f_k has image in C_{k}^{n}. Then if \varphi_{\#}([f])=(0,0,0,\dots), each f_k has degree 0 as a map S^n\to C_{k}^{n} and thus is null-homotopic in C_{k}^{n}. Let H_k:I^n\times I\to C_{k}^{n} be such a null-homotopy where H_{k}(\mathbf{t},0)=f_k(\mathbf{t}) and H_k(\mathbf{t},1)=b_0. Then we can define a null-homotopy H:I^n\times I\to \mathbb{E}_n of f so that H(\mathbf{t},s)=H_k(\mathbf{t},(1 + k) (1 + k (s-1))) for s\in \left[\frac{k-1}{k},\frac{k}{k+1}\right] and H(\mathbf{t},1)=b_0.

I do wonder if there is a simpler inverse limit type approach to proving that \varphi_{\#} is injective where the null-homotopy H=\varprojlim_{k}G_k is built as an inverse limit of null-homotopies G_k:I^n\times I\to \bigcup_{j\geq k}C_{j}^{n}. It would still be highly non-trivial, but would lend some insight into dealing with higher dimensional homotopy groups. If possible, one would likely have to inductively build the maps G_k by applying simplicial approximation at each step.

The homotopy groups \pi_k(\mathbb{E}_n), k> n

As noted in my problem list, these groups are still unknown. Of course, the higher homotopy groups of spheres are mysterious themselves here so the ultimate goal would be to express \pi_k(\mathbb{E}_n) in terms of homotopy groups of spheres.

From above, we may consider \mathbb{E}_n as a closed subspace of the direct product (S^n)^{\infty}=\prod_{\mathbb{N}}S^n over the naturals. The key is to notice that the argument for the “easy” part of the proof above applies to any dimension: for any k\in\mathbb{N}, the inclusion \mathbb{E}_n\to (S^n)^{\infty} induces a surjective homomorphism \varphi_{\#}:\pi_k(\mathbb{E}_n)\twoheadrightarrow \pi_k((S^n)^{\infty})=\prod_{\mathbb{N}}\pi_k(S^n). Moreover, by applying infinite commutativity, i.e. infinite Eckmann-Hilton, this surjection is a split epi.

Let’s look at the homotopy long exact sequence of the pair ((S^n)^{\infty},\mathbb{E}_n):

up5

The surjections tells us that we really have

up6

which splits up into short exact sequences:

up7

Since the epi \varphi_{\#}:\pi_k(\mathbb{E}_n)\twoheadrightarrow\prod_{k=1}^{\infty}\pi_k(S^n) always splits canonically, we have

\pi_k(\mathbb{E}_n)\cong \pi_{k+1}((S^n)^{\infty},\mathbb{E}_n)\oplus \prod_{\mathbb{N}}\pi_k(S^n)

This tells us at least that explicitly computing \pi_k(\mathbb{E}_n) requires computing \pi_k(S^n). Notice that the Eda-Kawamura computation only tells us that \pi_{n+1}((S^n)^{\infty},\mathbb{E}_n)=0.

An interesting case is

\pi_{3}(\mathbb{E}_2)\cong \pi_{4}((S^2)^{\infty},\mathbb{E}_2)\oplus \prod_{\mathbb{N}}\pi_{3}(S^2)\cong \pi_{4}((S^2)^{\infty},\mathbb{E}_2)\oplus \prod_{\mathbb{N}}\mathbb{Z}

since \pi_{3}(S^2)\cong \mathbb{Z} is generated by the Hopf map.

So the cheesesteak-worthy question is: what is \pi_{k+1}((S^n)^{\infty},\mathbb{E}_n) when k>n?

It will typically be non-trivial because of Whitehead products and infinite products of Whitehead products…just like in Barratt-Milnor! I have my own ideas about what the answer is in the case k=n+1, but there is one technical hurdle in the way from making it precise. There is a piece of technology missing that needs to be developed.

References:

[1] M.G. Barratt, J. Milnor, An example of anomalous singular theory, Proc. Amer. Math. Soc. 13 (1962) 293-297.

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

Added 3-12-20. I recently learned that the splitting of the exact sequence above appears in the following:

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

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1 Response to Higher Dimensional Earrings

  1. Pingback: Infinite Commutativity: Part II | Wild Topology

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