## Higher Dimensional Earrings

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.

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.

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 $f(1,t_2,t_3,\dots,t_n)=b_0$.

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 $((S^n)^{\infty},\mathbb{E}_n)$:

The surjections tells us that we really have

which splits up into short exact sequences:

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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