The n-dimensional earring

Fix an integer $n\geq 0$ . For $k\in\mathbb{N}$ , let $X_k$ be a homeomorphic copy of the unit $n$ -sphere $S^n=\{\mathbf{x}\in\mathbb{R}^{n+1}\mid \|\mathbf{x}\|=1\}$ with basepoint $x_k=(1,0,0,\dots,0)$ .

The $n$ -dimensional earring space $\mathbb{E}_n$ is the shrinking wedge (one-point union) of $n$ -spheres. Sometimes, this is just written as $\widetilde{\bigvee}_{k\in\mathbb{N}}S^n$ . In particular, the underlying set of $\mathbb{E}_n$ is the underlying set of the usual infinite one-point union $\bigvee_{k\in\mathbb{N}}X_k$ with wedgepoint $x_0$ . A set $U$ is open in $\mathbb{E}_n$ if and only if $U \cap X_k$ is open in $X_k$ for all $n\in\mathbb{N}$ and $X_k \subseteq U$ for all but finitely many $k$ whenever $x_0 \in U$ .

Often, the $n$ -dimensional earring is defined as the following subspace of $\mathbb{R}^{n+1}$ :

$\mathbb{E}_n= \bigcup_{k\in\mathbb{N}}\left\{(x_1,x_2,\dots,x_{n+1})\mid \left(x_1-\frac{1}{k}\right)^2+\sum_{i=2}^{n+1}x_{i}^{2}=\frac{1}{k^2}\right\}$

Visualizing low dimensional cases:

$\mathbb{E}_0= \{0,\dots ,1/4,1/3,1/2,1\}$ is a “convergent sequence” where the non-isolated point $0$ is the basepoint.

$\mathbb{E}_0$ is a convergent sequence. The basepoint, is the limit point illustrated as the larger blue point.

$\mathbb{E}_1$ is the usual earring space.

$\mathbb{E}_2$ is a shrinking bouquet of $2$ -spheres.

The 2-dimensional earring

Other Names: $\mathbb{E}_2$ is sometimes called the Barratt-Milnor Sphere. $\mathbb{E}_n$ has been referred to as the “n-dimensional Hawaiian earring” in the literature; however, there is substantial movement by experts to move away from using this term.

Alternative constructions:

Subspace of infinite product: There are retractions $r_k:\mathbb{E}_n\to X_k$ that collapse all but the $k$ -th sphere to a point. Together, these induce a map $\mathbb{E}_n\to \prod_{k\in\mathbb{N}}S^n$ to the infinite product of $n$ -spheres (with the product topology). This map is actually a closed embedding onto the wedge $\bigvee_{k\in\mathbb{N}}S^n\subseteq \prod_{k\in\mathbb{N}}S^n$ . In this way, we can view $\mathbb{E}_n$ canonically as a subspace of $\prod_{k\in\mathbb{N}}S^n$ whenever it is convenient to do so.
Inverse Limit: Let $X_m=\bigvee_{k=1}^{n}X_k$ be the union of the first $m$ spheres. Let $p_{m+1,m}:X_{m+1}\to X_m$ be the retraction that collapses the $(m+1)$ -th sphere to the basepoint and is the identity elsewhere. The canonical maps $p_{m}:\mathbb{E}_n\to X_m$ induce a homeomorphism $\mathbb{E}_n\to \varprojlim_{m}(X-m,r_{m+1,m})$ to the inverse limit. Hence, we may view $latex \mathbb{E}_n$ as this inverse limit whenever it is convenient to do so.

Topological Properties: $\mathbb{E}_n$ is a $n$ -dimensional, compact metric space that embeds in $\mathbb{R}^{n+1}$ . When $n\geq 1$ , $\mathbb{E}_n$ is $(n-1)$ -connected and locally $(n-1)$ -connected.

Homotopy Groups: Fix $n \geq 2$ . The space $\mathbb{E}_n$ is $(n-1)$ -connected and so $\pi_m(\mathbb{E}_n)=0$ for $0\leq m\leq n-1$ . The retractions $r_k:\mathbb{E}_n\to X_k$ induce a canonical homomorphism $\pi_{n}(\mathbb{E}_n)\to \prod_{k\in\mathbb{N}}\pi_n(X_k)$ . It is a highly non-trivial result of Eda-Kawamura that this homomorphism is actually an isomorphism. Hence, $\pi_{n}(\mathbb{E}_n)$ is isomorphic to the Specker group $\prod_{k\in\mathbb{N}}\mathbb{Z}=\mathbb{Z}^{\mathbb{N}}$ .

Higher homotopy groups: When $m>n$ , the isomorphism type of $\pi_m(\mathbb{E}_n)$ is unknown although it is known that it splits as $\prod_{k}\pi_m(S^n)\bigoplus \pi_m\left(\prod_{k}S^n,\mathbb{E}_n\right)$ .

Homology groups: Fix $n\geq 2$ .

$\widetilde{H}_m(\mathbb{E}_n)= \begin{cases} 0 , & 0 \leq m \leq n-1 \\ \mathbb{Z}^{\mathbb{N}}, & m=n \\ ??, & m>n \end{cases}$

Cech Homotopy groups: Fix $n\geq 2$ .

$\check{\pi}_m(\mathbb{E}_n)= \begin{cases} 0 , & 0 \leq m \leq n-1 \\ \mathbb{Z}^{\mathbb{N}} , & m=n,\, m=n+1=3 \\ (\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}} , & m=n+1\geq 4 \\ ??, & m>n+1 \end{cases}$

See [3] for the cases $m=n+1$ .

Cech Homology groups: $\check{H}_n(\mathbb{E}_n)= \begin{cases} 0, & n \neq m>0 \\ \mathbb{Z}^{\mathbb{N}} , & m=n \end{cases}$

Other Properties:

Reduced suspension increases dimension by one: $\Sigma \mathbb{E}_n\cong\mathbb{E}_{n+1}$ (the wedge point must be the basepoint).
The earring spaces behave nicely with respect to the smash product operation: $\mathbb{E}_m \wedge\mathbb{E}_n\cong\mathbb{E}_{m+n}$ . (see these unpublished notes)
Traditional Universal Covering Space: $\mathbb{E}_0$ is not path-connected and $\mathbb{E}_1$ is not semilocally simply connected so these two do not have simply connected covering spaces. When $n\geq 2$ , $\mathbb{E}_n$ is its own universal cover.
Generalized Universal Covering Space: Yes
Homotopically Hausdorff: Yes
Strongly (freely) Homotopically Hausdorff: Yes
Homotopically Path-Hausdorff: Yes
$1$ – $UV_0$ : Yes
$\pi_m$ -shape injective: Yes, for $0\leq m\leq n$