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.

2d Hawaiian Earring 02

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
  • 1UV_0: Yes
  • \pi_m-shape injective: Yes, for 0\leq m\leq n

See this blog post for some more details.

[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.

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