Fix an integer . For , let be a homeomorphic copy of the unit -sphere with basepoint .

The -dimensional earring space is the shrinking wedge (one-point union) of -spheres. Sometimes, this is just written as . In particular, the underlying set of is the underlying set of the usual infinite one-point union with wedgepoint . A set is open in if and only if is open in for all and for all but finitely many whenever .

Often, the -dimensional earring is defined as the following subspace of :

**Visualizing low dimensional cases:**

is a “convergent sequence” where the non-isolated point is the basepoint.

is the usual earring space.

is a shrinking bouquet of -spheres.

**Other Names:** is sometimes called the Barratt-Milnor Sphere. 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 that collapse all but the -th sphere to a point. Together, these induce a map to the infinite product of -spheres (with the product topology). This map is actually a closed embedding onto the wedge . In this way, we can view canonically as a subspace of whenever it is convenient to do so.*Inverse Limit:*Let be the union of the first spheres. Let be the retraction that collapses the -th sphere to the basepoint and is the identity elsewhere. The canonical maps induce a homeomorphism 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: ** is a -dimensional, compact metric space that embeds in . When , is -connected and locally -connected.

**Homotopy Groups: **Fix . The space is -connected and so for . The retractions induce a canonical homomorphism . It is a highly non-trivial result of Eda-Kawamura that this homomorphism is actually an isomorphism. Hence, is isomorphic to the Specker group .

**Higher homotopy groups:** When , the isomorphism type of is unknown although it is known that it splits as .

**Homology groups: **Fix **.**

**Cech Homotopy groups: **Fix .

See [3] for the cases .

**Cech Homology groups:**

**Other Properties:**

**Reduced suspension increases dimension by one:**(the wedge point must be the basepoint).**The earring spaces behave nicely with respect to the smash product operation:**. (see these unpublished notes)**Traditional Universal Covering Space:**is not path-connected and is not semilocally simply connected so these two do not have simply connected covering spaces. When , is its own universal cover.**Generalized Universal Covering Space:**Yes**Homotopically Hausdorff:**Yes**Strongly (freely) Homotopically Hausdorff:**Yes**Homotopically Path-Hausdorff:**Yes**–:**Yes**-shape injective:**Yes, for

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.