Let be a homeomorphic copy of the torus for all with basepoint corresponding to the identity element of . Let be the shrinking wedge of the sequence with wedge point . In particular, the underlying set is the usual one-point union . A set is open in if and only if is open in for all and for all but finitely many whenever .

This space can be embedded in as seen below.

**Other Names:** shrinking tori wedge

**Topological Properties:** 2-dimensional, path-connected, locally path-connected, compact metric space. Embeds in .

**Fundamental Group: **Let be the wedge of the first -tori and notice that is the free product of -copies of . There are retractions collapsing to that induce homomorphisms . These form an inverse system to which there is a canonical homomorphism . It follows from results in [3] that is injective and has image on the subgroup of locally eventually constant elements in .

Note that the space may also be constructed as a “shrinking CW-complex” by starting with the usual earring and attaching a sequence of 2-cells along the conjugating loops

,

,

,

,

…

Here, represents the loop traversing the -th circle of . We can then give this relative CW-structure the topology so that every neighborhood of the wedgepoint of contains all but finitely many of the 2-cells .

In this way, we see that sits naturally as , that is, between the infintie earring and infinite torus. This construction also makes it clear that can be realized as a quotient of the earring group , where any finite or infinite appearance of the attaching loops in an element of becomes trivialized. For instance, given our attaching loops , then becomes identified with in the quotient. To make this precise, it is necessary to trivialize conjugates of the loops and shrinking sequences of such conjugates. This is more intuitive than precise, but this is not the place for a precise treatment.

**Fundamental Group Properties:** Uncountable, torsion free, residually free. Not locally free (since is not locally free).

**Higher homotopy groups:** for all latex , i.e. is aspherical. This is proved in [1], which extended the ad-hoc approach for in [2].

**Homology groups: **

For , see [1, Theorem 1.6]. I imagine that first homology is abstractly isomorphic to but I don’t recall seeing this proven anywhere yet.

**Cech Homotopy groups:** .

**Cech Homology groups:** .

**Wild Set/Homotopy Type:** The wild set

**Other Properties:**

**Semi-locally simply connected:**No.**Traditional Universal Covering Space:**No**Generalized Universal Covering Space:**Yes**Homotopically Hausdorff:**Yes**Strongly (freely) Homotopically Hausdorff:**Yes**Homotopically Path-Hausdorff:**Yes**–:**Yes**-shape injective:**Yes

**Other notes:** If you collapse the bolded earring in the second image, you get the 2-dimensional earring space .

[1] J. Brazas, Homotopy groups of shrinking wedges of non-simply connected CW-complexes. https://arxiv.org/abs/2204.03751

[2] K. Eda, K.H. Umed, D. Repovs, A. Zastrow, *On snake cones, alternating cones and related constructions*. Glasnik Mat. 48 (2013) no. 1 115-135.

[3] J. Morgan, I. Morrison, *A van kampen theorem for weak joins*, Proc. London Math. Soc. 53 (1986) 562–576.