# Earring Mapping Torus

Let where is the circle of radius with basepoint . Let be the shift map, which takes homeomorphically onto . The **earring mapping torus** is the mapping torus of the map . In particular,

where .

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

**Fundamental Group:** Identify with the image of in . Let be the loop in traversing the n-th circle. If is the loop going once around the circle formed by the basepoint in the mapping torus, then we have the relation in . Using the van Kampen theorem and the fact that neighborhoods of deformation retract onto , one can show that where is the normal subgroup generated by .

The inner black loop represents the image of .

**Fundamental Group Properties:** Uncountable, torsion free, locally free. Not residually free.

**Higher homotopy groups:** for , i.e. is aspherical.

**Homology groups: **One non-intuitive thing about the singular homology of this space is that is infinite cyclic, but is highly non-trivial.

**Cech Homotopy groups:** . Moreover, is shape equivalent to .

**Cech Homology groups:**

**Wild Set/Homotopy Type:** The wild set homeomorphic to .

**Other Properties:**

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

**Other notes:** If you collapse the circle , then you get an infinitely shrinking spiral, which is contractible.

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