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,
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 .
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 .
- 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.