Let be the circle of radius centered at with basepoint . Let be the odd circles of the earring space and be the even circles so that the union is the usual earring space and . Embed in the xy-plane. Let be the cone over the odd circles with vertex and be the cone over the even circles with vertex . The **Griffiths Twin Cone** is .

See this post for other details about .

Griffiths Twin Cone

**Generic Construction:** Since and are homeomorphic copies of , one can construct by letting be the cone over the earring with basepoint , the image of in the quotient. Then is the wedge .

Griffiths Twin Cone

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

**Fundamental Group:** The inclusions and are sections and so induce injections on . Let and be the images of these homomorphisms. By the van Kampen Theorem, is isomorphic to where is the normal closure of .

Alternatively, let be the retractions that (respectively) collapse and to and are the identity elsewhere. Then is isomorphic to the coequalizer of the induced homomorphisms .

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

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

**Homology groups: **

Note: the isomorphism in dimension 1 is not constructive.

**Cech Homotopy groups:** for all . Moreover, is shape equivalent to a point.

**Cech Homology groups:** for all .

**Wild Set/Homotopy Type:** The wild set is a single point.

**Other Properties:**

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

**References: **

[1] H.B. Griffiths, *The fundamental group of two spaces with a common point*, Quart. J. Math. Oxford (2) 5 (1954) 175-190.

[2] H. Fischer, K. Eda, *Cotorsion-free groups from a topological viewpoint*, Topology Appl. 214 (2016) 21-34.

[3] K. Eda, *A locally simply connected space and fundamental groups of one point unions of cones*, Proceedings of the American Mathematical Society. 116 no. 1 (1992) 239-249.

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