Griffiths Twin Cone

Let C_n be the circle of radius 1/n centered at (1/n,0) with basepoint b_0=(0,0). Let \mathbb{E}_o=\bigcup_{k\in\mathbb{N}}C_{2k-1} be the odd circles of the earring space and \mathbb{E}_e=\bigcup_{k\in\mathbb{N}}C_{2k} be the even circles so that the union \mathbb{E}=\mathbb{E}_o\cup \mathbb{E}_e is the usual earring space and \mathbb{E}_o\cap \mathbb{E}_e=\{b_0\}. Embed \mathbb{E}\subseteq \mathbb{R}^3 in the xy-plane. Let C\mathbb{E}_o\subseteq \mathbb{R}^3 be the cone over the odd circles with vertex (0,0,1) and C\mathbb{E}_e\subseteq \mathbb{R}^3 be the cone over the even circles with vertex (0,0,-1). The Griffiths Twin Cone is \mathbb{G}=C\mathbb{E}_o\cup C\mathbb{E}_e.

See this post for other details about \pi_1.

griffiths2

Griffiths Twin Cone

Generic Construction: Since \mathbb{E}_o and \mathbb{E}_e are homeomorphic copies of \mathbb{E}, one can construct \mathbb{G} by letting C\mathbb{E}=\mathbb{E}\times [0,1]/\mathbb{E}\times\{1\} be the cone over the earring with basepoint x_0, the image of (b_0,0) in the quotient. Then \mathbb{G} is the wedge (C\mathbb{E},x_0)\vee(C\mathbb{E},x_0).

JBrazas

Griffiths Twin Cone

Topological Properties:  2-dimensional, path-connected, locally path-connected, compact metric space. Embeds in \mathbb{R}^3.

Fundamental Group: The inclusions i_o:\mathbb{E}_o\to\mathbb{E} and i_e:\mathbb{E}_e\to\mathbb{E} are sections and so induce injections on \pi_1. Let H_o=(i_o)_{\#}(\pi_1(\mathbb{E}_o,b_o)) and H_e=(i_e)_{\#}(\pi_1(\mathbb{E}_e,b_0)) be the images of these homomorphisms. By the van Kampen Theorem, \pi_1(\mathbb{G},b_0) is isomorphic to \pi_1(\mathbb{E},b_0)/N where N is the normal closure of H_o\cup H_e.

Alternatively, let f_o,f_e:\mathbb{E}\to\mathbb{E} be the retractions that (respectively) collapse \mathbb{E}_o and \mathbb{E}_e to b_0 and are the identity elsewhere. Then \pi_1(\mathbb{G},b_0) is isomorphic to the coequalizer of the induced homomorphisms (f_o)_{\#},(f_e)_{\#}:\pi_1(\mathbb{E},b_0)\to \pi_1(\mathbb{E},b_0).

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

Higher homotopy groups: \pi_n(\mathbb{G},b_0)=0 for n \geq 2, i.e. \mathbb{G} is aspherical.

Homology groups: \widetilde{H}_n(\mathbb{G})=\begin{cases} \prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}, & n=1\\ 0, & n\neq 1 \end{cases}

Note: the isomorphism in dimension 1 is not constructive.

Cech Homotopy groups: \check{\pi}_n(\mathbb{G})=0 for all n\geq 0. Moreover, \mathbb{G} is shape equivalent to a point.

Cech Homology groups: \check{H}_n(\mathbb{G})=0 for all n\geq 0.

Wild Set/Homotopy Type: The wild set \mathbf{w}(\mathbb{G})=\{b_0\} 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
  • 1–UV_0: No
  • \pi_1-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.