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 $\pi_1$.

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 $(C\mathbb{E},x_0)\vee(C\mathbb{E},x_0)$.

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.