# Shrinking Wedge of Tori

Let $X_n\cong S^1\times S^1$ be a homeomorphic copy of the torus for all $n\in\mathbb{N}$ with basepoint $x_n$ corresponding to the identity element of $S^1\times S^1$. Let $X=\widetilde{\bigvee}_{n\in\mathbb{N}}X_n$ be the shrinking wedge of the sequence $\{X_n\}$ with wedge point $x_0$. In particular, the underlying set is the usual one-point union $\bigvee_{n\in\mathbb{N}}X_n$. A set $U$ is open in $X$ if and only if $U\cap X_n$ is open in $X_n$ for all $n\in\mathbb{N}$ and $X_n\subseteq U$ for all but finitely many $n$ whenever $x_0\in U$.

This space can be embedded in $\mathbb{R}^3$ as seen below.

Shrinking wedge of tori

Other Names: shrinking tori wedge

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

Fundamental Group: Let $Y_n=\bigvee_{i=1}^{n}X_n$ be the wedge of the first $n$-tori and notice that $\pi_1(Y_n,x_0)\cong \ast_{i=1}^{n}\mathbb{Z}^2$ is the free product of $n$-copies of $\mathbb{Z}^2$. There are retractions $Y_{n+1}\to Y_n$ collapsing $X_{n+1}$ to $x_0$ that induce homomorphisms $\pi_1(Y_{n+1},x_0)\to \pi_1(Y_n,x_0)$. These form an inverse system to which there is a canonical homomorphism $\Phi:\pi_1(X,x_0)\to \varprojlim_{n}\pi_1(Y_nx_0)$. It follows from results in [3] that $\Phi$ is injective and has image on the subgroup of locally eventually constant elements in $\varprojlim_{n}\pi_1(Y_nx_0)$.

Note that the space $X$ may also be constructed as a “shrinking CW-complex” by starting with the usual earring $\mathbb{E}$ and attaching a sequence of 2-cells $e^{2}_{n}$ along the conjugating loops

$\ell_{1}\cdot\ell_{2}\cdot\ell_{1}^{-}\cdot \ell_{2}^{-}$,

$\ell_{3}\cdot\ell_{4}\cdot\ell_{3}^{-}\cdot \ell_{4}^{-}$,

$\ell_{5}\cdot\ell_{6}\cdot\ell_{5}^{-}\cdot \ell_{6}^{-}$,

$\ell_{7}\cdot\ell_{8}\cdot\ell_{7}^{-}\cdot \ell_{8}^{-}$,

Here, $\ell_n$ represents the loop traversing the $n$-th circle of $\mathbb{E}$. We can then give this relative CW-structure the topology so that every neighborhood of the wedgepoint $b_0$ of $\mathbb{E}$ contains all but finitely many of the 2-cells $e^{2}_{n}$.

The infinite earring as a subspace of the shrinking wedge of tori.

In this way, we see that $X$ sits naturally as $\mathbb{E}\subseteq X\subseteq (S^1)^{\mathbb{N}}$, that is, between the infintie earring and infinite torus. This construction also makes it clear that $\pi_1(X,x_0)$ can be realized as a quotient of the earring group $\pi_1(\mathbb{E},b_0)$, where any finite or infinite appearance of the attaching loops in an element of $\pi_1(\mathbb{E},b_0)$ becomes trivialized. For instance, given our attaching loops $\alpha_n=\ell_{2n-1}\cdot\ell_{2n}\cdot\ell_{2n-1}^{-}\cdot\ell_{2n}$, then $[\ell_1\cdot\alpha_1\cdot\ell_2\cdot\alpha_2\cdot\ell_3\cdot\alpha_3\cdots]$ becomes identified with $[\ell_1\cdot\ell_2\cdot\ell_3...]$ in the quotient.  To make this precise, it is necessary to trivialize conjugates of the loops $\alpha_n$ and shrinking sequences of such conjugates. This is more intuitive than precise, but this is not the place for a precise treatment.

Fundamental Group Properties: Uncountable, torsion free, residually free. Not locally free (since $\mathbb{Z}^2$ is not locally free).

Higher homotopy groups: $\pi_n(X,x_0)=0$ for all latex $n\geq 2$, i.e. $X$ is aspherical. This is proved in [1], which extended the ad-hoc approach for $n=2$ in [2].

Homology groups:

$\widetilde{H}_n(X)=\begin{cases} 0 , & n =0 \\ ?? , & n = 1 \\ \bigoplus_{\mathbb{N}}\mathbb{Z} , & n=2 \\ ?? , & n\geq 3 \end{cases}$

For $n=2$, see [1, Theorem 1.6]. I imagine that first homology is abstractly isomorphic to $H_1(\mathbb{E})$ but I don’t recall seeing this proven anywhere yet.

Cech Homotopy groups: $\check{\pi}_n(X,x_0)=\begin{cases} 0, & n=0 \\ \mathbb{Z}^{\mathbb{N}}, & n =1 \\ 0, & n\geq 2 \end{cases}$.

Cech Homology groups: $\check{H}_n(X)=\begin{cases} 0, & n=0 \\ \mathbb{Z}^{\mathbb{N}}, & n =1 \\ 0, & n\geq 2 \end{cases}$.

Wild Set/Homotopy Type: The wild set $\mathbf{w}(X)=\{x_0\}$

Other Properties:

• Semi-locally simply connected: No.
• Traditional Universal Covering Space: No
• Generalized Universal Covering Space: Yes
• Homotopically Hausdorff: Yes
• Strongly (freely) Homotopically Hausdorff: Yes
• Homotopically Path-Hausdorff: Yes
• $1$$UV_0$: Yes
• $\pi_1$-shape injective: Yes

Other notes: If you collapse the bolded earring in the second image, you get the 2-dimensional earring space $\mathbb{E}_2$.

[1] J. Brazas, Homotopy groups of shrinking wedges of non-simply connected CW-complexes. https://arxiv.org/abs/2204.03751

[2] K. Eda, K.H. Umed, D. Repovs, A. Zastrow, On snake cones, alternating cones and related constructions. Glasnik Mat. 48 (2013) no. 1 115-135.

[3]  J. Morgan, I. Morrison, A van kampen theorem for weak joins, Proc. London Math. Soc. 53 (1986) 562–576.