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.