# Harmonic Pants Space

The harmonic pants space is much like the harmonic archipelago but highlights even more striking phenomena. It is constructed by attaching a countably infinite number of `pairs of pants’ to the infinite earring in a particular way.

Let $p_0$ be the wild point of the earring latex $\mathbb{E}$ and let $\ell_n:S^1\to\mathbb{E}$ be the loop parameterizing the $n$-th circle of $\mathbb{E}$.

Consider a sequence of pairs of pants $P_1,P_2,P_3,...$ (a pair of pants is a 2-disk with two open 2-disks removed from the interior). Let $\alpha_n,\beta_n,\gamma_n:S^1\to \partial P_n$ parameterized the three boundary components of $P_n$ as illustrated below. Attach $P_n$ to $\mathbb{E}$ using the attaching map $f_n:\partial P_n\to \mathbb{E}$ defined so that $f_n\circ \alpha_n=\ell_n$, $f_n\circ\beta_n=\ell_{2n}$, and $f_n\circ\gamma_n=\ell_{2n+1}$. Let $\mathbb{P}$ be the resulting quotient of $\mathbb{E}\sqcup\coprod_{n\geq 1}P_n$.

Construction of the harmonic pants space

It’s helpful to think about it this way: for the $n$-th circle $C_n$ of $\mathbb{E}$, you have one pair of pants “coming out of” $C_n$ with the belt attached to $C_n$ and then, as long as $n>1$, there is one pair of pants “coming into” $C_n$ where ankle cuff from an earlier pair of pants is attached. For instance $C_{11}$ has $P_{11}$ attached at the belt and an ankle of $P_5$ attached.

Alternative Construction: Here’s another nice way to think about this space. For $n\in\mathbb{N}$ and $i\in\{1,2\}$, let $A_{n,i}$ be a copy of the unit circle. Let $\rho_{n,i}:S^1\to A_{n,i}$ be a parameterization of each circle. Consider the one-point union $\mathbb{E}\vee\bigvee_{(n,i)\in\mathbb{N}\times\{1,2\}}A_{n,i}$, which is latex $\mathbb{E}$ with a regular infinite one-point union of circles attached and note that this is not same as the earring. Now for every $n\in\mathbb{N}$, attach a 2-cell to $\mathbb{E}\vee\bigvee_{(n,i)\in\mathbb{N}\times\{1,2\}}A_{n,i}$ by the attaching loop

$(\rho_{n,1}\cdot\ell_{2n}\cdot\rho_{n,1}^{-}\cdot\rho_{n,2}\cdot\ell_{2n+1}\cdot\rho_{n,2}^{-})\cdot\ell_{n}^{-}$.

The resulting space is homeomorphic to $\mathbb{P}$.

Homotopy Type and Metric Construction: Since $\mathbb{P}$ is not first countable, it can’t be embedded into any finite-dimensional real space. However, by changing the topology at $p_0$, it is possible to construct a subset $\mathbb{P}^{\ast}\subseteq \mathbb{R}^3$ for which there is a bijective homotopy equivalence $\mathbb{P}\to \mathbb{P}^{\ast}$. This means that for all homotopy-theoretic purposes, you can replace $\mathbb{P}$ like a metric space. But for most arguments, it’s easier to deal with $\mathbb{P}$ in the weak topology.

Topological Properties: This space is path-connected, locally path-connected, 2-dimensional, and paracompact. In particular $\mathbb{P}\backslash \{p_0\}$ is a 2-manifold with boundary component $C_1\backslash\{p_0\}$.

The harmonic pants space $\mathbb{P}$ is a relative CW-complex, that is, it is obtained by attaching cells to a given space. With the weak topology with respect to $\{\mathbb{E},P_1,P_2,P_3,...\}$, $\mathbb{P}$ is not first countable at $b_0$. However, every compact subset of $\mathbb{P}$ must lie in the union of $\mathbb{E}$ and finitely many $P_n$.

Fundamental Group: Let $W=\bigvee_{(n,i)\in\mathbb{N}\times\{1,2\}}A_{n,i}$ and notice that $\pi_1(W,p_0)$ is a countably infinite free group generated by $\{[\rho_{n,i}]\mid n\in\mathbb{N},i\in\{1,2\}\}$. Since $W$ is first countable and locally contractible at $p_0$, we have $\pi_1(\mathbb{E}\vee W)\cong \pi_1(\mathbb{E},p_0)\ast \pi_1(W,p_0)$. Since $\mathbb{P}$ can be constructed by attaching 2-cells as described above, we have that $\pi_1(\mathbb{P},p_0)$ is isomorphic to the quotient of $\pi_1(\mathbb{E},p_0)\ast \pi_1(W,p_0)$ by the relation

$[\ell_n]=[\rho_{n,1}][\ell_{2n}][\rho_{n,1}]^{-1}[\rho_{n,2}][\ell_{2n+1}][\rho_{n,2}]^{-1}$.

In [1], it is shown that $\pi_1(\mathbb{P},b_0)$ is isomorphic to a certain direct limit of earring groups much like the harmonic archipelago.

Fundamental Group Properties: Uncountable, torsion free, locally free but not residually free.

Higher homotopy groups: $\pi_n(X)=0$ for $n \geq 2$, i.e. this space is aspherical.

Homology groups: There is no obvious description of $H_1(\mathbb{P})$, but this should end up being similar to that for the archipelago.

$\widetilde{H}_n(\mathbb{P})=\begin{cases} ? , & n=1 \\ 0, & n \neq 1 \end{cases}$

Cech Homotopy groups: $\check{\pi}_n(X)=\begin{cases} F_{\infty} & n=1 \\ 0, & n \neq 1 \end{cases}$ where $F_{\infty}$ denotes a countably infinite free group.

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

Cech homotopy and homology only detect the non-triviality of the loops $\rho_{n,i}$.

Wild Set/Homotopy Type: The 1-wild set consists of the single point set $\{p_0\}$.

Other Properties:

• Semi-locally simply connected: No, not at $p_0$
• 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
• Discrete monodromy property: Yes
• $\pi_1$-shape injective: No

References:

[1] J. Brazas, H. Fischer, On the failure of the first Cech homotopy group to register geometrically relevant fundamental group elements, To Appear in Bul. London Math. Soc.