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.

One nice way to think about this space is that you start with the

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.