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 be the wild point of the earring latex and let be the loop parameterizing the -th circle of .

Consider a sequence of pairs of pants (a pair of pants is a 2-disk with two open 2-disks removed from the interior). Let parameterized the three boundary components of as illustrated below. Attach to using the attaching map defined so that , , and . Let be the resulting quotient of .

It’s helpful to think about it this way: for the -th circle of , you have one pair of pants “coming out of” with the belt attached to and then, as long as , there is one pair of pants “coming into” where ankle cuff from an earlier pair of pants is attached. For instance has attached at the belt and an ankle of attached.

**Alternative Construction:** Here’s another nice way to think about this space. For and , let be a copy of the unit circle. Let be a parameterization of each circle. Consider the one-point union , which is latex with a regular infinite one-point union of circles attached and note that this is **not** same as the earring. Now for every , attach a 2-cell to by the attaching loop

.

The resulting space is homeomorphic to .

**Homotopy Type and Metric Construction:** Since is not first countable, it can’t be embedded into any finite-dimensional real space. However, by changing the topology at , it is possible to construct a subset for which there is a bijective homotopy equivalence . This means that for all homotopy-theoretic purposes, you can replace like a metric space. But for most arguments, it’s easier to deal with 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 is a 2-manifold with boundary component .

The harmonic pants space is a relative CW-complex, that is, it is obtained by attaching cells to a given space. With the weak topology with respect to , is not first countable at . However, every compact subset of must lie in the union of and finitely many .

**Fundamental Group: **Let and notice that is a countably infinite free group generated by . Since is first countable and locally contractible at , we have . Since can be constructed by attaching 2-cells as described above, we have that is isomorphic to the quotient of by the relation

.

In [1], it is shown that 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:** for , i.e. this space is aspherical.

**Homology groups: **There is no obvious description of , but this should end up being similar to that for the archipelago.

**Cech Homotopy groups:** where denotes a countably infinite free group.

**Cech Homology groups:**

Cech homotopy and homology only detect the non-triviality of the loops .

**Wild Set/Homotopy Type:** The 1-wild set consists of the single point set .

**Other Properties:**

**Semi-locally simply connected:**No, not at**Traditional Universal Covering Space:**No**Generalized Universal Covering Space:**Yes**Homotopically Hausdorff:**Yes**Strongly (freely) homotopically Hausdorff:**Yes**Homotopically Path-Hausdorff:**Yes**–:**Yes**Discrete monodromy property:**Yes**-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.