## The Harmonic Archipelago

Another wild space that has been receiving a lot of recent attention is the so-called Harmonic Archipelago $\mathbb{HA}$ which is the following subspace of $\mathbb{R}^3$.

Harmonic Archipelago

You can describe the construction like this: Start by drawing the usual Hawaiian earring $\mathbb{H}$ onto a solid disk $D$ in the xy-plane. Now between the 1st and 2nd hoops, draw a small disk and push it up so that it becomes a smooth hill with unit height. Do the same thing between the 2nd and 3rd hoops of $\mathbb{H}$  and then the 3rd and 4th hoops and so on. Notice that the Hawaiian earring $\mathbb{H}$ is naturally a subspace of $\mathbb{HA}$ and each hill is hollow underneath. Also, the diameters of the hills tend to $0$.

If you can believe it, the fundamental group of this space is even crazier than the fundamental group of the Hawaiian earring $\mathbb{H}$!

First, let’s get some notation down:

• Let $C_n=\left\{(x,y,0)|\left(x-\frac{1}{n}\right)^2+y^2=\frac{1}{n^2}\right\}$ be the n-th circle so that $\mathbb{H}=\bigcup_{n\geq 1}C_n$ with basepoint $x_0=(0,0,0)$.
• Let $\mathbb{H}_{\geq m}=\bigcup_{n\geq m}C_n$ be the smaller copies of the Hawaiian earring.
• Let $B_n\subset \mathbb{HA}$ be the open  disk between $C_n$ and $C_{n+1}$ which contains the n-th hill in the archipelago.

Now some observations:

1) $\mathbb{HA}$ is path-connected and locally path connected.

2) $\mathbb{HA}$  is non-compact since it does not include limit points on the z-axis. For instance, the sequence given by the top of the hills converges to $(0,0,1)$ which is not included. This means that if $X$ is compact and $f:X\to \mathbb{HA}$ is continuous, then the image $f(X)$ can only hit finitely many of the hills.

3) If $\ell_n:S^1\to C_n$ is the loop that traverses $C_n$ once counterclockwise in the xy-plane, then $\ell_{n}$ can be deformed over finitely many hills (but not infinitely many!). So the homotopy classes $[\ell_{n}]=[\ell_{n+1}]$ in $\pi_1(\mathbb{HA},x_0)$ are the same for all $n\geq 1$ but yet are non-trivial since deforming $\ell_1$ over every hill should violate continuity.

Lemma 1: Every based loop $\alpha:S^1\to \mathbb{HA}$ is homotopic to a loop $\beta_{m}\colon S^1 \to\mathbb{H}_{\geq m}$ for each $m\geq 1$.

Proof: As mentioned above, the image of any based loop $\alpha:S^1\to \mathbb{HA}$ can intersect at most finitely many hills $B_n$. Thus $\alpha$ must have image in one of the spaces that looks like this:

Notice the holes get smaller and smaller so this subspace is a deformation retract of one of the form

$B_1\cup B_2\cup\cdots\cup B_{m-1}\cup\mathbb{H}_{\geq m+1}.$

The retraction is given by expanding each little circle in the xy-plane to the entire hole usually present in the Hawaiian earring. At this point, we can choose $m$ to be as large as we want (by adding some hills back in). The resulting space looks like the one-point union of a smaller copy of the Hawaiian earring and a bumpy region that is homotopy equivalent to a circle.

Where there are no hills, we see a copy of the Hawaiian earring. So the region between the 3rd/4th, 4th/5th,… circles is empty.

Now deform the bumpy region $B_1\cup B_2\cup\cdots\cup B_{m-1}$ onto the smallest circle $C_m$. The composition of these deformation retracts provides a homotopy of $\alpha$ to a loop in $\mathbb{H}_{\geq m}$. Since we could choose $m$ to be arbitrarily large, the lemma is proven.$\square$

Lemma 1 basically says that every based loop is homotopic to arbitrarily small loops.

Corollary 2: The homomorphism on fundamental groups $\phi :\pi_1(\mathbb{H},x_0)\to\pi_1(\mathbb{HA},x_0)$ induced by inclusion is surjective and $\phi([\ell_m])=\phi([\ell_n])$ for all $n,m\geq 1$.

More generally, if $g_n=[\ell_n]\in\pi_1(\mathbb{H},x_0)$, then $g_{k_1}^{\epsilon_1}g_{k_2}^{\epsilon_2} \cdots g_{k_p}^{\epsilon_p}\in\ker\phi$ whenever $\sum_{j}\epsilon_{j}=0$. However, the fundamental group $\pi_1(\mathbb{H},x_0)$ is way bigger than the free subgroup $F_{\infty}=$ so we should not expect that these are the only elements of $\ker\phi$.

Let’s make sure no other surprising homotopies of loops can show up.

Let $s_{n}:\mathbb{H}_{\geq n}\to \mathbb{H}_{\geq n+1}$ be the natural retraction which collapses $C_n$ homeomorphically onto $C_{n+1}$. These maps induce retractions $c_n:\pi_1(\mathbb{H}_{\geq n},x_0)\to\pi_1(\mathbb{H}_{\geq n},x_0)$ which together form a directed system:

Notice that if $\phi_n:\pi_1(\mathbb{H}_{\geq n},x_0)\to\pi_1(\mathbb{HA},x_0)$ is the homomorphism induced by inclusion, then we have, by Corollary 2, that $\phi_{n+1}\circ c_n=\phi_n$ for each $n\geq 1$. Consequently, we get a canonical homomorphism $\Phi$ from the direct limit:

Theorem 3: $\Phi:\varinjlim_{n}\pi_1(\mathbb{H}_{\geq n},x_0)\to\pi_1(\mathbb{HA},x_0)$ is an isomorphism of groups.

Proof: Since $\phi$ is surjective (Corollary 2), so is $\Phi$. Since each $c_n$ is a retraction it suffices to show that if $[\alpha]\in\pi_1(\mathbb{H},x_0)=\pi_1(\mathbb{H}_{\geq 1},x_0)$ and $\phi([\alpha])=1$, then $c_{n-1}\circ c_{n-2}\circ\dots\circ c_1([\alpha])=1$ for some $n>1$. Since $\phi([\alpha])=1$, there is a homotopy $H:[0,1]\times[0,1]\to\mathbb{HA}$ contracting $\alpha$ to the constant loop at ${x_0}$. By compactness, the image of $H$ can intersect only finitely many hills. Apply the composition of deformation retracts from the proof of Lemma 1 to obtain an $n$ and a homotopy $G:[0,1]\times[0,1]\to\mathbb{H}_{\geq n}$ which contracts $s_{n-1}\circ s_{n-2}\circ\dots\circ s_1\circ\alpha$ to the constant loop ${x_0}$. Thus $c_{n-1}\circ c_{n-2}\circ\dots\circ c_1([\alpha])=1$ in $\pi_1(\mathbb{H}_{\geq n},x_0)$. $\square$

Identifying  $\pi_1(\mathbb{HA},x_0)$ as a direct limit illustrates a kind of “universal property.”

Corollary 4: Suppose $Y$ is a space which is first countable at it’s basepoint $y_0$. For every shrinking sequence of based loops $\beta_n\to y_0$ such that $\beta_n\simeq\beta_{n+1}$ for all $n\geq 1$, there is a unique induced homomorphism $f:\pi_1(\mathbb{HA},x_0)\to \pi_1(Y,y_0)$ such that $f(g_n)=[\beta_n]$.

Here is one last interpretation of Theorem 4: Recall that we can represent a homotopy class $[\alpha]\in \pi_1(\mathbb{H},x_0)$ as a sequence $(w_1,w_2,...)\in\varprojlim_{n}F_n$, i.e. where $w_n$ is the word in the free group $F_n$ on letters $g_1,...,g_n$ obtained by removing all appearances of the letter $g_{n+1}$ from $w_{n+1}$. Also, the number of times a given letter $g_k$ can appear in $w_n$ stabilizes as $n\to \infty$ (in other words, $(w_1,w_2,...)$ is locally eventually constant).

If $k, let $\sigma_n(w_k)=1$  and if $k\geq n$, let $\sigma_n(w_k)$ be the reduced word in $F_n$ obtained after each letter $g_1,...,g_k$ is replaced by $g_n$. Now let

$\sigma_n(w_1,w_2,...)=(\sigma_n(w_1),\sigma_n(w_2),...)=(1,1,...,1,\sigma_n(w_n),\sigma_n(w_{n+1}),...)$

where the first possible non-trivial word appears in the n-th position. It is pretty straightforward to check that $\sigma_n(w_1,w_2,...)$ is still a locally eventually constant element of $\varprojlim_{n}F_n$.

Corollary 5: If $[\alpha]\in\pi_1(\mathbb{H},x_0)$ corresponds to the sequence $(w_1,w_2,...)\in\varprojlim_{n}F_n$, then $[\alpha]\in \ker\phi$ if and only if there is an $n\geq 1$ such that $\sigma_n(w_1,w_2,...)=(1,1,...,1,\sigma_n(w_n),\sigma_n(w_{n+1}),...)$ is the trivial element of $\varprojlim_{n}F_n$.

Corollary 6: $\ker\phi$ is the conjugate closure of the free subgroup of $\pi_1(\mathbb{H})$ generated by the elements $g_{n}g_{n+1}^{-1}$, $n\geq 1$.

References.

Apparently the first appearance of the harmonic archipelago (where it was also named) was in the following unpublished note:

[1] W.A. Bogley, A.J. Sieradski, Universal Path Spaces, Unpublished preprint. http://people.oregonstate.edu/~bogleyw/research/ups.pdf

Some unpublished notes on understanding the fundamental group of the harmonic archipelago:

[2] P. Fabel, The fundamental group of the harmonic archipelago, preprint. http://arxiv.org/abs/math/0501426.