The Harmonic Archipelago

Another fascinating space that receives a lot of 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 earring space $\mathbb{E}$ 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 $\mathbb{E}$ and then the 3rd and 4th hoops and so on. Notice that the earring $\mathbb{E}$ is naturally a subspace of $\mathbb{HA}$ and each hill is hollow underneath. Although the height of the hills always remains at $1$ , the projections of the hills to the xy-plane have diameters that tend to $0$ .

The Harmonic Archipelago Space

If you can believe it, the fundamental group $\pi_1(\mathbb{HA},x_0)$ , which we might refer to as the archipelago group, is even crazier than the earring group $\pi_1(\mathbb{E})$ !

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 $\mathbb{E}=\bigcup_{n\geq 1}C_n$ with basepoint $x_0=(0,0,0)$ .
Let $\mathbb{E}_{\geq m}=\bigcup_{n\geq m}C_n$ be the smaller copies of the earring space.
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.
Let $D_n=\{(x,y,z)\in B_n \mid \, z>\frac{1}{2}$ be the open disk which is the upper half of the hill $B_n$ .

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 open disks $D_n$ .

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 $\beta_{m}\colon S^1 \to\mathbb{E}_{\geq m}$ 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 of the disks $D_n$ . Thus $\alpha$ must have image in one of the spaces that looks like this:

Mostly chopped Harmonic Archipelago

In such a space you can deformation retract each of the infinitely many chopped hills $B_n\backslash D_n$, which are now cylinders down to height $z=0$ . Therefore, $\alpha$ is homotopic rel. basepoint to a loop in a space that looks like this:

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

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

The retraction is given by expanding each little circle in the xy-plane to the entire hole usually present in the earring space. 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 earring and a bumpy region that is homotopy equivalent to a circle.

Now deform the bumpy region $B_1\cup B_2\cup\cdots\cup B_{m-1}$ onto the smallest circle which bounds the hills, namely $C_m$ . The composition of these deformation retracts provides a homotopy of $\alpha$ to a loop in $\mathbb{E}_{\geq m}$ . 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 an arbitrarily small loop.

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

More generally, if $g_n=[\ell_n]\in\pi_1(\mathbb{E},x_0)$ , 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 $\pi_1(\mathbb{E},x_0)$ is way bigger than the free subgroup $F_{\infty}=<g_n|n\geq 1>$ 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 $s_{n}:\mathbb{E}_{\geq n}\to \mathbb{E}_{\geq n+1}$ be the natural retraction which collapses $C_n$ homeomorphically onto $C_{n+1}$ . These maps induce retractions $c_n:\pi_1(\mathbb{E}_{\geq n},x_0)\to\pi_1(\mathbb{E}_{\geq n},x_0)$ which together form a directed system:

Notice that if $\phi_n:\pi_1(\mathbb{E}_{\geq n},x_0)\to\pi_1(\mathbb{HA},x_0)$ 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: $\Phi:\varinjlim_{n}\pi_1(\mathbb{E}_{\geq n},x_0)\to\pi_1(\mathbb{HA},x_0)$ 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 $[\alpha]\in\pi_1(\mathbb{E},x_0)=\pi_1(\mathbb{E}_{\geq 1},x_0)$ 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 $G:[0,1]\times[0,1]\to\mathbb{E}_{\geq n}$ 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 $\pi_1(\mathbb{E}_{\geq n},x_0)$ . $\square$

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 $[\alpha]\in \pi_1(\mathbb{E},x_0)$ 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 $n<m$ , let $\sigma_m(w_n)=1$ and if $n\geq m$ , let $\sigma_m(w_n)$ be the reduced word in $F_n$ obtained after each letter $g_1,\dots,g_{m-1}$ is replaced by $g_m$ . Now let

$\rho_m(w_1,w_2,w_3,\dots)=(\sigma_m(w_1),\sigma_m(w_2),\sigma_m(w_3),\dots)=(1,1,1,\dots,1,\sigma_m(w_m),\sigma_m(w_{m+1}),\dots)$

where the first possible non-trivial word appears in the m-th position. It is pretty straightforward to check that $\rho_m(w_1,w_2,w_3,\dots)$ is still a locally eventually constant element of $\varprojlim_{n}F_n$ .

Corollary 5: If $[\alpha]\in\pi_1(\mathbb{E},x_0)$ 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 $m\geq 1$ such that $\rho_m(w_1,w_2,w_3,\dots)=(1,1,1,\dots,1,\sigma_m(w_m),\sigma_m(w_{m+1}),\dots)$ is the trivial element of $\varprojlim_{n}F_n$ .

I’m going to call the next statement a Corollary because technically it does follow from Corollary 5 if you’re comfortable with which elements of the earring group are trivial. However, it does follow more simply from observing that $\mathbb{HA}$ is obtained by attaching a sequence of 2-cells to the subspace $A=\{(x,y,z)\in \mathbb{HA}\mid z\leq \frac{1}{2}\}$ , which deformation retracts on to $\mathbb{E}$ . Recall that when you attach 2-cells to a space $X$ along a set of attaching loops $\alpha_j$ , $j\in J$ , then an application of the van Kampen theorem is that the resulting space has fundamental group $\pi_1(X)/N$ where $N$ is the conjugate closure of $\{[\alpha_j]\mid j\in J\}$ . Combining this with some of the above ideas gives:

Corollary 6: $\ker\phi$ is the conjugate closure of the free subgroup of $\pi_1(\mathbb{E},x_0)$ generated by the elements $g_{n}g_{n+1}^{-1}$ , $n\in\mathbb{N}$ .

There’s also an important algebraic consequence of Theorem 3. A group $G$ is locally free if every finitely generated subgroup of $G$ is a free group. Even though the earring group $\pi_1(\mathbb{E},x_0$ is not free, it is locally free. It’s a nice exercise to show that any direct limit of locally free groups is a locally free group. Hence, we have the following.

Theorem 7: The archipelago group $\pi_1(\mathbb{HA},x_0)$ is locally free.

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.

Acknowledgements.

Thanks to Moaaz AlQady for sending me some corrections (11/29/20).

The Harmonic Archipelago

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