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

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.

haholes2

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}=<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{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:

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:

dirsys2

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<n, 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.

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This entry was posted in Algebraic Topology, Fundamental group, harmonic archipelago, Hawaiian earring and tagged , , , , . Bookmark the permalink.

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