Another wild space that has been receiving a lot of recent attention is the so-called Harmonic Archipelago which is the following subspace of .
You can describe the construction like this: Start by drawing the usual Hawaiian earring onto a solid disk 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 and then the 3rd and 4th hoops and so on. Notice that the Hawaiian earring is naturally a subspace of and each hill is hollow underneath. Also, the diameters of the hills tend to .
If you can believe it, the fundamental group of this space is even crazier than the fundamental group of the Hawaiian earring !
First, let’s get some notation down:
- Let be the n-th circle so that with basepoint .
- Let be the smaller copies of the Hawaiian earring.
- Let be the open disk between and which contains the n-th hill in the archipelago.
Now some observations:
1) is path-connected and locally path connected.
2) 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 which is not included. This means that if is compact and is continuous, then the image can only hit finitely many of the hills.
3) If is the loop that traverses once counterclockwise in the xy-plane, then can be deformed over finitely many hills (but not infinitely many!). So the homotopy classes in are the same for all but yet are non-trivial since deforming over every hill should violate continuity.
Lemma 1: Every based loop is homotopic to a loop for each .
Proof: As mentioned above, the image of any based loop can intersect at most finitely many hills . Thus 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
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 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.
Now deform the bumpy region onto the smallest circle . The composition of these deformation retracts provides a homotopy of to a loop in . Since we could choose to be arbitrarily large, the lemma is proven.
Lemma 1 basically says that every based loop is homotopic to arbitrarily small loops.
Corollary 2: The homomorphism on fundamental groups induced by inclusion is surjective and for all .
More generally, if , then whenever . However, the fundamental group is way bigger than the free subgroup so we should not expect that these are the only elements of .
Let’s make sure no other surprising homotopies of loops can show up.
Let be the natural retraction which collapses homeomorphically onto . These maps induce retractions which together form a directed system:
Notice that if is the homomorphism induced by inclusion, then we have, by Corollary 2, that for each . Consequently, we get a canonical homomorphism from the direct limit:
Theorem 3: is an isomorphism of groups.
Proof: Since is surjective (Corollary 2), so is . Since each is a retraction it suffices to show that if and , then for some . Since , there is a homotopy contracting to the constant loop at . By compactness, the image of can intersect only finitely many hills. Apply the composition of deformation retracts from the proof of Lemma 1 to obtain an and a homotopy which contracts to the constant loop . Thus in .
Identifying as a direct limit illustrates a kind of “universal property.”
Corollary 4: Suppose is a space which is first countable at it’s basepoint . For every shrinking sequence of based loops such that for all , there is a unique induced homomorphism such that .
Here is one last interpretation of Theorem 4: Recall that we can represent a homotopy class as a sequence , i.e. where is the word in the free group on letters obtained by removing all appearances of the letter from . Also, the number of times a given letter can appear in stabilizes as (in other words, is locally eventually constant).
If , let and if , let be the reduced word in obtained after each letter is replaced by . Now let
where the first possible non-trivial word appears in the n-th position. It is pretty straightforward to check that is still a locally eventually constant element of .
Corollary 5: If corresponds to the sequence , then if and only if there is an such that is the trivial element of .
Corollary 6: is the conjugate closure of the free subgroup of generated by the elements , .
Apparently the first appearance of the harmonic archipelago (where it was also named) was in the following unpublished note:
 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:
 P. Fabel, The fundamental group of the harmonic archipelago, preprint. http://arxiv.org/abs/math/0501426.