Another fascinating space that receives a lot of 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 earring space 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 earring is naturally a subspace of and each hill is hollow underneath. Although the height of the hills always remains at , the projections of the hills to the xy-plane have diameters that tend to .
If you can believe it, the fundamental group , which we might refer to as the archipelago group, is even crazier than the earring group !
First, let’s get some notation down:
- Let be the n-th circle so that with basepoint .
- Let be the smaller copies of the earring space.
- Let be the open disk between and which contains the n-th hill in the archipelago.
- Let be the open disk which is the upper half of the hill .
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 open disks .
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 of the disks . Thus must have image in one of the spaces that looks like this:
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 . Therefore, 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
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 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 onto the smallest circle which bounds the hills, namely . 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 an arbitrarily small loop.
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 m-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 .
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 is obtained by attaching a sequence of 2-cells to the subspace , which deformation retracts on to . Recall that when you attach 2-cells to a space along a set of attaching loops , , then an application of the van Kampen theorem is that the resulting space has fundamental group where is the conjugate closure of . Combining this with some of the above ideas gives:
Corollary 6: is the conjugate closure of the free subgroup of generated by the elements , .
There’s also an important algebraic consequence of Theorem 3. A group is locally free if every finitely generated subgroup of is a free group. Even though the earring group 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 is locally free.
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.
Thanks to Moaaz AlQady for sending me some corrections (11/29/20).