Theorem 1 (de Smit [1]):  is not isomorphic to a free group.

The Hawaiian earring group is complicated enough that this should not be completely obvious. In this post I’ll fully hash out the details of Bart de Smits proof in [1]. Working through all these details has certainly helped me understand it better.

Theorem 1 is in contrast with the fact that the fundamental group on a countable wedge of circles (the Hawaiian earring with the CW-topology) is isomorphic to the free group on a countably infinite set of generators.

The Hawaiian earring group

In a previous post, I discussed how to begin understanding the algebraic structure of the fundamental group of the Hawaiian earring , where  is the circle of radius centered at . The basepoint is the origin , the one point where the shrinking circles meet.

We decided that the fundamental group  is an uncountable group that could be understood a subgroup of the inverse limit

where  is the free group on the generators and the map identifies the letter to the identity element (or empty word). The inverse limit consists of sequences  where is obtained from by removing all occurrences of the letter . The fundamental group corresponds to a certain subgroup of .

Definition: Suppose  is a reduced word in the letters . Reduced means that and for each . The k-weight of is

.

Essentially,  is the number of times or appears in .

If , then the sequence  (for fixed ) is non-decreasing since the projections only delete letters. We wish to consider the elements of the inverse limit where each such sequence is also bounded (and thus eventually constant). Let

be the subgroup of  consisting of sequences where every k-weight is eventually constant. These sequences are usually called locally eventually constant sequences and the group  is often called the free -product of .

In a previous post, we found the following canonical group isomorphism.

Theorem 2: .

So to study the properties of the Hawaiian earring group, we can focus our attention on the purely algebraic structure of . For instance, is certainly uncountable and also must be torsion free since it is a subgroup of the torsion free group .

Some homomorphisms out of 

First let’s exploit the inverse limit structure of  to construct some interesting self-homomorphisms of . For simplicity of notation, we’ll just write for the element where the first non-trivial term is in the k-th position. More generally, we could also identify  with it’s image under the canonical embedding of the infinite free group.

Consider a sequence  of locally eventually constant sequences. The should satisfy the follow two properties:

• for all
• For all and , we have

For instance, we could have something like:

  which corresponds to an infinite word

   which corresponds to an infinite word 

   which corresponds to an infinite word 

and so on…

What is important is that  1) the first terms of are trivial and that 2) the letters don’t show up in .

Condition 1) means precisely that the sequence must converge to the identity element when  has the inverse limit topology, i.e. as a subspace of .

Lemma 3: There is a self-homomorphism where .

Proof. Define a homomorphism on the free group by (obviously for ). Since

and

the following diagram commutes.

Consequently, we get a self-homomorphism on the inverse limit such that . We check that and then use the restriction of to prove the lemma. If then for each , we have . Now for fixed :

The inequality must be there since when we replace each letter  of with , we may have some word reduction to do. Our restriction that  for all  and means that

Since, by assumption,  , we have . Thus

showing that is locally eventually constant.

Now let’s see what happens when we map to the additive group of integers .

Lemma 4: If  is a homomorphism, then there is an such that for all .

Proof. Suppose for  and let

By Lemma 3, there is a homomorphism  such that for all . Thus for each . We might as well now replace with so from now on let’s assume that

.

Ok, now let’s define a special sequence  that will help us arrive at a contradiction. We define the n-th term  to be

\begin{cases} 1 & \text{ if }n<j\\ g_j & \text{ if }n=j \\ g_jg_{j+1}^{a_j} & \text{ if }n=j+1 \\ g_j(g_{j+1}g_{j+2}^{a_{j+1}})^{a_j} & \text{ if }n=j+2 \\ g_j(g_{j+1}(g_{j+2}g_{j+3}^{a_{j+2}})^{a_{j+1}})^{a_j} & \text{ if }n=j+3 \\ … & … \end{cases}

So the general form for is

.

Notice that removing from gives  and that for (i.e. the letter never appears more than times despite the fact that the appearances get further and further apart). Thus  is a well defined element of .

The main feature of this sequence is that so that when we apply , we get

Iterating this formula times for  gives

So if we let

   and    ,

then we see that and thus

Let’s make a few more observations about  and :

Proof: this one is pretty obvious since .

Proof: Since , we have  and inductively if , . 

Proof: Here we also use the fact that for each . Notice

,

,

, and so on.

If we recursively define the increasing sequence , , we get , which does the trick. 

Added Remark: de Smit’s construction uses , however, the inductive proofs of these three bullet points also seem to work if you only assume . So actually, I believe this slight simplification can be made.

Now, let’s finally finish the proof of Lemma 4 by showing that satisfies way to many modular equations.

In general, suppose  where . If , then we must have and if , then we must have .

Consequently, if , then we must have  for all since  and . But this contradicts the fact that . On the other hand, if , then we must have  for all but this contradicts the fact that .

References.

[1] B. de Smith, The fundamental group of the Hawaiian earring is not free, International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33-37.

Harmonic Archipelago

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.

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 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, in some sense, it’s “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 an 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 .

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.

The Hawaiian earring

This space is the first step into the world of “wild” topological spaces. This post is meant to be an introduction into how one can understand the fundamental group of this space, often just called the Hawaiian earring group.

The Hawaiian earring is usually defined as the following planar set: let be the circle of radius centered at . Now take the union with the subspace topology of .

The key feature of this space is that if is any open neighborhood of the “wild” point , then there is an such that for all . Note that the Hawaiian earring has the same underlying set as the infinite wedge   of circles, however the topology of  is finer than that of .  So there is a canonical continuous bijection  which is not a homeomorphism.

Topological facts: is a one-dimensional, compact, locally path connected, metric space.

Other ways to construct :

1. As a one-point compatification: is homeomorphic to the one-point compactification of a countable disjoint union of open intervals.
2. As a subspace of : View as a subspace of in the obvious way and give it the subspace topology. The resulting space is homeomorphic to .
3. As an inverse limit: Let . If , there is a retraction which collapses the circles , to . These maps form an inverse system . The inverse limit of this inverse system is homeormorphic to .

The really interesting things happen when you start considering loops and their homotopy classes, i.e. the fundamental group . For each consider the loop , where which traverses the n-th circle once in the counterclockwise direction (and is based at ). Let’s write for the reverse loop which goes around in the opposite direction. The loop  is definitely not homotopic to the constant loop (for a proof of this, consider the retraction collapsing all other circles to ). It seems that together, the homotopy classes should “generate”  in some way but these will not be group generators in the usual sense.

A space  is semilocally simply connected at a point if there is an open neighborhood of such that every loop in based at is homotopic to the constant loop at in (but not necessarily by a homotopy in ). This definition is very important in covering space theory. In particular, one must typically require a space to be semilocally simply connected in order to guarantee the existence of a universal covering.

Proposition: is not semilocally simply connected.

Proof. Every neighborhood of contains all but finitely many of the circles and therefore the non-trivial loops .

In fact, the Hawaiian earring does not have a universal covering (though there is a known suitable replacement) and one must attack the fundamental group using other methods.

Wild loops: The combinatorial structure of  is complicated by the fact that we can form “infinite” concatenations of loops. For instance, we can define a loop by defining to be on the interval and . This loop is continuous because of the topology of at . In this way we obtain an infinite “word”  . What is intuitive but (formally) less obvious is that is not in the free subgroup of  generated by the set .

With all these wild loops floating around, we have a pretty big group on our hands.

Proposition:  is uncountably generated.

Proof. If  were countably generated, then  would be countable. Thus it suffices to show  is uncountable. Recall that the infinite product of the cyclic group  of order is uncountable. For any sequence , we construct a loop  by defining to be constant on if and to be on if . We also define . In this way we obtain an uncountable family of homotopy class . It suffices to show whenever . Suppose . Then, without loss of generality, we have and for some .  We again call upon the retraction which collapses all circles but . If , then in .  But  is trivial  and  is non-trivial, which is a contradiction. Therefore .

One might be tempted to think that all elements of  can be understood as infinite sequences of shrinking loops like but alas this is also too much to hope for. Not only is this too much to hope for, but the combinatorial structure of  is far from free [3] since we can have “infinite” cancellations of the letters when we multiply two elements. As a first example, notice that can be thought of as the infinite word and the product is the identity element. In more geometric terms, this means we can construct a null-homotopy of by nesting “small null-homotopies” of the loops inside of each other.

You can take this one-step further by considering the following iterative construction. Start with

Now insert more trivial pairs, but make the index of the s get larger at each step so the construction is actually represented by a continuous loop.

             (cont. on next line)

At every stage and in the limit, this construction should represent the identity element of the group, however, in the “transfinite word” which is the limit, there are no straightforward cancellation pairs to be found anywhere! This is because we went on to put new letters between each pair. So the cancellations that go on in can be quite subtle. How could you possibly define a loop representing the above word? Well, if you look closely at where new pairs are inserted, you can see that it has a “Cantor set-ish” feel to it.

To describe loops representing all elements of , we call upon the middle-third Cantor set . There are countably many open intervals . We can define a loop by defining and defining on  to either be the constant loop or to be one of the loops  for some . We have one restriction to ensure that is continuous. We must ensure that for each , we only have for finitely many . Otherwise, we would admit infinite concatenations like which clearly cannot be continuous. It turns out that any element of  is represented by a loop constructed in this way. You can convince yourself of this by first noticing that for any loop , the preimage is a countable union of disjoint open intervals.

We’ve yet to really compute . We could argue exactly what I mean by “compute” here but I really mean “identify the isomorphism class as a reasonably familiar group so that we can make formal algebraic arguments about the group structure without appealing to loops.” This is done using shape theory. Before we do this, I should mention that this shape theoretic approach can fail to provide an explicit characterization of when you start considering subsets of .

Recall that one way to construct is as an inverse limit  where where is the union of the first n-circles. Note that is the free group on the generators . If we apply the fundamental group to the entire inverse system

,

we get an inverse system of free groups

where the homomorphism collapses to the identity. The inverse limit  is the first shape group of . To be fair, the shape group cannot always be constructed in this way but this is a nice way to understand the one-dimensional case.

We also have projections which collapse to the basepoint for . The induced homomorphisms clearly agree with the bonding homomorphisms   in the inverse system of free groups so we get an induced homomorphism  to the first shape group.

The inverse limit of free groups is constructed as a subgroup of . Specifically, consists of the sequences of words such that . This means we can think of elements of as sequences of words where the word (in letters ) is obtained from the word  (in letters ) by removing all instances of the letter . The homomorphism is defied as where .

The key to understanding is the following theorem which originally appeared in a paper of H.B. Griffiths [1]. Griffiths’ proof apparently had some sort of error in it; a corrected proof was given by Morgan and Morrison [2] and many have since appeared.

Theorem:  is injective.

The main idea in the proof of this theorem is to use the data of infinite “word reduction” to construct a null-homotopy of a loop such that (equivalently   for each ). It is helpful to imagine doing this for the example above where we kept inserting trivial pairs between trivial pairs and so on. The details of the full proof are somewhat non-trivial so I’ll skip it for now (but plan to come back to it later). The upshot of the theorem is that we can now understand elements of as sequences of words in .

The question then remains: what is the image of ?

Proposition:  is not surjective.

Consider the sequence of commutators . Note that as the number of appearances of grows without bound. But we can’t have a loop that corresponds to this element  since no continuous loop can traverse infinitely many times. This geometric restriction suggests which subgroup we should be looking for.

Definition: If and , let be the number of times  appears in the reduced word . We say an element is locally eventually constant if for each , the sequence  is eventually constant (as ). Let be the subgroup of locally eventually constant sequences.

If is not locally eventually constant, then we’d have some where the number of times appears is unbounded and this contradicts the continuity of . On the other hand, our method of using the Cantor set to construct loops provides a nice way to represent every locally eventually constant sequence by a continuous loop. We conclude that the locally eventually constant sequences are precisely the sequences corresponding to continuous loops.

Theorem [2]:  embeds isomorphically onto .

The group is sometimes called the free -product in infinite group theory.

References.

[1] H.B. Griffiths, Infinite products of semigroups and local connectivity, Proc. London Math. Soc. (3), 6 (1956), 455-485.

[2]  J. Morgan, I. Morrison, A van kampen theorem for weak joins, Proc. London Math. Soc. 53 (1986) 562–576.

[3] B. de Smit, The fundamental group of the Hawaiian earring is not free, Internat. J. Algebra Comput. 2 (1) (1992) 33–37.

Another great reference on the Hawaiian earring group is

[4] J.W. Cannon, G.R. Conner, The combinatorial structure of the Hawaiian earring group, Topology Appl. 106 (2000) 225-271.

Heavier machinery is required to study the structure of these spaces which, locally, are more complicated than CW-complexes. One of the most traditional approaches to this problem is shape theory. A common technique in mathematics is to approximate complicated objects by simpler ones. For instance, approximating functions in calculus by Taylor polynomials of increasing degree. This is basically the approach of shape theory: approximate complicated spaces by simpler ones, in particular polyhedra (spaces built out of lines, triangles, tetrahedra, etc…). Borsuk, the inventor of shape theory, first used ANR’s to study the topology of compact metric spaces. A more modern approach pioneered by Segal and Mardesic [1] is categorical in nature and makes use of inverse systems of polyhedra.

Thought shape theory helps a great deal in our understanding of complicated spaces, it has its limits (pun intended); later on, we’ll run into some spaces where shape theory breaks down.

So where do we start?

Let’s look at the Cech expansion of a space . This is supposed to let us approximate spaces by simpler ones. We’ll start with an open cover of our space . Even when you forget about the points in each set , the open cover still gives a vague picture of . For instance, take the following cover of the unit circle as a subspace of the plane.

Now forget about the points in the space.

The shape left still closely resembles that of a circle. We can even recover the circle from this data: replace each open set with a point. If two open sets intersect, draw a line segment between the two corresponding points.

We get a polyhedron homeomorphic to the circle. It isn’t usually true that we will get back the original space; this only happens in very special cases.

The nerve of an open cover

Definition: An abstract simplicial complex is a set and a set consisting of finite subsets of such that if and , then . A vertex or 0-simplex is a singleton and an n-simplex is a set containing elements. The n-skeleton of is the set

Now if is an open cover of , we construct an abstract simplicial complex called the nerve of . An element of  is a finite set such that . The geometric realization is a geometric complex obtained by pasting simplices together using  as instructions.

Definition: The geometric realization of an abstract simplicial complex with vertex set is the topological space defined as a subset of the product of functions . In particular, is the set of functions such that

1.  (in particular all but finitely many are zero)

2.

Give   the weak (or induced) topology so that is open in iff is open in for all finite sets . is topologized with the subspace topology of .

Sometimes we’ll write the simplex in spanned by vertices in as .

Back to the nerve:

While is defined as the geometric realization of the nerve, it is a bit more intuitive to think of it in the following way.

Here is the cover:

An open cover

0 skeleton – A vertex of  is a set

0-skeleton

1 skeleton – If , then place an edge (1-simplex) between and .

1-skeleton

2 skeleton – If , then there are three edges joining each pair of the vertices . Place a triangle (or 2-simplex) so that the edges of the triangle match up with these three edges. In the picture, fill in the each empty triangle with a triangle.

3 skeleton – If , attach a tetrahedron to fill in the boundary that exists from the four triangles.

tetrahedron

Input the tetrahedron into the place where it obviously goes (on the right).

In our example, we would stop here and leave it embedded in but, in general, you would continue to add higher dimensional simplices and give the resulting geometric simplicial complex the weak topology. In addition, the space might not be compact and open covers would typically contain infinitely many sets.

Refinements:

The nerve is supposed to be an “approximation” of the original space . What if it is a bad approximation? Well…take a “closer look” at by covering with smaller open sets.

Definition: An open cover is a refinement of another cover if for each there is a such that .

If  refines , then is “larger” than  since there are more sets in . It makes sense to think of  as being a better approximation to since if we collapse the appropriate simplices of , we get back something homotopy equivalent to . This is captures in the next proposition.

Proposition: If   is a refinement of another cover , there is a there is a simplicial map . The map induced on geometric realizations is unique up to homotopy.

proof. First define on vertices (i.e. elements of ). If  and for , define . If , then clearly so we define on the 1-simplex spanned by and to the 1-simplex spanned by and . Any map defined in this way is called a projection.

The same goes for higher simplices; if , then  and we send the simplex to . This gives a well-defined simplicial map on the nerves.

Though it seems like there is a lot of freedom in defining a projection, it is not too hard to show that any two projections induce contiguous maps on geometric realizations. But contiguous maps of simplicial complexes are homotopic, proving the proposition.

Note that if is a refinement of  and  is a refinement of , the composition  is a canonical map. Thus if is another projection, then there is a homotopy . Now if we let denote the homotopy class of , we have strict equality . Therefore, since open covers of form a directed set , we have an inverse system of homotopy classes of nerves of covers.

Definition: For a paracompact Hausdorff space , the Cech expansion of is the inverse system in the homotopy category of polyhedra.

Of course, even if is not paracompact Hausdorff, you still get an inverse system; the problem with non-paracompact spaces is that it is much harder to relate to the inverse system without “enough” partitions of unity…more on this later.

References:

[1] S. Mardsic and J. Segal, Shape theory, North-Holland Publishing Company, 1982.

If I’m thinking about wild topology, I’m probably thinking about topological spaces (geometric-type objects) which have points where non-trivial structure is visible no matter how far you zoom in. Though I’m not so interested in generating fractals, these pretty little things do reach the point of being “complex on an infinitely small scale.”

Since we are talking about homotopy, we are allowed to bend and stretch the objects (but not cut or puncture them) and consider it to be the same object. The object of homotopy theory is to classify spaces up to homotopy equivalence: Two spaces and are homotopy equivalent if there are maps and such that the compositions and .

When spaces look “simple” on a local level (like manifolds or CW-complexes), this classification boils down to using discrete gadgets (like homotopy groups) that look for algebraic structure hiding within. Things become much more complicated when you begin to allow local pathologies; if your space has important structure as you zoom in you’d better make sure it’s still there when you zoom in on your newly deformed object!

The tip of the iceberg

Even in the usual Euclidean plane we learn about in middle and high school it is all too easy to construct such objects. One of the simplest examples is the Hawaiian earring which can be constructed by taking a sequence of circles that get smaller and smaller and all converge up on a single point of intersection.

We can realize the Hawaiian earring as a compact planar set: If

,

then

so the point is the intersection .

Note the Hawaiian earring is distinct from the countable wedge of circles only at a single point. In fact, there is a continuous bijection

which is a local homeomorphism at every point except the intersection point. This one point makes a world of difference. For one, the Hawaiian earring is compact and  is not. For another, the two are not homotopy equivalence; we’ll investigate the details later.

Despite the simple appearance, has an enormous amount of combinatorial group theory and topological algebra hiding within…and this is pretty much the simplest example of a space where covering space theory falls apart.

The rest

More generally, you could ask: What does it take to classify connected subspaces of the plane up to homotopy equivalence? What if we add conditions like locally path connectedness or compactness? What if we extend to subspaces of ?

One of the blossoming approaches to these questions makes use of a classical tool in topology: the fundamental group . This is likely to be the topic of upcoming posts if I can manage to find the time to write them…