The Harmonic Archipelago

Posted on May 1, 2014 by Jeremy Brazas

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

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).

Posted in Algebraic Topology, earring space, Fundamental group, harmonic archipelago | Tagged , , , , , | 11 Comments

The fundamental group of the earring space

Posted on November 23, 2013 by Jeremy Brazas

Here is one of my favorite spaces: The earring space, i.e. the “shrinking wedge of circles.”

The Hawaiian earring

The earring space

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, which I will just refer to as the earring group. I’m no longer using “Hawaiian” on purpose. In fact, this is not just a regular group. If you can imagine groups (like the complex numbers) with geometrically relevant, natural infinite product operations, and is a “free” object of this type.

The earring space \mathbb{E} is usually defined as the following planar set: let C_n=\left\{(x,y)\in \mathbb{R}^2\Big|\left(x-\frac{1}{n}\right)^2+y^2=\frac{1}{n^2}\right\} be the circle of radius \frac{1}{n} centered at \left(\frac{1}{n},0\right). Now take the union \mathbb{E}=\bigcup_{n\geq 1}C_n with the subspace topology of \mathbb{R}^2.

he

The key feature of this space is that if {U} is any open neighborhood of the “wild” point x_0=(0,0), then there is an {N}\geq {1} such that {C_n}\subset {U} for all {n}\geq {N}. Note that the earring space has the same underlying set as the infinite wedge \bigvee_{n=1}^{\infty} S^1  of circles, however the topology of \bigvee_{n=1}^{\infty} S^1 is finer than that of \mathbb{E}.  So there is a canonical continuous bijection \bigvee_{n=1}^{\infty} S^1\to \mathbb{E}, which is not a homeomorphism.

Topological facts: \mathbb{E} is a connected, one-dimensional, locally path connected, compact metric space.

Other ways to construct \mathbb{E}:

  1. As a one-point compactification: {\mathbb{E}} is homeomorphic to the one-point compactification of a countable disjoint union \coprod_{n=1}^{\infty}(0,1) of open intervals.
  2. As a subspace of \prod_{n=1}^{\infty} S^1: View \bigvee_{n=1}^{\infty} S^1 as a subspace of \prod_{n=1}^{\infty} S^1 in the obvious way and give it the subspace topology. The resulting space is homeomorphic to \mathbb{E}.
  3. As an inverse limit: Let X_n=\bigcup_{k=1}^{n}C_k. If n>m, there is a retraction r_{n,m}:X_n\to X_m which collapses the circles C_k, m<k\leq n to x_0. These maps form an inverse system \cdots \to X_{n+1}\to X_{n} \to X_{n-1}\to \cdots \to X_1. The inverse limit \varprojlim_{n}X_n of this inverse system is homeormorphic to \mathbb{E}.

The really interesting things happen when you start considering loops and their homotopy classes, i.e. the fundamental group \pi_1(\mathbb{E}). For each n\geq 1 consider the loop \ell_n:[0,1]\to \mathbb{E}, where \ell_n(t)=\left(\frac{1}{n}\cos(2\pi t-\pi)+\frac{1}{n},\frac{1}{n}\sin(2\pi t-\pi)\right) which traverses the n-th circle {C_n} once in the counterclockwise direction (and is based at x_0). Let’s write \ell_{n}^{-1} for the reverse loop \ell_{n}^{-1}(t)=\ell_{n}(1-t) which goes around in the opposite direction. The loop \ell_{n} is definitely not homotopic to the constant loop (for a proof of this, consider the retraction q_n:\mathbb{E}\to C_n collapsing all other circles to x_0). It seems that together, the homotopy classes g_n=[\ell_n] should “generate” \pi_1(\mathbb{E}) in some way but these will not be group generators in the usual sense.

A space X  is semilocally simply connected at a point {x}\in {X} if there is an open neighborhood {U} of {x} such that every loop in {U} based at {x} is homotopic to the constant loop at {x} in {X} (but not necessarily by a homotopy in {U}). 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: {\mathbb{E}} is not semilocally simply connected.

Proof. Every neighborhood of the wild point x_0 contains all but finitely many of the circles C_n and therefore the non-trivial loops \ell_n. {\square}

In fact, the earring space does not have a universal covering (though there is a known suitable replacement) and one must attack the fundamental group \pi_1(\mathbb{E}) using other methods.

Wild loops: The combinatorial structure of \pi_1(\mathbb{E}) is complicated by the fact that we can form “infinite” concatenations of loops. For instance, we can define a loop {\alpha}{:}{[0,1]}\to \mathbb{E} by defining \alpha to be \ell_n on the interval \left[\frac{n-1}{n},\frac{n}{n+1}\right] and \alpha(1)=x_0. This loop is continuous because of the topology of \mathbb{E} at {x_0}. In this way we obtain an infinite “word”  g_1 g_2 g_3 ...\in\pi_1(\mathbb{E}). What is intuitive but (formally) less obvious is that [\alpha]=g_1 g_2 g_3 ... is not in the free subgroup of {\pi}_{1}(\mathbb{E}) generated by the set \{g_1,g_2,g_3,...\}.

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

Proposition: \pi_1(\mathbb{E}) is uncountably generated.

Proof. If \pi_1(\mathbb{E}) were countably generated, then \pi_1(\mathbb{E}) would be countable. Thus it suffices to show \pi_1(\mathbb{E}) is uncountable. Recall that the infinite product \prod_{n=1}^{\infty}\mathbb{Z}/2\mathbb{Z} of the cyclic group {\mathbb{Z}}/{2}\mathbb{Z}=\{0,1\} of order {2} is uncountable. For any sequence s=(a_n)\in\prod_{n=1}^{\infty}\mathbb{Z}/2\mathbb{Z}, we construct a loop {\alpha_s}:[0,1]\to\mathbb{E} by defining {\alpha_s} to be constant on \left[\frac{n-1}{n},\frac{n}{n+1}\right] if {a_n}={0} and {\alpha_s} to be \ell_n on \left[\frac{n-1}{n},\frac{n}{n+1}\right] if a_n=1. We also define \alpha_s(1)=x_0. In this way we obtain an uncountable family of homotopy class [\alpha_s]\in {\pi_1}(\mathbb{E}). It suffices to show {[\alpha_s]}\neq{[\alpha_t]} whenever {s}\neq{t}. Suppose {s}={(a_n)}\neq {(b_n)}={t}. Then, without loss of generality, we have {a_N}={1} and {b_N}={0} for some {N}.  We again call upon the retraction {q_N}:\mathbb{E}\to C_N which collapses all circles but {C_N}. If {[\alpha_s]}={[\alpha_t]}, then {[q_N\circ\alpha_s]}={[q_N\circ\alpha_t]} in {\pi_1}{(C_N)}={\mathbb{Z}}.  But {[q_N\circ\alpha_t]}={0}\in{\mathbb{Z}} is trivial  and {[q_N\circ\alpha_s]}={[q_N\circ\ell_N]}={1}\in{\mathbb{Z}} is non-trivial, which is a contradiction. Therefore {[\alpha_s]}\neq{[\alpha_t]}. {\square}

Uncountability and the Specker group: Another way to show that \pi_1(\mathbb{E}) is uncountable is to show \pi_1(\mathbb{E}) surjects onto the uncountable infinite product \prod_{n=1}^{\infty}\mathbb{Z} which is usually called the Specker group and happens to be the first Cech homology group of \mathbb{E}. Each map q_n:\mathbb{E}\to C_n collapsing all but the n-th circles to the basepoint induces a retraction of groups (q_n)_{\ast}:\pi_1(\mathbb{E})\to\pi_1(C_n)=\mathbb{Z} which essentially picks out the “winding number” around the n-th circle. Together, these winding numbers uniquely induce a homomorphism \epsilon:\pi_1(\mathbb{E})\to\prod_{n=1}^{\infty}\mathbb{Z} given by \epsilon([\alpha])=((q_1)_{\ast}([\alpha]),(q_2)_{\ast}([\alpha]),...). To check surjectivity, convince yourself that \epsilon sends the homotopy class of a loop \alpha defined as (\ell_{n})^{a_n} on the interval \left[\frac{n-1}{n},\frac{n}{n+1}\right] and \alpha(1)=x_0 to the generic sequence (a_n)\in\prod_{n=1}^{\infty}\mathbb{Z}.

One might be tempted to think that all elements of \pi_1(\mathbb{E}) can be realized as products of infinite sequences of shrinking loops like [\alpha]=g_1g_2g_3... but alas, this is also too much to hope for. Not only is this too much to hope for, but the combinatorial structure of \pi_1(\mathbb{E}) is far from free [3] since we can have “infinite” cancellations of the letters {g_n} when we multiply two elements. See this post for a proof of non-freeness. As a first example, notice that [\alpha]^{-1} can be thought of as the infinite word {...}{g}_{3}^{-1} {g_{2}^{-1}}{g_{1}^{-1}} and the product g_1 g_2 g_3......g_{3}^{-1} g_{2}^{-1} g_{1}^{-1}=[\alpha][\alpha]^{-1}=e is the identity element. In more geometric terms, this means we can construct a null-homotopy of {\alpha}{\cdot}{\alpha}^{-1} by nesting “small null-homotopies” of the loops \ell_n\cdot \ell_{n}^{-1} inside of each other.

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

g_{1}g_{1}^{-1}

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

g_{1}(g_{2}g_{2}^{-1})(g_{3}g_{3}^{-1})g_{1}^{-1}

g_{1}(g_{2}(g_{4}g_{4}^{-1})(g_{5}g_{5}^{-1})g_{2}^{-1})(g_{3}(g_{6}g_{6}^{-1})(g_{7}g_{7}^{-1})g_{3}^{-1})g_{1}^{-1}

g_{1}(g_{2}(g_{4}(g_{8}g_{8}^{-1})(g_{9}g_{9}^{-1})g_{4}^{-1})(g_{5}(g_{10}g_{10}^{-1})(g_{11}g_{11}^{-1})g_{5}^{-1})g_{2}^{-1})              (cont. on next line)

(g_{3}(g_{6}(g_{12}g_{12}^{-1})(g_{13}g_{13}^{-1})g_{6}^{-1})(g_{7}(g_{14}g_{14}^{-1})(g_{15}g_{15}^{-1})g_{7}^{-1})g_{3}^{-1})g_{1}^{-1}

\cdots

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 g_{k}g_{k}^{-1} to be found anywhere! This is because we went on to put new letters in the middle of every such pair. So the cancellations that go on in \pi_1(\mathbb{E}) 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 {\pi}_1(\mathbb{E}), we call upon the middle-third Cantor set {C}\subset {[0,1]}. There are countably many open intervals {[0,1]}{\backslash}{C}=\bigcup_{k\geq 1}(a_k,b_k). We can define a loop {\alpha}{:}{[0,1]}\to \mathbb{E} by defining \alpha(C)=x_0 and defining {\alpha} on [a_k,b_k] to either be the constant loop or to be one of the loops \ell_{n_k}^{\pm 1} for some {n_k}\geq {1}. We have one restriction to ensure that {\alpha} is continuous. We must ensure that for each {n}\geq {1}, we only have {n_k}={n} for finitely many {k}. This means for fixed n, the loops \ell_{n} and \ell_{n}^{-1} can only be used finitely many times. Otherwise, we would admit infinite concatenations like {\ell_1}\cdot {\ell_1}\cdot {\ell_1}{\cdots} which clearly cannot be continuous. It turns out that any element of {\pi_1}(\mathbb{E}) is represented by a loop constructed in this way. You can convince yourself of this by first noticing that for any loop \alpha, the preimage \alpha^{-1}(\mathbb{E}\backslash \{x_0\}) is a countable union of disjoint open intervals.

We’ve yet to really compute \pi_1(\mathbb{E}). 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 continue, I should mention that this shape theoretic approach can fail to provide an explicit characterization of \pi_1 when you start considering subsets of \mathbb{R}^3.

Recall that one way to construct {\mathbb{E}} is as an inverse limit \varprojlim_{n}X_n where where {X_n}=\bigvee_{i=1}^{n}S^1 is the union of the first n-circles. Note that {\pi}_{1}(X_n)=F_n is the free group on the generators {g_1},{g_2},{...},{g_n}. If we apply the fundamental group {\pi_1} to the entire inverse system

\cdots \to X_{n+1}\to X_{n} \to X_{n-1}\to \cdots \to X_1,

we get an inverse system of free groups

\cdots \to F_{n+1}\to F_{n} \to F_{n-1}\to \cdots \to F_1

where the homomorphism h_n:F_{n+1}\to F_n collapses {g}_{n+1} to the identity. The inverse limit \check{\pi}_1(\mathbb{E})= \varprojlim_{n}F_n is the first shape group of \mathbb{E}. 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 {p_n}{\colon}\mathbb{E}=\varprojlim_{n}X_n\to {X_n} which collapse {C_k} to the basepoint for {k}>{n}. The induced homomorphisms (p_n)_{\ast}:\pi_1(\mathbb{E})\to \pi_1(X_n)=F_n clearly agree with the bonding homomorphisms h_n:F_{n}\to F_{n-1}  in the inverse system of free groups so we get an induced homomorphism {\Psi }:{\pi_1}\left(\mathbb{E}\right)\to\varprojlim_{n}{F_n} to the first shape group.

helimit

The inverse limit of free groups {\varprojlim_n}F_n is constructed as a subgroup of \prod_{n=1}^{\infty}F_n. Specifically, {\varprojlim_n}F_n consists of the sequences (w_n) of words w_n\in F_n such that h_n(w_n)=w_{n-1}. This means we can think of elements of {\varprojlim_n}F_n as sequences of words (w_n) where the word w_{n-1} (in letters g_1,..,g_{n-1}) is obtained from the word w_n (in letters g_1,..,g_{n}) by removing all instances of the letter g_n. The homomorphism \Psi is defied as \Psi([\alpha])=(w_n) where w_n=[p_n\circ\alpha]\in F_n.

The key to understanding \pi_1(\mathbb{E}) 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.

Shape Injectivity Theorem: {\Psi }:{\pi_1}\left(\mathbb{E}\right)\to\varprojlim_{n}{F_n} is injective.

One way to prove this theorem is to use the data of infinite “word reduction” to construct a null-homotopy of a loop \alpha such that \Psi([\alpha])=1 (equivalently  [p_n\circ \alpha]=1\in \pi_1(X_n)=F_n for each n\geq 1). It is helpful to imagine doing this for the example above where we kept inserting trivial pairs g_kg_{k}^{-1} between trivial pairs and so on. The details of a 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 {\pi}_{1}\left(\mathbb{E}\right) as sequences of words in {\varprojlim_{n}}{F_n}.

The question then remains: what is the image of \Psi?

Proposition: \Psi : \pi_{1}\left(\mathbb{E}\right) \to \varprojlim_{n}F_n is not surjective.

Consider the sequence (w_n)\in\varprojlim_{n}F_n of commutators w_n = (g_{1} g_{2} g_{1}^{-1} g_{2}^{-1} )(g_{1} g_{3} g_{1}^{-1} g_{3}^{-1} )(g_{1} g_{4} g_{1}^{-1} g_{4}^{-1}) \cdots (g_{1} g_{n} g_{1}^{-1} g_{n}^{-1}). Note that as {n}{\to}{\infty} the number of appearances of {g_1} grows without bound. But we can’t have a loop {\alpha} :{[0,1]}\to\mathbb{E} that corresponds to this element  since no continuous loop can traverse {C_1} infinitely many times. This geometric restriction suggests which subgroup we should be looking for.

Definition: If {1}\leq {k}\leq {n} and {w\in F_n}, let {\#}_{k}(w) be the number of times g_{k}^{\pm 1} appears in the reduced word w. We say an element (w_n)\in\varprojlim_{n}F_n is locally eventually constant if for each {k} \geq {1}, the sequence \#_{k}(w_n)  is eventually constant (as n\to\infty). Let \#_{\mathbb{N}}\mathbb{Z}\leq\varprojlim_{n}F_n be the subgroup of locally eventually constant sequences.

If \Psi ([\alpha])=(w_n) is not locally eventually constant, then we’d have some k where the number of times g_{k}^{\pm 1} appears is unbounded and this contradicts the continuity of \alpha. 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]: {\Psi} : {\pi}_{1}\left(\mathbb{E}\right) \to {\varprojlim_{n}}{F_n} embeds {\pi}_{1}\left(\mathbb{E}\right) isomorphically onto \#_{\mathbb{N}}\mathbb{Z}.

The group {\pi_1}\left(\mathbb{E}\right){\cong}\#_{\mathbb{N}}\mathbb{Z} is sometimes called the free {\sigma}-product of \mathbb{Z} in infinite group theory.

Summary

Let’s sum up this combinatorial description of \pi_1(\mathbb{E}): The fundamental group of the earring space \pi_1(\mathbb{E}) is isomorphic to the group of sequences (w_n) where

  1. w_n\in F_n is a reduced word in the free group on letters g_1,...,g_n,
  2. removing the letter g_n from w_n gives the word w_{n-1},
  3. for each k\geq 1, the number of times the letter g_k appears in w_n stabilizes at n\to\infty (i.e. the sequence \#_k(w_1),\#_k(w_2),... is eventually constant for each k).

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 earring group is

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

Posted in Algebraic Topology, earring group, earring space, Fundamental group, Homotopy theory | Tagged , , , , , , | 21 Comments

The Cech Expansion: nerves of open covers

Posted on October 30, 2012 by Jeremy Brazas

The Whitehead theorem in homotopy theory basically says that to fully understand the homotopy type of a CW-complex one only needs to know about the homotopy groups (really, the weak homotopy type). It is very easy to produce spaces to which Whitehead’s theorem doesn’t apply. For instance, consider an infinite wedge X=\bigvee_{n\in\mathbb{N}}S^1 of circles and the one-dimensional space Y pictured below:

The space Y: a wedge of converging circles

There is clearly a continuous bijection X\to Y, which induces an isomorphism on all homotopy groups. However, X and Y are not homotopy equivalent because the topological fundamental group of X, which is a homotopy invariant, is discrete while the topological fundamental group of Y is not discrete.

Heavier machinery is required to differentiate spaces which are more complicated on a local level than CW-complexes. One of the most traditional approaches to this problem is Shape Theory, initially developed by Carl Borsuk and refined and expanded in the 1970’s by many, including the authors of [1].

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 line segments, triangles, tetrahedra, etc.). Borsuk, the inventor of Shape Theory, first used ANR’s to study the topology of compact metric spaces. The more modern approach pioneered by Segal and Mardesic [1] is categorical in nature and makes heavy use of inverse systems of polyhedra.

Though Shape Theory helps a great deal in our understanding of complicated spaces; however, it has its limits (pun intended). For instance Shape Theory tells you absolutely nothing about the Griffiths Twin Cone because it is “shape equivalent” to a point!

So where do we start?

Let’s look at the Cech Expansion of a space X. This is supposed to let us approximate spaces by simpler ones. We’ll start with an open cover \mathscr{U} of our space X. Even when you forget about the points in each set U\in\mathscr{U}, the open cover still gives a vague picture of X. For instance, take the following cover of the unit circle S^1 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 and for very fine open covers.

The nerve of an open cover

Definition: An abstract simplicial complex is a set S and a set K consisting of finite subsets of S such that if A\in K and B\subset A, then B\in K. A vertex or 0-simplex is a singleton \{s\}\in K and an n-simplex is a set \{s_1,...,s_{n+1}\}\in K containing n+1 elements. The n-skeleton of K is the set K_n=\{A\in K||A|=n+1\}

Now if \mathscr{U} is an open cover of X, we construct an abstract simplicial complex N(\mathscr{U}) called the nerve of \mathscr{U}. An element of \mathscr{U} is a finite set \{U_1,U_2,...,U_n\}\subset \mathscr{U} such that \bigcap_{i=1}^{n}U_i\neq \emptyset . The geometric realization |N(\mathscr{U})| is a geometric complex obtained by pasting simplices together using N(\mathscr{U}) as instructions.

Definition: The geometric realization of an abstract simplicial complex K with vertex set K_0 is the topological space |K| defined as a subset of the product P=[0,1]^{K_0} of functions f:K_0\to [0,1]. In particular, |K| is the set of functions f\in P such that

  1. \{v\in K_0|f(v)>0\}\in K  (in particular all but finitely many f(v) are zero),
  2. \sum_{v} f(v)=1.

Give  P the weak (or induced) topology so that U is open in P if and only if U\cap [0,1]^{F} is open in [0,1]^{F} for all finite sets F\subset K_0. |K| is topologized with the subspace topology of P.

It is common to write the simplex in |K| spanned by vertices s_1,...,s_n in |K| as [s_1,...,s_n]=\{f\in |K||f(s_i)>0\text{ for some }1\leq i\leq n\}.

While |N(\mathscr{U})| 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 N(\mathscr{U}) is a set U\in \mathscr{U}

0-skeleton

1 skeleton – If U_1\cap U_2 \neq \emptyset , then place an edge (1-simplex) between U_1 and U_2.

1-skeleton

2 skeleton – If U_1\cap U_2\cap U_3\neq \emptyset, then there are three edges joining each pair of the vertices U_1,U_2,U_3. 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 U_1\cap U_2\cap U_3\cap U_4\neq \emptyset, 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 stop here and leave it embedded in \mathbb{R}^3. 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.

Here’s an observation we can go ahead and make. The order of an open cover \mathscr{U} of X (if it exists) is the smallest number n\in\mathbb{N} such that every point x\in X lies in n elements of \mathscr{U}. Hence, if the order of \mathscr{U} is n, then at most n distinct elements can intersect and N(\mathscr{U}) can only have simplices of dimension \leq n.

Proposition: If the order of \mathscr{U} is n, then dim(N(\mathscr{U}))=n.

Refinements:

The nerve |N(\mathscr{U})| is supposed to be an “approximation” of the original space X. What if it is a bad approximation? You should take a “closer look” at X by using a finer open cover of X, i.e. one consisting of smaller open sets.

Definition: An open cover \mathscr{V} is a refinement of another cover \mathscr{U} if for each V\in \mathscr{V} there is a U\in \mathscr{U} such that V\subseteq U.

If \mathscr{V} refines \mathscr{U}, then |N(\mathscr{V})| is “larger” than |N(\mathscr{U})| since there are more sets in \mathscr{V}. It makes sense to think of |N(\mathscr{V})| as being a better approximation to X since if we collapse the appropriate simplices of |N(\mathscr{V})|, we get back something homotopy equivalent to |N(\mathscr{U})|. This is captured in the next proposition.

Proposition: If  \mathscr{V} is a refinement of another cover \mathscr{U}, there is a there is an onto simplicial map p_{\mathscr{V}\mathscr{U}}:N(\mathscr{V})\to N(\mathscr{U}). The map |p_{\mathscr{V}\mathscr{U}}|:|N(\mathscr{V})|\to |N(\mathscr{U})| induced on geometric realizations is unique up to homotopy.

Proof. First define p_{\mathscr{V}\mathscr{U}} on vertices (i.e. elements of \mathscr{V}). If V\in \mathscr{V} and V\subseteq U_{V} for U_{V}\in\mathscr{U}, define p_{\mathscr{V}\mathscr{U}}(V)=U_{V}. If V\cap V'\neq \emptyset, then clearly U_{V}\cap U_{V'}\neq \emptyset so we define p_{\mathscr{V}\mathscr{U}} on the 1-simplex [V,V'] spanned by V and V' to the 1-simplex \left[U_{V},U_{V'}\right] spanned by U_{V} and U_{V'}. Any map defined in this way is called a projection.

The same goes for higher simplices; if \bigcap_{i=1}^{n}V_i\neq \emptyset, then \bigcap_{i=1}^{n}U_{V_{i}}\neq \emptyset and we send the simplex [V_1,\dots,V_n] to [U_{V_{1}},...,U_{V_{n}}]. This gives a well-defined simplicial map on the nerves.

Though it seems like there is a lot of freedom in defining a projection, the choice is independent of homotopy. By construction, any two projections N(\mathscr{V})\to N(\mathscr{U}) induce contiguous maps on geometric realizations. But contiguous maps of simplicial complexes are homotopic, proving the proposition. \square

Note that if \mathscr{W} is a refinement of \mathscr{V} and \mathscr{V} is a refinement of \mathscr{U}, the composition p_{\mathscr{V}\mathscr{U}} \circ p_{\mathscr{W}\mathscr{V}} is a canonical map. Thus if p_{\mathscr{W}\mathscr{U}} is another projection, then there is a homotopy \left|p_{\mathscr{W}\mathscr{U}}\right| \simeq |p_{\mathscr{V}\mathscr{U}}| \circ |p_{\mathscr{W}\mathscr{V}}|. Now if we let [p_{\mathscr{V}\mathscr{U}}] denote the homotopy class of |p_{\mathscr{V}\mathscr{U}}|, we have strict equality [p_{\mathscr{W}\mathscr{U}}] = [p_{\mathscr{V}\mathscr{U}}] \circ [p_{\mathscr{W}\mathscr{V}}]. Therefore, since open covers of X form a directed set \mathcal{O}(X), we have an inverse system \left(|N(\mathscr{U})|,[p_{\mathscr{V}\mathscr{U}}],\mathcal{O}(X)\right) of homotopy classes of nerves of covers.

Definition: For a paracompact Hausdorff space X, the Cech Expansion of X is the inverse system \left(|N(\mathscr{U})|,[p_{\mathscr{V}\mathscr{U}}],\mathcal{O}(X)\right) in the homotopy category of polyhedra.

Of course, even if X is not paracompact Hausdorff, you still get an inverse system; the problem with non-paracompact spaces is that it is much harder to relate X to the inverse system without “enough” partitions of unity to build maps X\to |N(\mathscr{U})|.

References:

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

Posted in Cech expansion, Shape theory, Simplicial complexes | Tagged , , , , , | 4 Comments