## The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 3)

We saw in the previous post that the problem of producing our isomorphism is solved provided we can produce a sufficiently large coherent collection of coi triples. But how is this to be accomplished? For example, given a (perhaps quite complicated) word $W \in \textbf{Red}_T$, is there a way to find some $U \in \textbf{Red}_H$ and coi $\iota$ from $W$ to $U$ so that the one-element collection $\{\text{coi}(W, \iota, U)\}$ is coherent? More challengingly, if we have already defined a coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ of coi triples and we are given a word $W \in \textbf{Red}_T$ then can we find $\iota$ and $U \in \textbf{Red}_H$ so that the slightly larger collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is again coherent? And even if we can surmount this challenge for a reasonable coherent collection, might we still fail to produce a sufficiently large coherent collection on account of the fact that

$\beth_T(\text{P-fine}(\{U_x\}_{x \in X})) = \textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle$

but

$\beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \neq \textbf{Red}_H/\langle\langle\textbf{Pure}_H\rangle\rangle$.

In other words, we may have exhausted the codomain but have failed to fully extend the homomorphism to have the appropriate domain. The reverse problem could also occur: we could exhaust the codomain before producing the isomorphism.

The last two potential problems are solved by alternately considering the elements of $\textbf{Red}_H$ and $\textbf{Red}_T$, ensuring that no $[[\cdot]]$-classes of words are left out of the homomorphism by a transfinite induction. The addition of “just one more coi” can require a great deal of technical care, and we will attempt to give the big ideas behind the ability to do so. We let $\|W\| = \frac{1}{n +1}$ where $n$ is the smallest subscript on a letter in $W \in \textbf{Red}_H$ (and $\|E\| = 0$) and similarly $\|U\| = \frac{1}{n + 1}$ where $n$ is the smallest second subscript of a letter in the word $U \in \textbf{Red}_T$.

To begin our collection of coi we notice that $\{\text{coi}(W, \iota_W, E)\}_{W \in \textbf{Pure}_H}$ is coherent (each $\iota_W$ is obviously the empty function). So far our collection is countable (since $|\textbf{Pure}_H| = \aleph_0$) and more particularly of cardinality less than $2^{\aleph_0}$. Next one can prove the following (we’ll number lemmas within this post).

Lemma 1. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent and that $\epsilon > 0$.

(1) If $W \in \text{P-fine}(\{W_x\}_{x \in X})$ then we can find $U \in \textbf{Red}_T$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)$ is coherent, and $\|U\| \leq \epsilon$, and $U \not\equiv E$ provided $W \not\equiv E$.

(2) If $U \in \text{P-fine}(\{U_x\}_{x \in X})$ then we can find $W \in \textbf{Red}_H$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)$ is coherent, and $\|W\| \leq \epsilon$, and $W \not\equiv E$ provided $U \not\equiv E$.

The proof of this not-very-surprising lemma uses the fact that changing finitely many pure p-chunks of a word does not change the $[[\cdot]]$ equivalence class. Next we tackle infinitary concatenations of order type $\mathbb{N}$ (and we will need to use the crucial fact that the coi collection is not very large).

Lemma 2. Suppose that $\{ \text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine}(\{W_x\}_{x \in X})$ and we can write $\text{p-index}(W) \equiv \prod_{n \in \mathbb{N}} I_n$ with each $I_n \neq \emptyset$ and $W\upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$, then we can find $U \in \textbf{Red}_{T}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X})$ and we can write $\text{p-index}(U) \equiv \prod_{n \in \mathbb{N}} I_n$ with each $I_n \neq \emptyset$ and $U \upharpoonright_p I_n \in \text{P-fine}(\{U_x\}_{x \in X})$, then we can find $W \in \textbf{Red}_H$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

To prove part (1) we inductively use Lemma 1 (1) to produce a coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W\upharpoonright_p I_n, \iota_n, U_n)\}_{n \in \mathbb{N}}$ so that $U_n \neq E$ and $\|U_{n+1}\| \leq \frac{\|U_n\|}{2}$. Now an obvious candidate for $U$ would be $\prod_{n \in \mathbb{N}}U_n$, and this infinitary concatenation is indeed a word by the requirement $\|U_{n+1}\| \leq \frac{\|U_n\|}{2}$, but it may not be reduced. Therefore we instead will introduce a sequence of words $\{V_n\}_{n \in \omega} \subseteq \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$ with $\|V_n\| = \|U_n\|$ and $|\text{p-index}(V_n)| \in \{1, 2\}$ and so that each concatenation $U_nV_nU_{n + 1}$ is reduced. The ability to make such a selection is guaranteed be the fact that the number of pure elements in $\text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$ is at most $(|X| + |\mathbb{N}|) \cdot \aleph_0 < 2^{\aleph_0}$. The fact that

$U \equiv U_0V_0U_1V_1\cdots$

is reduced uses the fact that each subword $U_nV_nU_{n + 1}$ was reduced (and we allowed $\text{p-index}(V_n)$ to have cardinality either $1$ or $2$ depending on how the word $U_n$ ends and how the word $U_{n + 1}$ begins). The function $\iota$ will be given in the obvious way: $\iota = \bigcup_{n \in \mathbb{N}} \iota_n$ and the tedious check that

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W\upharpoonright_p I_n, \iota_n, U_n)\}_{n \in \omega} \cup \{\text{coi}(W, \iota, U)\}$

is coherent (and therefore so is $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$) uses the fact that $\{V_n\}_{n \in \omega} \subseteq \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$.

The proof for part (2) is somewhat similar: one inductively extends to a larger coherent collection

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W_n, \iota_n, U \upharpoonright_p I_n )\}_{n \in \omega}$

using Lemma 1 (2), but “buffer” words $V_n \in \textbf{Red}_H$ are selected during the induction to be of form $V_n \equiv h_{k_n}^{m_n}$. The sequences $\{k_n\}_{n \in \mathbb{N}}$ and $\{m_n\}_{n \in \mathbb{N}}$ are selected so that for each $n$ we have

$W_nV_nW_{n + 1}V_{n + 1}\cdots \notin \text{P-fine}(\{W_x\}_{x\in X} \cup \{W_n\}_{n \in \mathbb{N}})$

(this selection makes use of the fact that $|X| < 2^{\aleph_0}$).

Another difficult situation arises with concatenations which are of order type $\mathbb{Q}$.

Lemma 3. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine}(\{W_x\}_{x \in X})$ and we can write $\text{p-index}(W) \equiv \prod_{q \in \mathbb{Q}} I_q$ with each $I_q \neq \emptyset$ and $W \upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$ and $I_q$ is a maximal such interval, then we can find $U \in \textbf{Red}_{T}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x \in X})$ and we can write $\text{p-index}(U) \equiv \prod_{q \in \mathbb{Q}} I_q$ with each $I_q \neq \emptyset$ and $U \upharpoonright_p I_n \in \text{P-fine}(\{U_x\}_{x \in X})$ and $I_q$ is a maximal such interval, then we can find $W \in \textbf{Red}_{H}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

For (1) we make a list $\{W_n\}_{n\in \mathbb{N}}$ so that for each $q\in \mathbb{Q}$ exactly one of $W \upharpoonright_p I_q$ or $(W \upharpoonright_p I_q)^{-1}$ appears in the enumeration. As in Lemma 2 we produce a coherent collection

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W_n, \iota_n, U_n )\}_{n \in \omega}$

by inductively using Lemma 1 and the sequence $\{U_n\}_{n \in \mathbb{N}}$ is again selected to satisfy nice properties; for example the values $\|U_n\|$ shrink to $0$ quite rapidly. Now we select two buffer words $V_{n, a}, V_{n, b}$, this time for both the front and tail of the word $U_n$, so that $V_{n, a}U_nV_{n, b}$ is reduced and some other technical properties hold. Now define the word $U \equiv \prod_{q\in \mathbb{Q}} (V_{f(q), a}U_f(q)V_{f(q), b})^{\delta_q}$ where $W \upharpoonright_p I_q \in \{W_{f(q)}^{\pm 1}\}$ and $\delta_q \in \{\pm 1\}$ with $\delta_q = 1$ if and only if $W \upharpoonright_p I_q \equiv W_{f(q)}$. From how cleverly the buffer words were selected, one argues that $U$ is reduced, and a coi $\iota$ is produced from the collection $\{\iota_n\}_{n \in \mathbb{N}}$ in the natural way. Part (2) requires similar modifications as those used in Lemma 2 (2). In both (1) and (2) the ability to select suitably nice buffer words makes essential use of the fact that $|X| < 2^{\aleph_0}$.

Lemma 4. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine(}\{W_x\}_{x \in X})$ then there exists $U \in \textbf{Red}_T$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x \in X})$ then there exists $W \in \textbf{Red}_{H}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

The proof of part (2) is essentially that of part (1), with obvious modifications. For (1) we ask whether there exists a sequence of intervals $\{I_n\}_{n \in \mathbb{N}}$ in $\text{p-index}(W)$ where all $I_n$ have the same minimum or all have the same maximum, $I_n$ is properly included into $I_{n + 1}$, $W \upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$ for all $n$, and $W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n \notin \text{P-fine}(\{W_x\}_{x \in X})$. If such an interval does not exist then we proceed to the next paragraph. If it does exist, then we extend the coi collection so as to include $W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n$ using Lemma 2 (1) (applied to $(W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n)^{-1}$ in case all the $I_n$ have a common maximum) and we once again ask whether such a sequence exists for the new collection. We do this over and over again, taking unions of the previously defined coherent collections at limit ordinals. Using certain parameters to keep track of how many times this process iterates, we deduce that it can only be executed countably many times. Thus we move on to the next step.

If $W$ is in $\text{P-fine}(\{W_y\}_{y\in Y})$, where $\{\text{coi}(W_y, \iota_y, U_y)\}_{y \in Y}$ is the slightly enlarged coi collection, then we produce $U$ and $\iota$ using Lemma 1 (1). Else we can write $\text{p-index}(W) \equiv \prod_{\lambda \in \Lambda} I_{\lambda}$ where $\Lambda$ is infinite dense-in-itself and each interval $I_{\lambda}$ is nonempty and maximal such that $W \upharpoonright_p I_{\lambda} \in \text{P-fine}(\{W_y\}_{y\in Y})$. The set $\Lambda$ may have a maximum and/or minimum, so we let $\Lambda' \subseteq \Lambda$ be the subset excluding such elements. Then $\Lambda' \equiv \mathbb{Q}$ and we use Lemma 4 (1) to extend to a collection, say, indexed by $Y'$, so that $W\upharpoonright_p \bigcup_{\lambda \in \Lambda'} I_{\lambda} \in \text{P-fine}(\{W_y\}_{y \in Y'})$ and by applying Lemma 1 (1) perhaps once or twice (in case we have a maximum and/or minimum in $\Lambda$) we then obtain the $U$ and $\iota$ so that the collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

Now that we are armed with Lemma 4 we can define a suitable collection by induction over $2^{\aleph_0}$. Let $\prec_H$ (respectively $\prec_T$) be a well-ordering of $\textbf{Red}_H$ (resp. $\textbf{Red}_T$) such that each element has fewer than $2^{\aleph_0}$ predecessors. We already have $\{\text{coi}(W_n, \iota_n, E)\}_{m \in \mathbb{N}}$ in our collection, where $\textbf{Pure}_H = \{W_m\}_{m \in \mathbb{N}}$ is an enumeration. Recall that each ordinal $\alpha$ can be expressed uniquely as $\alpha = \gamma + n$ where $\gamma$ is either zero or a limit ordinal and $n \in \mathbb{N}$; in particular each ordinal can be considered either even or odd depending on the number $n \in \mathbb{N}$.

Suppose that we have already defined a coherent collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \zeta}$ for all $\zeta < \alpha$ where $\mathbb{N} \leq \alpha$ is an ordinal below $2^{\aleph_0}$. Then the collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \alpha}$ is coherent (this is easy to check). If $\alpha$ is even then we select $W \in \textbf{Red}_H$ such that $[[W]] \notin \beth_H(\text{Pfine}(\{W_{\beta}\}_{\beta < \alpha}))$ (such a $W$ exists using a cardinality argument) which is minimal under $\prec_H$ and by Lemma 4 (1) we choose suitable $U$ and $\iota$ to coherently extend and write $W_{\alpha} \equiv W$, $\iota_{\alpha} = \iota$, and $U_{\alpha} \equiv U$. In case $\alpha$ is odd we instead select $U \in \textbf{Red}_T$ with $[[U]] \notin \beth_T(\text{P-fine}(\{U_{\beta}\}_{\beta < \alpha}))$ which is minimal under $\prec_T$, use Lemma 4 (2) and extend accordingly. Thus we obtain a larger coherent collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \alpha + 1}$.

Perform the process on all $\alpha < 2^{\aleph_0}$ and it is clear that $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < 2^{\aleph_0}}$ is coherent and

$\beth_H(\text{P-fine}(\{W_{\beta}\}_{\beta < 2^{\aleph_0}})) = \textbf{Red}_H/\langle\langle\textbf{Pure}_H\rangle\rangle$

and similarly

$\beth_T(\text{P-fine}(\{U_{\beta}\}_{\beta < 2^{\aleph_0}})) = \textbf{Red}_T/\langle\langle\textbf{Pure}_T\rangle\rangle$.

The argument is finished.

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