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.

This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s