The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 2)

This is Part 2 of a guest post by Sam Corson, who is a Heibronn Fellow at the University of Bristol. It will be helpful to read Part 1 first.

We will furthermore overload the notation used for word concatenation and apply it to totally ordered sets. For example, we will write \Lambda_0 \equiv \Lambda_1 for totally ordered sets \Lambda_0 and \Lambda_1 provided there exists an order isomorphism between them. The concatenation of totally ordered sets \Lambda_0 and \Lambda_1 is denoted \Lambda_0\Lambda_1 and is the disjoint union \Lambda_0 \sqcup \Lambda_1 under the obvious order. If \{\Lambda_{\lambda}\}_{\lambda \in \Lambda} is a collection of totally ordered sets, indexed by a totally ordered set \Lambda, then the concatenation \prod_{\lambda \in \Lambda} \Lambda_{\lambda} is the totally ordered set which is the disjoint union \bigsqcup_{\lambda \in \Lambda} \Lambda_{\lambda} under the natural order.

Given a word W \in \textbf{Red}_H and t \in \overline{W} there exists a maximal interval I \subseteq \overline{W} such that t \in I and W \upharpoonright I is H-pure. We can therefore write \overline{W} \equiv \prod_{\lambda \in \Lambda} I_{\lambda} where each I_{\lambda} is a maximal nonempty interval in \overline{W} for which W \upharpoonright I_{\lambda} is H-pure, and the totally ordered set \Lambda is unique up to \equiv. This gives rise to a decomposition of the word W as W \equiv \prod_{\lambda \in \Lambda} W_{\lambda} where W_{\lambda} \equiv W \upharpoonright I_{\lambda}. This decomposition we call the p-decomposition and write W \equiv_p \prod_{\lambda \in \Lambda} W_{\lambda} to express that the p-decomposition of W is the concatenation \prod_{\lambda \in \Lambda} W_{\lambda}. We let \text{p-index}(W) denote the totally ordered set \Lambda, which is well-defined up to \equiv. Of course, \text{p-index}(E) \equiv \emptyset. As an example, one can consider the word

W\equiv a_0^1a_1^{-2}a_2^{3}a_3^{-4}a_4^{-5}\cdots

where one has \text{p-index}(W) \equiv \mathbb{N} and W \equiv_p \prod_{n \in \mathbb{N}} W_n where W_n \equiv a_n^{(n+1)(-1)^n}.

A word W' is a p-chunk of the word W \equiv_p \prod_{\lambda \in \text{p-index}(W)} W_{\lambda} provided there exists some interval J \subseteq \text{p-index}(W) such that W' \equiv \prod_{\lambda \in J} W_{\lambda} (we may indeed write W' \equiv_p \prod_{\lambda \in J} W_{\lambda}). Thus a p-chunk of a word is a subword which respects the p-decomposition. Given an interval J \subseteq \text{p-index}(W) we write W\upharpoonright_p J for the p-chunk \prod_{\lambda \in J} W_{\lambda}. An H-pure p-chunk of a word W \equiv_p \prod_{\lambda \in \text{p-index}(W)}W_{\lambda} will clearly either be E or will be one of the W_{\lambda}. Given W \in \textbf{Red}_H we will let \text{p-chunk}(W) denote the set of all p-chunks of W. Note that this set might be uncountable (if \text{p-index}(W) \equiv \mathbb{Q} then consider the p-chunks associated with the Dedekind cuts). Given a subset \{W_x\}_{x \in X} \subseteq \textbf{Red}_H we let \text{P-fine}(\{W_x\}_{x \in X}) denote the generated subgroup \langle \bigcup_{x \in X} \text{p-chunk}(W_x) \rangle \leq \textbf{Red}_H. One can prove that this subgroup is closed under taking p-chunks of elements.

For U \in \textbf{Red}_T there similarly exists a decomposition of \overline{U} into maximal nonempty intervals \overline{U} \equiv \prod_{\lambda \in \Lambda} I_{\lambda} where U \upharpoonright I_{\lambda} is T-pure. Thus we obtain a decomposition, which we again call the p-decomposition and use the same notation \equiv_p and again write \text{p-index}(U) to identify the p-index.

If W \in \textbf{Red}_H with W \equiv_p \prod_{\lambda \in \text{p-index}(W)}W_{\lambda} and F \subseteq \text{p-index}(W) is finite then we have [[W]] = [[\textbf{Red}(\prod_{\lambda \in \text{p-index}(W) \setminus F}W_{\lambda})]], and similarly for a word U \in \textbf{Red}_T. This fact that the [[\cdot]] class is preserved under deleting finitely many elements of the p-index and then reducing provides the motivation for the essential idea in constructing the isomorphism. We recount the idea now.

Given a totally ordered set \Lambda we will say that a subset \Lambda' \subseteq \Lambda is close in \Lambda if for every infinite interval I \subseteq \Lambda we have I \cap \Lambda' \neq \emptyset. For example, if \Lambda is finite then every subset of \Lambda, including \emptyset, is close in \Lambda. If \Lambda \equiv \mathbb{N} then any infinite subset \Lambda' \subseteq \Lambda is close in \Lambda. If \Lambda \equiv \mathbb{Q} then every dense subset of \Lambda will be close in \Lambda. If \Lambda_0 and \Lambda_1 are totally ordered sets and \Lambda_0' \subseteq \Lambda_0 and \Lambda_1' \subseteq \Lambda_1 are each close then we call an order isomorphism \iota: \Lambda_0' \rightarrow \Lambda_1' a close order isomorphism (abbreviated coi) from \Lambda_0 to \Lambda_1.

A close order isomorphism \iota from \Lambda_0 to \Lambda_1 defines a correspondence (not necessarily one-to-one) between the intervals of \Lambda_0 and those in \Lambda_1: given interval I \subseteq \Lambda_0 we obtain interval \varpropto(I, \iota) := \bigcup_{\lambda < \lambda'; \lambda, \lambda' \in \iota(I)} [\lambda, \lambda'] \subseteq \Lambda_1. For an interval I \subseteq \Lambda_1 we define \varpropto(I, \iota^{-1}) similarly. Many nice properties hold for this correspondence; for instance, we have \varpropto(\varpropto(I, \iota), \iota^{-1}) is a subinterval of I and there exist (possibly empty) finite subintervals I_0, I_2 \subseteq I such that I \equiv I_0 \varpropto(\varpropto(I, \iota), \iota^{-1}) I_2.

If W \in \textbf{Red}_H and U \in \textbf{Red}_T and \iota is a coi from \text{p-index}(W) to \text{p-index}(U) then we write \text{coi}(W, \iota, U) and call such a triple a coi triple. A collection \{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} of coi triples is coherent if for any choice of x_0, x_1 \in X, intervals I_0 \subseteq \text{p-index}(W_{x_0}) and I_1 \subseteq \text{p-index}(W_{x_1}), and i \in \{-1, 1\} such that

W_{x_0} \upharpoonright_p I_0 \equiv (W_{x_1} \upharpoonright_p I_1)^i

we get that

[[U_{x_0} \upharpoonright_p \varpropto(I_0, \iota_{x_0})]] = [[(U_{x_1} \upharpoonright_p \varpropto(I_1, \iota_{x_1}))^i]]

and also for any choice of x_2, x_3 \in X, intervals I_2 \subseteq \text{p-index}(U_{x_2}) and I_3 \subseteq \text{p-index}(U_{x_3}), and j \in \{-1, 1\} such that

U_{x_2} \upharpoonright_p I_2 \equiv (U_{x_3}\upharpoonright_p I_3)^j

we get that

[[W_{x_2} \upharpoonright_p \varpropto(I_2, \iota_{x_2}^{-1})]] = [[(W_{x_3} \upharpoonright_p \varpropto(I_3, \iota_{x_3}^{-1}))^j]].

Note that it is possible that a collection of coi triples which has only one element can fail to be coherent, since the above definition allows that x_0 = x_1 and I_0 \neq I_1. Thus, great care must be taken in producing a coherent collection of coi triples. For a coherent example, one can take

W \equiv a_0^1a_1^{-2}a_2^{3}a_3^{-4}a_4^{-5}\cdots


U \equiv t_{0, 0}t_{1, 1}t_{0, 2}t_{1, 3}t_{0, 4}\cdots

where we clearly have \text{p-index}(W) \equiv \mathbb{N} \equiv \text{p-index}(U) and let \iota be the unique order isomorphism between \text{p-index}(W) and \text{p-index}(U). Of course, one needs to check that the appropriate conditions hold in order to conclude that \{\text{coi}(W, \iota, U)\} is coherent. As a hint in this easy example, one can see that if I_0, I_1 \subseteq \text{p-index}(W) are intervals and W \upharpoonright_p I_0 \equiv W\upharpoonright_p I_1 and either of I_0 or I_1 is finite, then both are finite and [[U \upharpoonright_p \varpropto(I_0, \iota)]] = [[E]] = [[U \upharpoonright_p \varpropto(I_1, \iota)]]. If at least one of I_0 or I_1 is infinite then both must be infinite and in fact W \upharpoonright_p I_0 \equiv W\upharpoonright_p I_1 implies I_0 = I_1 and the desired equality once again holds. One cannot have W\upharpoonright_p I_0 \equiv (W\upharpoonright_p I_1)^{-1} if either of I_0 or I_1 is infinite (by considering the order type), and if either of I_0 or I_1 is finite we again see that [[U \upharpoonright_p \varpropto(I_0, \iota)]] = [[E]] = [[(U \upharpoonright_p \varpropto(I_1, \iota))^{-1}]]. The check for intervals in \text{p-index}(U) is comparable.

One can imagine that the check for coherence becomes annoying when the collection has more elements and words become more complicated. The payoff for producing such a collection, however, is hinted at in the following:

Proposition. Let \beth_T: \textbf{Red}_T \rightarrow \textbf{Red}_T/\langle\langle \textbf{Pure}_T\rangle\rangle and \beth_H: \textbf{Red}_H \rightarrow \textbf{Red}_H/\langle\langle \textbf{Pure}_H \rangle\rangle denote the respective quotient maps. A coherent collection \{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} of coi triples induces an isomorphism

\Phi: \beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \rightarrow \beth_T(\text{P-fine}(\{U_x\}_{x \in X})).

If we can produce a coherent collection of coi triples \{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} which is plentiful enough that

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


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

then we have obtained an isomorphism.

Although the proposition is very intuitive, the proof is technical. One first shows that from a coherent collection one obtains well-defined homomorphisms \text{P-fine}(\{W_x\}_{x\in X}) \rightarrow \textbf{Red}_T/\langle\langle \textbf{Pure}_T\rangle\rangle and \text{P-fine}(\{U_x\}_{x\in X}) \rightarrow \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle. That these natural maps (extending W\upharpoonright_p I \mapsto [[U\upharpoonright_p \varpropto (I, \iota)]]) are well-defined requires some effort, since a word W \in \text{P-fine}(\{W_x\}_{x\in X}) might decompose in many distinct ways as a finitary concatenation of elements in \bigcup_{x\in X} \text{p-chunk}(\{W_x^{\pm 1}\}_{x \in X}), and coherence is essential to the argument. Once these homomorphisms are in hand, it is easier to see that they descend to homomorphisms \beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \rightarrow \textbf{Red}_T/\langle\langle \textbf{Pure}_T\rangle\rangle and \beth_T(\text{P-fine}(\{U_x\}_{x \in X})) \rightarrow \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle and that these homomorphisms are inverse to each other.

Posted in Fundamental group, Griffiths twin cone, harmonic archipelago, Infinite Group Theory, Infinite products, Order Theory | Tagged , , | Leave a comment

The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 1)

This is a guest post by Sam Corson, who is a Heibronn Fellow at the University of Bristol.

This first post will provide background on the infinite word combinatorics which are used in the description of the fundamental group of each of the spaces in question. The Griffiths twin cone space \textbf{TC} first appeared in print in H. B. Griffith’s paper The fundamental group of two spaces with a common point, Quart. J. Math. Oxford 2 (1954), 175-190. The first appearance of the harmonic archipelago \textbf{HA} seems to be in the work of W. A. Bogley and A. J. Sieradski Weighted combinatorial group theory and wild metric complexes, Groups-Korea ’98 (Pusan), de Gruyter, Berlin, 2000, 53-80. For more background into these two spaces, you can consult some of Brazas’ old blog posts: harmonic archipelago and Griffiths twin cone. The conjecture that \pi_1(\textbf{TC}) \simeq \pi_1(\textbf{HA}) originated with James W. Cannon and Gregory R. Conner.

Recall that the earring space \textbf{E} is the shrinking wedge of countably infinitely many circles. More formally if p \in \mathbb{R}^2 we let C(p, r) denote the circle centered at p of radius r. The subspace \textbf{E} \subseteq \mathbb{R}^2 is given by \textbf{E} = \bigcup_{n\in \mathbb{N}} C((0, \frac{1}{n + 1}), \frac{1}{n + 1}) (this post of Brazas gives some nice background). It is well-known that the fundamental group of a wedge of circles is a free group (with each circle corresponding to a free generator), and so one would expect that the the fundamental group \pi_1(\textbf{E}) will be like a free group. While this is true, it is emphasized that \pi_1(\textbf{E}) is not a free group. This is best illustrated by the curious fact that | \pi_1(\textbf{E})| =2^{\aleph_0} and \pi_1(\textbf{E}) cannot homomorphically surject onto a free group of infinite rank (for this latter result, see Theorem 1 of G. Higman, Unrestricted free products and topological varieties, J. London Math. Soc. 27 (1952), 73-81.)

Let A = \{a_n^{\pm 1}\}_{n \in \mathbb{N}} be a countably infinite collection of symbols, which we will call letters, which is equipped with formal inverses. Usually the superscript 1 is not written. A word W is a finite-to-one function W: \overline{W} \rightarrow A where the domain \overline{W} is a totally ordered set (finite-to-one means in this case that for each n \in \mathbb{N} and \epsilon \in \{\pm 1\} the set \{t \in \overline{W} : W(t) = a_n^{\epsilon}\} is finite). It follows that the domain \overline{W} of a word W must be countable (possibly finite or empty). As an example the infinite string

a_0^{-1}a_1a_2^{-1}a_3a_4^{-1}a_5 \cdots

is a word; more formally it is the word W: \mathbb{N} \rightarrow A given by W(n) = a_n^{{(-1)}^{n + 1}} (notice that each element of the alphabet A is utilized at most once in the word). The infinite string

a_0a_1a_0a_3a_0a_5 \cdots

given by the rule n\mapsto \begin{cases}a_0 \text{ if }n\text{ is even}\\a_n\text{ if }n\text{ is odd } \end{cases} is not a word since the letter a_0 is used infinitely often. Let E denote the empty word, i.e. the word with empty domain. A word can have more exotic domain than \mathbb{N}: any finite-to-one function W: \mathbb{Q} \rightarrow A is a word. As a technical aside, we consider two words W_0 and W_1 to be equivalent, and write W_0 \equiv W_1, provided there exists an order isomorphism \iota: \overline{W_0} \rightarrow \overline{W_1} such that W_0(t) = W_1(\iota(t)) for all t\in \overline{W_0}. We form the concatenation of two words W_0 and W_1, denoted W_0W_1, by declaring that W_0W_1 has domain which is the disjoint union \overline{W_0} \sqcup \overline{W_1} with the elements in \overline{W_0} being ordered below those in \overline{W_1} and having

W_0W_1(t) = \begin{cases}W_0(t)\text{ if }t\in \overline{W_0}\\W_1(t)\text{ if }t\in \overline{W_1}\end{cases}

Analogously, given a totally ordered set \Lambda and collection of words \{W_{\lambda}\}_{\lambda \in \Lambda} indexed by \Lambda we can form a function whose domain is the disjoint union \bigsqcup_{\lambda \in \Lambda}\overline{W_{\lambda}}, ordered in the natural way, and defined by t \mapsto W_{\lambda}(t) where t \in \overline{W_{\lambda}}. This function we denote \prod_{\lambda} W_{\lambda} and it is a word provided it is finite-to-one.

A word W has an inverse, which is denoted W^{-1}, given by letting \overline{W^{-1}} be the set \overline{W} under the reverse order and W^{-1}(t) = (W(t))^{-1}. For example the inverse of the word

a_0^{-1}a_1a_2^{-1}a_3a_4^{-1}a_5 \cdots

will be the word

\cdots a_5^{-1}a_4a_3^{-1}a_2a_1^{-1}a_0

Given N \in\mathbb{N} and word W we let p_N(W) be the finite word given by the restriction W \upharpoonright\{t\in \overline{W}: W(t) \in \{a_0^{\pm 1}, \ldots, a_N^{\pm 1}\}\}. Given words W_0, W_1 we write W_0 \sim W_1 if for each N \in \mathbb{N} the words p_N(W_0) and p_N(W_1) are equal as elements in the free group. For example, the word W

a_0a_1^2a_4a_5a_6a_7a_8a_9 \cdots \cdots a_9^{-1}a_8^{-1}a_7^{-1}a_6^{-1}a_5^{-1}a_4^{-1}a_1^{-3}a_0

has p_0(W) \equiv a_0^2, p_1(W) \equiv a_0a_1^2a_1^{-3}a_0 \equiv p_2(W) \equiv p_3(W) and for N \geq 4 we get

p_N(W) \equiv a_0a_1^2a_4 \cdots a_N a_N^{-1}\cdots a_4^{-1}a_1^{-3}a_0

It is easy to see that a_0a_1^{-1}a_0 \sim W.

The group \pi_1(\textbf{E}) is isomorphic to the collection of equivalence classes over \sim. The binary operation is given by concatenation: (W_0/\sim) * (W_1/\sim) = (W_0W_1)/\sim and the \sim class of the empty word E plays the role of the group identity. Inverses in the group are predictably defined by (W/\sim)^{-1} = W^{-1}/\sim.

Analogously to a free group, there are specific words with which we prefer to work. Given a word W we say that W_1 is a subword of W if there exist words W_0, W_2 (either or both of which may be empty) such that W \equiv W_0W_1W_2. Moreover W_1 is an initial (respectively terminal) subword provided W_0 (resp. W_2) in the above writing is empty. Finally a word W is reduced if for every subword W_1 we have W_1\sim E implies W_1 \equiv E. Clearly every subword of a reduced word is itself reduced. The proof of the following result is far more difficult than that of the free group analogue:

Lemma. Every \sim class contains a reduced word which is unique up to \equiv. Letting \textbf{Red}(W) denote the reduced representative of the \sim class of word W we have for all words W_0, W_1, W_2 that \textbf{Red}(W_0 \textbf{Red}(W_1W_2)) \equiv \textbf{Red}(\textbf{Red}(W_0W_1)W_2). Moreover, given reduced words W, W' there exist words W_0, W_1, W_0', W_1' such that

(1) W \equiv W_0W_1;

(2) W' \equiv W_0'W_1';

(3) (W_1)^{-1} \equiv W_0';

(4) W_0W_1' is reduced.

For further reading on (reduced) words see Section 1 of K. Eda, Free \sigma-products and noncommutatively slender groups, J. Algebra 148 (1992), 243-263.

The nice qualities of reduced words motivate one to consider the earring group as the set \textbf{Red} of reduced words with binary operation W_0*W_1 \equiv \textbf{Red}(W_0W_1). We introduce two alphabets with formal inverses:

H = \{h_n^{\pm 1}\}_{n \in \mathbb{N}} (with H for “h”armonic archipelago); and

T = \{t_{i, n}^{\pm 1}\}_{i \in \{0, 1\}, n \in \mathbb{N}} (with T for “t”win cone).

Define words, concatenation, \sim, reduced word, etc. just as before for each of these new alphabets and let \textbf{Red}_H and \textbf{Red}_T denote the respective sets of reduced words. These two sets are each groups under the binary operation W_0*W_1 \equiv \textbf{Red}(W_0W_1) and both are isomorphic to \textbf{Red} (the isomorphism with \textbf{Red}_H is given by the word mapping which extends a_n^{\pm 1} \mapsto h_n^{\pm 1} and the isomorphism with \textbf{Red}_T is given by a_n^{\pm 1} \mapsto t_{i, m}^{\pm 1} where n = 2m + i).

A word W \in \textbf{Red}_T is (0, T)-pure if the first subscript in each of the letters appearing in W is 0, and (1, T)pure is defined analogously. A word is Tpure provided it is either (0, T)-pure or (1, T)-pure. For i \in \{0, 1\} every subword of a (i, T)-pure word is again (i, T)-pure, and the only word which is both (0, T)-pure and (1, T)-pure is E. Let \textbf{Pure}_T denote the set of T-pure words. The group \pi_1(\textbf{TC}) is isomorphic to \textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle, where the notation \langle\langle \cdot \rangle\rangle denotes the smallest normal subgroup which includes the input. This isomorphism can be seen by two applications of van Kampen’s Theorem (see e.g. Section 4 in K. Eda, H. Fischer, Cotorsion-free groups from a topological viewpoint, Topology Appl. 214 (2016), 21-34.)

A word W \in \textbf{Red}_H is (n, H)pure, where n \in \mathbb{N}, provided all subscripts of letters appearing in W are n (i.e. W is of form h_n^j where j \in \mathbb{Z}). A word is Hpure provided it is (n, H)-pure for some n \in \mathbb{N} and we let \textbf{Pure}_H denote the set of H-pure words. The group \pi_1(\textbf{HA}) is isomorphic to \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle (see Theorem 5 of G. R. Conner, W. Hojka, M. Meilstrup, Archipelago groups, Proc. Amer. Math. Soc. 143 (2015), 4973-4988.)

Now the task of establishing the isomorphism \pi_1(\textbf{TC}) \simeq \pi_1(\textbf{HA}) is reduced to producing an isomorphism between \textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle and \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle. This is not an easy task. It’s a nice exercise to check that any continuous function f: \textbf{TC} \rightarrow \textbf{HA} induces a trivial homomorphism f_*: \pi_1(\textbf{TC}) \rightarrow \pi_1(\textbf{HA}) (using the fact that \textbf{TC} is a Peano continuum and any continuous Hausdorff image of a Peano continuum is again a Peano continuum). While it is possible to give a continuous function f: \textbf{HA} \rightarrow \textbf{TC} so that f_* is surjective, it is not possible to make such an f_* injective as well. Thus, the natural (spacial) homomorphisms are ruled out. The fact that each element of \textbf{Red}_T is a (possibly infinitary) concatenation of T-pure words and similarly each element of \textbf{Red}_H is a (possibly infinitary) concatenation of H-pure words should be used in some way. A confounding issue is that |\textbf{Pure}_T| = 2^{\aleph_0} and |\textbf{Pure}_H| = \aleph_0. We will continue in Part 2.

Posted in Fundamental group, Griffiths twin cone, Group theory, harmonic archipelago, Infinite Group Theory | Tagged | 2 Comments

Homotopically Reduced Paths (Part III)

In this last post about reduced paths, I’m going to work through the details of one of the most useful results in wild topology. Writing this post helped me work out my own way of proving this result and hopefully will help bring together some ideas from the literature in a unique way in a way that is helpful to folks trying to learn about some of the techniques of the field.

Unique Reduced Path Theorem: If X is a one-dimensional Hausdorff space, then every path \alpha:[0,1]\to X is path-homotopic to a reduced path \beta:[0,1]\to X that is unique up to reparameterization. Moreover, the homotopy between \alpha and \beta has image in Im(\alpha).

Although stated a little differently, this was basically proven in [2]. The contemporary version appears in [1]. This theorem changed the game for me. Instead of using inverse limits all of the time, this allowed me to understand and prove things using order theory and unique reduced representatives.

First, it might be helpful to give more explanation of the statement itself.

  1. What does one-dimensional mean? Actually, you can use any of the three standard notions of dimension (Lebesgue covering, small inductive, or large inductive) and this theorem would still be true. Generally, Lebesgue covering dimension is the typical choice.
  2. Recall from Part I that a path \beta:[0,1]\to X being reduced means that \beta has no null-homotopic subloops, i.e. there is no [a,b]\subseteq [0,1] such that \beta|_{[a,b]} is a null-homotopic loop.
  3. What does “unique up to reparameterization” mean? A path \beta:[a,b]\to X is a reparameterization of a path \gamma:[c,d]\to X if there is an increasing homeomorphism h:[c,d]\to [a,b] such that \beta\circ h=\gamma. I use the notation \beta\equiv\gamma when this occurs. Reparameterization is an equivalence relation on the set of paths in a space. So “unique up to reparameterization” means that if \alpha and \beta are homotopic reduced paths in a 1-dim. space, then \alpha\equiv \beta.
  4. The last statement of the theorem implies that deforming \alpha to its reduced representative only requires deleting portions of \alpha. As a special case, if \alpha is a null-homotopic loop, then it contracts in its own image.

One-dimensional Peano continua

We need to dive into some terminology and “well-known” results from General Topology here. I’ve used these terms and results before but here’s a reminder:

An arc (respectively, simple closed curve) in a space X is a subspace of X that is homeomorphic to [0,1] (respectively, S^1). If h:[0,1]\to X is an embedding onto the arc A=h([0,1]), we call h(0) and h(1) the endpoints of the arc. Notice that an “arc” is technically not the same thing as an embedding [0,1]\to X. Rather an arc is the image of such an embedding with the subspace topology.

A space X is uniquely arcwise connected, if given any distinct a,b\in X, there is a unique arc A\subseteq X with endpoints a and b.

A Peano continuum is a connected, locally path-connected compact metric space. The Hahn-Mazurkiewicz Theorem says that a space X is a Peano continuum if and only if X is Hausdorff and there exists a continuous surjection [0,1]\to X.

A dendrite is a Peano continuum that has no simple closed curves. I go into some of the theory of dendrites in this old post on shape injectivity. In particular, I mention a well-known structure theorem, which says that a dendrite D is homeomorphic to an inverse limit \varprojlim_{n}T_n  of trees T_n where T_{n+1} is T_n with a single edge attached and each bonding map T_{n+1}\to T_n collapses that edge to the vertex at which it is attached. In Part II of the Shape Injectivity Post, we used this structure theorem to prove that dendrites are contractible.

To put these old posts to work here’s a theorem that I mentioned at the end of Part II.

Inverse Limit Representation Theorem [3, Theorem 1]: Every one-dimensional path-connected compact metric space X can be written as an inverse limit \varprojlim_{n}X_n of finite graphs X_n.

In fact, this is improved a bit in [4] where it is shown that it is possible to improve a given inverse system to ensure that all bonding maps X_{n+1}\to X_n map X_{n+1} surjectively and simplicially onto some finite subdivision of X_n.

Dendrite Factorization Lemma

The next lemma is at the heart of why we can do so much in wild one-dimensional spaces like the earring space, Menger Cube, and higher-dimensional constructions that start with one-dimensional spaces, e.g. the Harmonic Archipelago. The proof for a general space Y is the same as that for loops where Y=S^1 so I went ahead and wrote out the general proof.

Lemma: If X is a one-dimensional Hausdorff space, Y is a Peano continuum, and \alpha:Y\to X is a null-homotopic map based at x_0\in X, then \alpha factors through a dendrite, that is, there is a dendrite D a map \beta:Y\to D and a map g:D\to X such that \alpha=g\circ\beta.

Proof. Assuming that \alpha is non-constant, notice that Im(\alpha) is one-dimensional and Hausdorff (as a subspace of a one-dimensional Hausdorff space that admits a non-constant path). Since Im(\alpha) is a Hausdorff continuous image of a Peano continuum, it is a Peano continuum by the Hahn-Mazurkiewicz Theorem. Hence, we may assume that X=Im(\alpha) and that X is a Peano continuum.

Using the Inverse Limit Representation Theorem, write X=\varprojlim_{n}X_n  with bonding maps f_n:X_{n+1}\to X_n and finite graphs X_n. If g_n:X\to X_n are the projections, we take x_n=g_n(x_0) to be the basepoints in the graphs. Let p_n:T_n\to X_n be the universal covering map where T_n is a tree. As we did in Part II of the Shape Injectivity Post, once we choose basepoints t_n\in p_{n}^{-1}(x_n), the maps f_{n}\circ p_{n+1}:T_{n+1}\to X_n induce unique based maps \widetilde{f}_n:T_{n+1}\to T_n that give the following inverse system of covering maps.

Since \alpha is null-homotopic in X and each latex p_n is a Hurewicz fibration, the map \alpha_n=g_n\circ \alpha:Y\to X_n is null-homotopic in the graph X_n and has a unique lift to a based map \beta_n:Y\to T_n. These based lifts agree with the bonding maps \widetilde{f}_n and give the inverse system of covering maps you see below. The universal property of inverse limits gives a unique map \beta:Y \to \varprojlim_{n}T_n based at t_0=(t_1,t_2,t_3,\dots) such that. \alpha=\varprojlim_{n}p_n\circ\beta. In fact, Hurewicz fibrations are closed under inverse limits so \varprojlim_{n}p_n is also a Hurewicz fibration!

Let D=Im(\beta). Since the inverse limit of trees \varprojlim_{n}T_n is clearly Hausdorff, D is a Peano continuum. Moreover, I gave a detailed proof in Part I of the Shape Injectivity Post that an inverse limit of trees contains no simple closed curves. Since D is a Peano continuum with no simple closed curves, it must be a dendrite! Taking g to be the restriction of \varprojlim_{n}p_n to D, completes the proof. \square

We could replace S^1 with any Peano continuum in the next Corollary, but I’ll try to keep it focused.

Corollary: Every null-homotopic loop f:S^1\to X in a one-dimensional Hausdorff space X contracts in its own image.

Proof. Suppose \alpha:S^1\to X is a null-homotopic map. By the previous Lemma, we have \alpha=f\circ\beta for a map \beta:S^1\to D and map f:D\to X where D is a dendrite. Set d_0=\beta((1,0))\in D. Since D is contractible, there is a null-homotopy latex H:S^1\times [0,1] \to D with H(y,0)=\beta(y), H(y,1)=d_0. Now f\circ H is a null-homotopy of \alpha. \square

The null-homotopy H in the last proof is a “free” null-homotopy but since S^1 is well-pointed, you could just as easily construct a basepoint-preserving homotopy.

Getting back on track, we’d like to apply the general results in Part II, which says that homotopy classes of paths will have reduced representatives if our space has well-defined transfinite \Pi_1-products. So let’s make sure that happens.

Proposition: Every one-dimensional Hausdorff space has well-defined transfinite \Pi_1-products.

Proof. Let X be a one-dimensional Hausdorff space. Suppose A\subseteq [0,1] is a closed set containing \{0,1\} and \alpha,\beta:[0,1]\to X are paths such that \alpha|_{A}=\beta|_{A} and such that for every connected component J of [0,1]\backslash A, we have \alpha|_{\overline{J}}\simeq \beta|_{\overline{J}}. We must show that \alpha\simeq\beta. For each component J of [0,1]\backslash A, the loop \alpha|_{\overline{J}}\cdot\beta|_{\overline{J}}^{-} is null-homotopic and therefore (by the last Corollary) contracts by a null-homotopy in \alpha(\overline{J})\cup\beta(\overline{J}). In particular, there exists an endpoint-relative homotopy H_{I}:\overline{J}\times [0,1]\to X, where

  • H_J (s,0)=\alpha(s),
  • H_J (s,1)=\beta(s),
  • H_J (a,t)=\alpha(a)=\beta(a) and H_J(b,t)=\alpha(b)=\beta(b),
  • and Im(H_J )\subseteq \alpha(\overline{J})\cup\beta(\overline{J}).

We just need to put these all together! Define H:[0,1]^2\to X by

H(s,t)=\begin{cases} \alpha(s),  & \text{ if }s\in A, \\ H_J(s,t), & \text{ if }s\in J\text{ for a component }J\text{ of }[0,1]\backslash A \end{cases}

Notice that H|_{A\times[0,1]} is the constant homotopy at \alpha|_{A}=\beta|_{A}.

The homotopy H is the constant homotopy on A\times [0,1] (shaded in blue). For components J of [0,1]\backslash A, H is defined as H_J on \overline{J}\times [0,1] (the white boxes).

Checking continuity of H would be considered “routine” for those who make these constructions a lot so sometimes these things are skipped in the literature. But a blog is a good place to lay out the details for those who are still getting used to the proof techniques. So let’s do it!

Fix (s,t)\in [0,1]^2 and an open neighborhood U of H(s,t) in X. Since each H_J is continuous, H is continuous at (s,t) if s\in J for some component J of [0,1]\backslash A. So we may assume s\in A. Since \alpha and \beta are continuous at s, there exists \delta>0 such that \alpha([s-\delta,s+\delta])\cup \beta([s-\delta,s+\delta])\subseteq U. We now have H([([s-\delta,s+\delta]\cap A)\times [0,1])=\alpha([s-\delta,s+\delta]\cap A)\subseteq U. If J\subseteq [s-\delta,s+\delta], then H(\overline{J}\times [0,1])=Im(H_J)\subseteq \alpha(\overline{J})\cup\beta(\overline{J})\subseteq U by the definition of H. What remains is to consider components J having s as an endpoint.

If s is not an endpoint of a component of [0,1]\backslash A, then we may find c,d\in A such that s\in (c,d)\subseteq [s-\delta,s+\delta]. If s is a right endpoint of a component latex J=(a,s), then we may choose c small enough so that H((c,s]\times[0,1])=H_J((c,s]\times[0,1])\subseteq U (by the continuity of H_J). Similarly, if s is a left endpoint of a component J'=(s,b), then we may choose d small enough so that H([s,d)\times[0,1])=H_{J'}([s,d)\times[0,1])\subseteq U. In any case, we can find an open interval (c,d) of s such that H((c,d)\times[0,1])\subseteq U. \square

The main point of Part II, was to show that every path \alpha in a Hausdorff space with well-defined transfinite \Pi_1-products is path-homotopic to a reduced path. This path-homotopy was defined by “deleting” null-homotopic subloops on a maximal cancellation and therefore had image in the image of \alpha. Combining this old stuff with the above proposition, it must be that every path in a one-dimensional Hausdorff space is path-homotopic to a reduced path by a homotopy that takes place in the image of that path itself. This proves the existence portion of the Unique Reduced Path Theorem as well as the last statement about the size of the homotopy required.

Uniqueness of reduced paths in one-dimensional spaces

Dendrites have their infinitely many little fingers all over this content. We’ll need them again to finish the proof of uniqueness.

Lemma: If X is a one-dimensional Hausdorff space and \alpha,\beta:[0,1]\to X are reduced and path-homotopic to each other, then \alpha\equiv \beta, i.e. there exists an increasing homeomorphism h:[0,1]\to [0,1] such that \alpha=\beta\circ h.

Proof. By replacing X with the image of a homotopy from \alpha to \beta, we may assume that X is a Peano continuum. Now \alpha\cdot\beta^{-} is a null-homotopic loop and so by Dendrite factorization, there exists a dendrite D, a loop \gamma:[0,1]\to D, and a map f:D\to X such that \alpha\cdot\beta^{-}=f\circ \gamma. Write \gamma=\gamma_1\cdot \gamma_{2}^{-} so that f\circ\gamma_1=\alpha and f\circ\gamma_2=\beta. Let d_0=\gamma_1(0)=\gamma_2(0) and d_1=\gamma_1(1)=\gamma_2(1). Recall that dendrites are uniquely arcwise connected and so there is a unique arc A in D with endpoints d_0 and d_1.  Now \gamma_1 is a path in D from d_0 to d_1.  We check that \gamma_1 is injective and therefore a parameterization of A. If 0\leq a<b\leq 1 such that \gamma_1(a)=\gamma_1(b), then (\gamma_1)|_{[a,b]} would be a loop. Since dendrites are contractible, (\gamma_1)|_{[a,b]} is a null-homotopic loop. Then \alpha|_{[a,b]}=f\circ(\gamma_1)|_{[a,b]} must also be a null-homotopic loop.  However, this violates the assumption that \alpha is reduced. Since \beta is also reduced, applying the same argument to \gamma_2 shows that \gamma_2 also parameterizes A. Since both \gamma_1,\gamma_2:[0,1]\to A are homeomorphisms with the same orientation, we consider the increasing homeomorphism h=\gamma_{2}^{-}\circ\gamma_{1}:[0,1]\to [0,1]. Now \beta\circ h=f\circ\gamma_{2}\circ \gamma_{2}^{-}\circ\gamma_{1}=f\circ\gamma_1=\alpha, which completes the proof. \square.

What’s the Takeaway?

Imagine you’ve got a based loop \alpha:[0,1]\to X where X is the earring space or, more amazingly, the Menger cube. The homotopy class [\alpha] in the fundamental group is represented by a “tightest” loop \beta that has absolutely no homotopical redundancy. Every point \beta(t) is crucial to that homotopy class and the order in which those points are traced out in X is completely unique.

This also tells you about the operation in the fundamental groupoid \Pi_1(X) too. Suppose you’ve got two composable path-homotopy classes g,h\in \Pi_1(X). Write g=[\alpha] and h=[\beta] for reduced paths \alpha and \beta. Then the product gh is represented by the concatenation \alpha\cdot\beta. However, \alpha\cdot\beta may not be reduced. But, it’s still homotopic to some reduced path \gamma and that reduced representative is obtained by deleting null-homotopic subloops on a maximal cancellation. But wait! There’s only one possible way for this to happen because the entirety of \alpha and \beta are both reduced. A maximal cancellation of \alpha\cdot\beta can only contain one interval, which must contain the concatenation point 1/2. Hence, there exists 0<s<1 and 0<t<1 such that \alpha|_{[s,1]}\simeq \beta|_{[0,t]}^{-}. There’s more! If a path is reduced, then all of its subpaths are reduced too. Since \alpha|_{[s,1]} and \beta|_{[0,t]}^{-} are homotopic reduced paths, which means they are actually reparameterizations of each other.

In one-dimensional spaces, a concatenation \alpha\cdot\beta of reduced paths can only reduce by a cancelation of a terminal subpath of \alpha and an initial subpath of  \beta. Each of these cancelling subpaths (in red) must be a reparameterization of the reverse of the other. The resulting reduced path is shown in black and blue is the unique reduced representative of the product [\alpha][\beta].

Theorem: Suppose \alpha,\beta:[0,1]\to X are reduced paths in a one-dimensional Hausdorff space satisfying \alpha(1)=\beta(0). Then either \alpha\cdot\beta is reduced or there exists unique 0<s<1 and 0<t<1 such that \alpha|_{[s,1]}\equiv \beta|_{[t,1]}^{-} and \alpha|_{[0,s]}\cdot\beta|_{[t,1]} is a reduced path representing [\alpha][\beta].

[1] J.W. Cannon, G.R. Conner, On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006) 2648–2672.

[2] M.L. Curtis, M.K. Fort, Jr., The fundamental group of one-dimensional spaces, Proc. Amer. Math. Soc. 10 (1959) 140–148.

[3] Mardešic, S., Segal, J., \epsilonMappings onto polyhedra. Trans. Am. Math. Soc. 109, 146–164 (1963)

[4] Rogers, J.W. Jr., Inverse limits on graphs and monotone mappings. Trans. Am. Math. Soc. 176, 215–225 (1973)

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