Analyzing a fundamental group using basic descriptive set theory (Part 2)

In this installment I’ll give a brief review of some fundamental concepts of descriptive set theory. For further information I recommend A Course on Borel Sets by Srivastava, or Classical Descriptive Set Theory by Kechris. Some results about subgroups are then outlined.

Descriptive set theory is the study of subsets of a nice space that have a topological description. The nice space is generally Polish (separable, completely metrizable.) One of the initial motivating questions in descriptive set theory was the continuum hypothesis- the question of whether there exists an uncountable set of cardinality less than $2^{\aleph_0}$ . Cantor was able to show that a closed subset of the real line cannot be a counterexample to the continuum hypothesis (this follows from the Cantor-Bendixson theorem). What the proof really shows is that no Polish space can be a counterexample to the continuum hypothesis (topology hadn’t come into its own at this point, and so Polish spaces as such weren’t yet a subject of study). To describe further results will require more definitions.

Recall that a $\sigma$ -algebra on a set $X$ is a subset of the powerset of $X$ which contains $X$ and is closed under countable unions and under complementation. By considering the intersection of all $\sigma$ -algebras containing a collection of subsets of $X$ , we see that for any collection of subsets of $X$ there is a minimal $\sigma$ -algebra containing that collection. If $X$ is a topological space we let $\mathcal{B}(X)$ denote the class of Borel sets– the smallest $\sigma$ -algebra containing the open subsets of $X$ . Thus open sets, closed sets, countable unions of closed sets, etc., are Borel sets. When $X$ is a metric space, the open sets are a countable union of closed sets and the closed sets are a countable intersection of open sets. In this case the Borel sets can be arranged into a nice hierarchy of complexity classes. Let $\Sigma_1^0(X)$ denote the collection of open sets (the topology) of $X$ , $\Pi_1^0(X)$ denote the collection of closed subsets and $\Delta_1^0(X)$ denote the set of clopen subsets. For $1< \alpha< \omega_1$ we let $\Sigma_{\alpha}^0(X)$ denote the collection of countable unions of sets in $\bigcup_{\beta<\alpha} \Pi_{\beta}^0(X)$ , $\Pi_{\alpha}^0(X)$ denote the collection of countable intersections of sets in $\bigcup_{\beta<\alpha} \Sigma_{\beta}^0(X)$ , and $\Delta_{\alpha}^0(X) = \Sigma_{\alpha}^0(X) \cap \Pi_{\alpha}(X)$ . A transfinite induction shows that $\bigcup_{\alpha< \omega_1}\Sigma_{\alpha}^0(X) = \bigcup_{\alpha<\omega_1}\Pi_{\alpha}^0(X) = \bigcup_{\alpha<\omega_1}\Delta_{\alpha}^0(X) \subseteq \mathcal{B}(X)$ , and $\mathcal{B}(X) \subseteq \bigcup_{\alpha<\omega_1} \Sigma_{\alpha}^0(X)$ is seen by proving that $\bigcup_{\alpha<\omega_1} \Sigma_{\alpha}^0(X)$ is a $\sigma$ -algebra. These complexity classes arrange nicely into an array, with inclusions going left:

$\begin{matrix} & & \Sigma_1^0(Z) & & & \Sigma_2^0(Z) & & \cdots & & \Sigma_{\alpha}^0(Z) \\ \\ \\ \Delta_1^0(Z) & & & &\Delta_2^0(Z) & \cdots & & \Delta_{\alpha}^0(Z)\\ \\ \\ & & \Pi_1^0(Z) & & & \Pi_2^0(Z) & & \cdots & & \Pi_{\alpha}^0(Z) \end{matrix}$

For example $\Pi_{\alpha}^0(Z)\subseteq \Delta_{\alpha+1}^0(Z)$ .

Polish spaces are either finite (and therefore discrete), countable, or of cardinality $2^{\aleph_0}$ . In case a Polish space is countable we know that every subset is a $\Sigma_2^0$ as a countable union of singletons, and thus the Borel hierarchy stabilizes and is not especially interesting. In case a Polish space is uncountable we get a non-stabilizing hierarchy, with strict containments going left (i.e. $\Pi_{\alpha}^0\subsetneq \Delta_{\alpha+1}^0$ .) In a Polish space, all Borel sets are either countable (including finite) or contain a homeomorph of the Cantor set. For example, no Borel subset of $\mathbb{R}$ can violate the continuum hypothesis.

While Borel sets are closed under many set-theoretic operations, they are not closed under continuous images. There exist Polish spaces $X, Y$ , a closed subset $Z \subseteq X$ , and a continuous function $f: X \rightarrow Y$ such that $f(Z)$ is not Borel in $Y$ . Such a set is called analytic, and the class of analytic sets is denoted $\Sigma_1^1$ . The class consisting of complements of analytic sets is denoted $\Pi_1^1$ , and we denote $\Delta_1^1 = \Sigma_1^1\cap\Pi_1^1$ . Since the identity map is continuous we see that every Borel set in a Polish space is $\Delta_1^1$ . That all $\Delta_1^1$ sets are Borel is a theorem of Souslin (Sur une definition des ensembles B sans nombres transfinis, C. R. Acad. Sciences, Paris, 164 (1917), 88-91.) The notation suggests that we will define a hierarchy, in this case we shall only use subscripts in the natural numbers and not in all of $\omega_1$ . If $\Delta_n^1, \Sigma_n^1, \Pi_n^1$ have been defined for all $n<m$ we let $\Sigma_m^1$ denote the class of continuous images of sets in $\Pi_{m-1}^1$ , $\Pi_m^1$ is the class of complements of sets in $\Sigma_m^1$ and $\Delta_m^1 = \Sigma_m^1 \cap \Pi_m^1$ . A similar arrangement, with strict inclusions in case the Polish space is uncountable, holds for these so called projective pointclasses:

$\begin{matrix} & & \Sigma_1^1 & & & & \Sigma_2^1 & & \cdots\\ \\ \\ \Delta_1^1 & & & &\Delta_2^1 & & \cdots\\ \\ \\ & & \Pi_1^1 & & & & \Pi_2^1 & & \cdots \end{matrix}$

As some points of interest, $\Sigma_1^1$ sets are either countable or include a homeomorph of the Cantor set, and so are never a counterexample to the continuum hypothesis. The $\Sigma_1^1$ , and therefore also the $\Pi_1^1$ , subsets of $\mathbb{R}$ are always Lebesgue measurable. The $\Pi_1^1$ sets may be countable, of cardinality $\aleph_1$ , or of cardinality $2^{\aleph_0}$ and there exist models of ZFC modelling $\aleph_1< 2^{\aleph_0}$ in which there exists a $\Pi_1^1$ set of cardinality $\aleph_1$ . Other strange phenomena happen further up the hierarchy in various models of ZFC.

Let’s move on to some information on subgroups. We shall assume $X$ is a path connected Polish space for what follows and sometimes add additional hypotheses when stating theorems. Recall that a subgroup $G\leq \pi_1(X, x)$ is open, closed, Borel, etc. provided the set of loops $\bigcup G$ is an open, closed, Borel, etc. subset of the loop space $L_x$ under the $\sup$ metric. We first notice that many of the very natural subgroups of $\pi_1(X, x)$ are analytic (we state a specialization of Theorem 3.10 in On definable subgroups of the fundamental group):

Proposition 2.1 Let $f:(X, x) \rightarrow (Y, y)$ be a continuous function between Polish spaces. The following hold:

If $H\leq \pi_1(Y, y)$ is $\Sigma_1^1$ then $f_*^{-1}(H)\leq \pi_1(X, x)$ is $\Sigma_1^1$ .
If $G\leq \pi_1(X, x)$ is $\Sigma_1^1$ then $f_*(G)\leq \pi_1(Y, y)$ is $\Sigma_1^1$ .
The subgroups $1$ and $\pi_1(X, x)$ are analytic in $\pi_1(X, x)$ .
If $G_n \leq \pi_1(X, x)$ are $\Sigma_1^1$ then so are $\displaystyle\bigcap_{n\in \omega} G_n$ and $\displaystyle\langle \bigcup_{n\in \omega} G_n \rangle$ .
Countable subgroups of $\pi_1(X, x)$ are $\Sigma_1^1$ .
If $G\leq \pi_1(X, x)$ is $\Sigma_1^1$ then so is $\langle \langle G \rangle\rangle$ .
If $G\leq \pi_1(X,x)$ is $\Sigma_1^1$ then so is any conjugate of $G$ .
If $w(x_0, \ldots, x_k)$ is a reduced word in the free group $F(x_0, \ldots, x_k)$ and the groups $G_0, \ldots, G_k \leq \pi_1(X, x)$ are $\Sigma_1^1$ then so is the subgroup $\langle \{w(g_0, g_1, \ldots, g_k)\}_{g_i \in G_i}\rangle$ .
If $G, H \leq \pi_1(X, x)$ are $\Sigma_1^1$ then so is the subgroup $[G, H]$ .
If $G\leq \pi_1(X, x)$ is $\Sigma_1^1$ then each countable index term of the derived series $G^{(\alpha)}$ and each term of the lower central series $G_n$ is $\Sigma_1^1$ .

Here are some more theorems (also from On definable subgroups of the fundamental group):

Proposition 2.2 Let $G \leq \pi_1(X,x)$ .

If $G$ is $\Pi_1^1$ then the index $[\pi_1(X,x):G]$ is either $\leq \aleph_0$ or $2^{\aleph_0}$ .
If $G$ is $\Sigma_1^1$ then the index $[\pi_1(X, x):G]$ is either $\leq \aleph_1$ or $2^{\aleph_0}$ .

Theorem 2.3 Suppose $X$ is locally path connected. The following groups are of cardinality $2^{\aleph_0}$ or $\leq \aleph_0$ , and in case $X$ is compact they are of cardinality $2^{\aleph_0}$ or are finitely generated:

$\pi_1(X)$
$\pi_1(X)/(\pi_1(X))^{(\alpha)}$ for any $\alpha<\omega_1$ (derived series)
$\pi_1(X)/(\pi_1(X))_n$ for any $n\in \omega$ (lower central series)
$\pi_1(X)/N$ where $N$ is the normal subgroup generated by squares of elements, cubes of elements, or $n$ -th powers of elements, or is simply any $\Sigma_1^1$ normal subgroup

In case $X$ is compact then countability of the fundamental group is equivalent to being finitely presented.

Some general remarks about these results are in order. Proposition 2.1 is verified using direct computation of topological complexity. Proposition 2.2 follows from a theorem of Burgess and a theorem of Silver. Theorem 2.3 is proved using Baire category. I give an example in Definable subgroups of a compact path connected subspace $F$ of $\mathbb{R}^2$ for which there exists a model of ZFC which has $\aleph_1 <2^{\aleph_0}$ and normal subgroup $G \unlhd \pi_1(F, x)$ which is $\Sigma_1^1$ and for which $\pi_1(F, x)/G$ is of cardinality $\aleph_1$ . Thus in an appropriate model an intermediate cardinality $\aleph_1<2^{\aleph_0}$ can obtain and the hypothesis “locally path connected” in Theorem 2.3 (4 ) cannot be relaxed. The quotients in Theorem 2.3 are countable precisely when the normal subgroup is open, or in other words, when the subgroup has a covering space.

I’ll mention one more result (Theorem C from On subgroups of first homology). Recall that the Hawaiian earring is the subspace $E=\bigcup_{n\in \omega} C((0, \frac{1}{n+2}),\frac{1}{n+2})\subseteq \mathbb{R}^2$ , with $C(p,r)$ the circle centered at point $p$ of radius $r$ . We’ll say a set is true $\Sigma_{\alpha}^0$ (respectively $\Pi_{\alpha}^0$ ) provided it is not $\Pi_{\alpha}^0$ (resp. $\Sigma_{\alpha}^0$ ) and that a set is true $\Sigma_1^1$ provided it is not Borel.