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:

  1. If H\leq \pi_1(Y, y) is \Sigma_1^1 then f_*^{-1}(H)\leq \pi_1(X, x) is \Sigma_1^1.
  2. If G\leq \pi_1(X, x) is \Sigma_1^1 then f_*(G)\leq \pi_1(Y, y) is \Sigma_1^1.
  3. The subgroups 1 and \pi_1(X, x) are analytic in \pi_1(X, x).
  4. 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.
  5. Countable subgroups of \pi_1(X, x) are \Sigma_1^1.
  6. If G\leq \pi_1(X, x) is \Sigma_1^1 then so is \langle \langle G \rangle\rangle.
  7. If G\leq \pi_1(X,x) is \Sigma_1^1 then so is any conjugate of G.
  8. 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.
  9. If G, H \leq \pi_1(X, x) are \Sigma_1^1 then so is the subgroup [G, H].
  10. 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).

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

  1. \pi_1(X)
  2. \pi_1(X)/(\pi_1(X))^{(\alpha)} for any \alpha<\omega_1 (derived series)
  3. \pi_1(X)/(\pi_1(X))_n for any n\in \omega (lower central series)
  4. \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.

Theorem 2.4   The fundamental group \pi_1(E) has normal subgroups of the following types:

  1. true \Sigma_{\gamma}^0 for \omega_1>\gamma \geq 2
  2. true \Pi_{\gamma}^0 for each \omega_1 > \gamma \neq 2
  3. true \Sigma_1^1

These normal subgroups can be selected so that the quotient groups are of exponent 2.

 

 

Analyzing a fundamental group using basic descriptive set theory (Part 1)

The fundamental group is a useful tool in understanding a topological space.  In this installment I’ll present a very brief review of the concept of fundamental group, review the sup metric on the space of loops and introduce the notion of topological descriptions of subgroups of the fundamental group.

Recall that the fundamental group of a space X with distinguished point x, denoted \pi_1(X, x), is the set of all loops based at x modulo homotopy rel the endpoints.  The binary operation is defined by loop concatenation, the trivial element is the homotopy class of the constant loop at x, and inverses are given by taking the “backwards” loop.  As any loop based at x must stay within the path component of x, fundamental groups are most frequently used in path connected spaces.  For the remainder of this discussion I shall assume path connectedness of all spaces mentioned for which a fundamental group is computed, unless stated otherwise.  Given a path \rho in X from x to x' one obtains a natural isomorphism between \pi_1(X, x) and \pi_1(X, y), and thus one often drops the distinguished point x and speaks of the fundamental group \pi_1(X), which is well defined up to isomorphism.  A continuous map f: (X, x) \rightarrow (Y, y) induces a homomorphism f_*: \pi_1(X, x) \rightarrow \pi_1(Y, y) defined by letting f_*([l]) = [f\circ l].  The fundamental group is a homotopy invariant and so if spaces X and Y are homotopy equivalent there is an isomorphism \pi_1(X) \simeq \pi_1(Y) given by the functions witnessing homotopy equivalence.

Example  The fundamental group of a space consisting of one point is isomorphic to the trivial group, as is the fundamental group of any contractible space.

Example  The fundamental group of the circle S^1 is isomorphic to the group \mathbb{Z}.  Intuitively, declare one of the directions around the circle to be positive and given a loop l ask the net number of times that the loop has gone around the circle in the positive direction.

The fundamental group factors through products, meaning that \pi_1(\prod_{i\in I} X_i) \simeq \prod_{i\in I}\pi_1(X_i).  Thus one can compute the fundamental group of a two dimensional torus by noting that \pi_1(S^1 \times S^1) \simeq \pi_1(S^1) \times \pi_1(S^1) \simeq \mathbb{Z} \times \mathbb{Z}.

Many of the spaces that interest topologists are metrizable, and a metric on X induces a topology on the set of loops which underlies the fundamental group.  More precisely, let d be a metric on the space X and x \in X be a distinguished point.  Let L_x denote the set of all loops based at x.  One can topologize L_x using the sup metric which defines the distance between loops l_1, l_2 \in L_x to be \sup_{s\in [0, 1]}d(l_1(s), l_2(s)).  It is important to remember that we are not topologizing \pi_1(X, x) but rather the set L_x.

Now one can define a subgroup G \leq \pi_1(X, x) to be open (respectively closed) if the set \{l\in L_x: [l]\in G\} = \bigcup G is open (resp. closed) as a subset of L_x.  Change of base point isomorphisms preserve open-ness and closed-ness.  As is the case in topological groups, all open subgroups are also closed (the complement of an open subgroup is a union of left cosets of the group, which are also open subsets of L_x.)  If X is also assumed to be separable then L_x is also separable, and so any open subgroup of \pi_1(X, x) is of at most countable index by noticing that the set of left cosets gives a partition of L_x into nonempty open sets.  We recall that a space X is semi-locally simply connected if for every x\in X and open neighborhood U of x there exists a neighborhood V of x such that the map induced by inclusion \iota_*: \pi_1(V, x) \rightarrow \pi_1(X, x) is the trivial map (here, V might not be path connected).  Among path connected, locally path connected spaces the condition of semi-local simple connectivity is equivalent to having a universal cover.  We have the following (see On definable subgroups of the fundamental group, Proposition 2.10):

Fact  If X is metric and locally path connected the following are equivalent:

  1. The trivial subgroup of \pi_1(X) is open
  2. All subgroups of \pi_1(X) are clopen
  3. X is semilocally simply connected

This shows, among other things, that if we want to find topologically fascinating subgroups in the fundamental group of a locally path connected space we need to consider a “wild” space (i.e. one that has no universal cover).  As a matter of interest, we can say something a bit stronger (see Chapter 6 in my dissertation Subgroups and quotients of fundamental groups, Theorems 6.0.0.51 and 6.0.0.54):

Theorem  If X is metric, locally path connected and G \unlhd \pi_1(X, x) the following are equivalent:

  1. G is open
  2. There exists a covering space p: (\tilde{X}, \tilde{x}) \rightarrow (X, x) associated to G (i.e. p_*(\pi_1(\tilde{X}, \tilde{x})) = G)

To gain information about how \bigcup G sits inside of the loop space L_x, it helps to have at least one further assumption on X.  If d was assumed to be a complete metric then the sup metric on L_x is also complete.  Thus if X is a separable completely metrizable space (a Polish space) we have an accompanying separable completely metrizable topology on L_x.  Descriptive set theory is the field concerned with understanding “topological complication” of subsets of a Polish space.  Given a description of a subgroup G \leq \pi_1(X, x) using logic and/or topology it may be possible to know something about the index [\pi_1(X, x): G], and if G is normal we might gain insight into the quotient \pi_1(X, x)/G.

Welcome to Topology and Group Theory

Thank you for visiting!  This is a website mostly for posting my math.  I received a PhD in mathematics in May 2016 from Vanderbilt University under the advisement of Mark Sapir.  I do research in topology and group theory and have side interests in (descriptive, combinatorial) set theory and logic.

I got hooked on mathematics when I was exposed to Cantor’s diagonalization argument in high school.