Analyzing a fundamental group using basic descriptive set theory (Part 3)

In this third and final installment I’ll provide some more theorems regarding the fundamental group and first homology that can be derived using descriptive set theory. We have already seen in Part 2 that there are quick and easy dichotomies that can be drawn using either selection theorems from logic or a diagonalization using the Baire category theorem. By working a little harder we get the following results, which can be interpreted as compactness theorems (see On definable subgroups of the fundamental group Theorem B, and On subgroups of first homology Theorem 4, respectively):

Theorem 3.1 If $X$ is a Peano continuum there does not exist a strictly increasing sequence of analytic normal subgroups $\{G_n\}_{n\in \omega}$ of $\pi_1(X)$ such that $\bigcup_{n\in \omega} G_n = \pi_1(X)$ .

Theorem 3.2 If $X$ is a Peano continuum with $H_1(X)$ of cardinality $< 2^{\aleph_0}$ there exists $\epsilon>0$ and $N\in \omega$ such that any loop of diameter $<\epsilon$ is of commutator length $\leq N$ .

Theorem 3.1 implies that for $X$ a Peano continuum, if there is no finite subset $S\subseteq \pi_1(X)$ such that the normal closure $\langle\langle S\rangle\rangle$ is $\pi_1(X)$ , then for any countable $S \subseteq \pi_1(X)$ we again have $\langle\langle S\rangle\rangle \neq \pi_1(X)$ . There exist Peano continua with uncountable fundamental group which is normally generated by one element, and there exist Peano continua which are not finitely (ergot not countably) normally generated. Theorem 3.1 has numerous other applications to first homology (see On subgroups of first homology). For Theorem 3.2 we recall that the commutator length of an element $g$ of a group $G$ is the smallest number $n$ such that $g$ may be written as a product of $n$ commutators, and defined to be $\infty$ provided $g$ is not in the derived subgroup of $G$ . One can understand Theorem 3.2 by its contrapositive: if there are arbitrarily small loops of arbitrarily high commutator length, then first homology is of cardinality $2^{\aleph_0}$ . This is a strengthening of a result of Greg Conner and myself: a Peano continuum has first homology of cardinality $2^{\aleph_0}$ if there exist arbitrarily small loops of infinite commutator length (see On the first homology of Peano continua).

These ideas also lend themselves to very simple characterizations of topologically defined subgroups of the fundamental group. For example, the shape kernel of a path connected space $X$ is the intersection of the kernels of all maps to nerves over $X$ . Thus, the shape kernel represents the set of loops which unavoidably perish when one maps to a simplicial complex. One gets the following characterization of this subgroup under some mild assumptions (see Definable, Theorem 5.1):

Theorem 3.3 If $X$ is a metric space which is path connected, locally path connected then the shape kernel is the intersection of all open normal subgroups of $\pi_1(X)$ .

This gives one a very short, clean proof of the following (ibid Theorem 5.4):

Theorem 3.4 If $X$ is a metric space which is path connected, locally path connected then the shape kernel is equal to the Spanier group.

This is a slightly weaker result than that of Brazas and Fabel (see their work Thick Spanier groups and the first shape group), where they give the same conclusion and instead of “metric” they assume “paracompact.”

One can cook up easy examples of fundamental groups which have a “nonconstructive” decomposition into a direct sum. For example, by taking a countably infinite product of projective places one obtains a Peano continuum $X$ which has fundamental group isomorphic to $\prod_{\omega} \mathbb{Z}/2$ . It is not difficult to find subgroups $G$ and $H$ such that $\pi_1(X) = G\oplus H$ and $G$ is not Borel, analytic, or anything close to nice. The situation with free product decompositions is much nicer (Definable, Theorem C):

Theorem 3.5 Suppose $X$ is path connected locally path connected Polish and $\pi_1(X) \simeq *_{i\in I}G_i$ with each $G_i$ nontrivial. The following hold:

$|I| \leq \aleph_0$
Each retraction map $r_j:*_{i\in I}G_i \rightarrow G_j$ has analytic kernel.
Each $G_j$ is of cardinality $\leq \aleph_0$ or $2^{\aleph_0}$ .
The map $*_{i\in I} G_i \rightarrow \bigoplus_{i\in I} G_i$ has analytic kernel.

Local path connectedness cannot be dropped from the hypotheses, else one can violate the first three conditions.

Certain decompositions into products also behave nicely. Recall that a group $G$ is noncommutatively slender group if for any homomorphism from the fundamental group of the Hawaiian earring $\phi: \pi_1(E) \rightarrow G$ there exists $N\in \omega$ such that $\phi \circ p_N = \phi$ (here $p_N$ is the induced retraction to the outer $N$ circles). In other words, any homomorphism from the fundamental group of the Hawaiian earring to a noncommutatively slender group kills all circles of sufficiently small size. Free groups, free abelian groups, Thompson’s group $F$ , Baumslag-Solitar groups and torsion-free word hyperbolic groups are examples of noncommutatively slender groups (see A note on automatic continuity). We have the following:

Theorem 3.6 If $X$ is a path connected, locally path connected $\kappa$ -Lindelof metric space and $\{G_i\}_{i\in I}$ is a collection of nontrivial noncommutatively slender groups with $|I| >\kappa$ then there is no epimorphism $\phi:\pi_1(X) \rightarrow \prod_{i\in I} G_i$ .

If one assumes the generalized continuum hypothesis then the noncommutative slenderness of the groups can be dropped, since the product $\prod_{i\in I} G_i$ would simply be of higher cardinality than the set of all loops in the space $X$ . However the noncommutative slenderness cannot be dropped in general: in a universe where $2^{\aleph_0} = 2^{\aleph_1}$ you can take $X$ to be the countably infinite product of projective planes and notice that $\pi_1(X) \simeq \prod_{\omega} \mathbb{Z}/2 \simeq \bigoplus_{2^{\aleph_0}} \mathbb{Z}/2 \simeq \prod_{\aleph_1} \mathbb{Z}/2$ .

I’ll conclude by noting that extra set theoretic assumptions can enhance the results that one derives. Hypotheses about analyticity of a subgroup of a fundamental group can be relaxed (in the statement of Theorem 3.1, for example) to be $\Sigma_2^1$ or $\Sigma_3^1$ or whatever other semi-algebra is guaranteed to be in the algebra of subsets satisfying the property of Baire.

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