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:

The trivial subgroup of $\pi_1(X)$ is open
All subgroups of $\pi_1(X)$ are clopen
$X$ 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:

$G$ is open
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 $\pi_1(X, x)/G$ .

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