Topological puzzle: configuration space

I came across a problem which looks like an exercise from Topology by Munkres, yet I can’t figure it out.

Let X be a connected topological space. The product X\times X is also a topological space, and is connected as well. (Why? Because each “horizontal” fiber X\times \{x\} and each “vertical” fiber \{x\}\times X must lie within some connected component, but since they intersect, there is just one component.) However, if we remove the diagonal D=\{(x,x)\colon x\in X\} from the product, the connectedness may be lost. For example, (X\times X)\setminus D is disconnected when X is a line.

So far so easy. Now let’s take a quotient of (X\times X)\setminus D, identifying each pair (x_1,x_2) with the reordered pair (x_2,x_1). In other words, the symmetric group S_2 acts on (X\times X)\setminus D by permutation of coordinates, and we take the space of orbits. This new space is denoted C_2(X) and is called the configuration space of X.

Is C_2(X) connected?

It is in all examples that I can think of. Of course, this is not much evidence (for one thing, my thinking does not go beyond Hausdorff spaces). I’m happy to assume that X is metrizable, in which case C_2(X) carries the Hausdorff metric d(\{a,b\},\{a',b'\}) = \min(\max(|a-a'|,|b-b'|),\max(|a-b'|,|b-a'|)). Here I’m using the |\cdot -\cdot | notation for the metric to reduce the clutter.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s