In my mathematical reading and thoughts, I keep running across the notion of configuration spaces, and while I essentially understand the idea behind them, I don't have much intuition for them (not even simple ones like the configuration space of two (indistinguishable) points on a line), and I don't know how computations are done with them. Is there a textbook or paper that would serve as a gentle introduction to configuration spaces, to someone with perhaps a semester of graduate algebraic topology (or maybe a little more?)

## 4 Answers

The canonical reference is the textbook of Fadell and Husseini:

Fadell, Edward R.; Husseini, Sufian Y. *Geometry and topology of configuration spaces.* Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001.

Computations are mainly done using various fiberings which take a configuration to some initial subset of its points. This idea goes back at least as far as the paper of Fadell and Neuwirth:

Fadell, Edward; Neuwirth, Lee
*Configuration spaces.*
Math. Scand. 10 (1962) 111--118.

which would be a good place to start.

My recommendation is that you first get used to configurations of points on the disc, where the fundamental group is the braid group. A good place to start is Birman's book "Braids, Links, and Mapping Class Groups". This contains a pretty nice exposition of the foundational work of Fadell and Neuwirth that Mark mentioned in his answer. It also has some intro material about configuration spaces and braid groups on other manifolds, though of course at this point its bibliography is pretty outdated.

The configuration space of $n$ points in a topological space $X$ is usually defined to be,

$$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$

A theorem which may be enlightening to your intuitive understanding of $C_{\hat{n}}(X)$ would be the following:

$$C_{\hat{n}}([0,1]) = \coprod_{i=1}^{n!} \Delta_{i}^{n}$$

Where $\coprod$ denotes disjoint union, and $\Delta_{i}^{n}$ denotes the $i^{th}$ copy of the $n$-simplex $\Delta^{n}$. We see that there are $n!$ $\Delta^{n}$'s because the symmetric group (which has order $n!$) acts freely on $C_{\hat{n}}(X)$, permuting the coordinates in each $\vec{z}=(z_{1},...,z_{n}) \in C_{\hat{n}}(X)$.

In fact, we can define the orbit space $C_{n}(X) := C_{\hat{n}}(X) / \Sigma_{n}$ as configuration space modded out by the symmetric group on $n$ elements $\Sigma_{n}$. Then it can be shown that $$\pi_{1}(C_{\hat{n}},\vec{p}) = PB_{n}, \; \text{and} \; \pi_{1}(C_{n},\tau(\vec{p})) = B_{n}$$ Where, $\tau: C_{\hat{n}}(X) \rightarrow C_{n}(X)$ is an $n!$-sheeted covering map called the orbit space projection, and $PB_{n}$ and $B_{n}$ are the pure braid group and braid group on $n$ strands, respectively.

If this sounds interesting to you, then I suggest you read the following paper:

Let me add one more introductory book.

Hansen, Vagn Lundsgaard. Braids and coverings: selected topics. With appendices by Lars Gæde and Hugh R. Morton. London Mathematical Society Student Texts, 18. Cambridge University Press, Cambridge, 1989. x+191 pp. ISBN: 0-521-38479-6 MR1247697

If I remember correctly, this small book does not require much and explains basic techniques to handle configuration spaces of manifolds, such as the fibration sequence given by selecting points.

Note, however, that most classical books and papers, including this book and the one by Fadell and Husseini, only deal with configuration spaces of manifolds. In order to study configurations spaces of singular spaces, we need completely different techniques.

For example, a graph $\Gamma$, regarded as a 1-dimensional cell complex, is a singular space. We cannot use the basic fibration technique. I recommend this expository article by Abrams and Ghrist for such configuration spaces.

The homology of the little disks operad(arxiv.org/abs/math/0610236), which is about configuration spaces of points in $\mathbb{R}^n$. $\endgroup$