Configuration space of unordered points in the plane

Definition

Fix a natural number $n$ This topological space is defined as the configuration space of $n$ unordered points in the Euclidean plane ${\mathbb{R}}^2$, which can also be thought of as the space of complex numbers $\mathbb{C}$.

It is a classifying space for the braid group of degree $n$. In particular, the homology groups of this space are the homology groups of the braid group of degree $n$, viewed as a group.

Related notions

• Configuration space of ordered points in the plane: This can also be thought of as the classifying space for the pure braid group.