Configuration space of ordered points in the plane

Definition

Fix a natural number $n$ This topological space is defined as the configuration space of $n$ ordered 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 pure braid group of degree $n$. In particular, the homology groups of this space are the homology groups of the pure braid group of degree $n$, viewed as a group.

Related notions

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

Particular cases

$n$	Braid group $B_{n}$	Pure braid group $P_{n}$	Dimension of configuration space of $n$ ordered points in the plane as a real manifold	Smallest dimension of manifold that it's homotopy equivalent to	Dimension
1	trivial group	trivial group	2	one-point space	0
2	group of integers	group of integers (even integers, viewed as a subgroup)	4	circle	1
3	braid group:B3	pure braid group:P3	6	?	?
4			8	?	?