What is an interpretation of the relation in the cohomology of the pure braid groups? In 1968, Arnol'd proved that the integral cohomology of the pure braid group $P_n$ is isomorphic to the exterior algebra generated by the collection of degree-one classes $\omega_{i,j}\ (1 \le i < j \le n)$, subject to the following relation:
$$
\omega_{k,l} \omega_{l,m} + \omega_{l,m}\omega_{m,k} + \omega_{m,k}\omega_{k,l} = 0.
$$
The classes $\omega_{k,l}$ are realizable as differential forms on the pure configuration space of $n$ points in $\mathbb C$ as follows:
$$
\omega_{k,l} = \frac{1}{2 \pi i} \frac{dz_k - dz_l}{z_k - z_l},
$$
from which it can be seen that the classes $\omega_{k,l}$ are computing the winding number of the $k^{th}$ point around the $l^{th}$. In his paper (which can be found here), Arnol'd merely remarks that the above relation can be seen to hold for the forms $\omega_{k,l}$ by direct computation.
My question is, does the above relation have an interpretation in the context of winding numbers? How would I know to find this relation if I were trying to compute the cohomology of the pure braid group on a desert island?
 A: You know some kind of relation has to hold, since the configuration space of three points in the plane $C_3 \mathbb R^2$ has the homotopy-type of $(S^1 \vee S^1) \times S^1$, so $H^2$ only has rank $2$.  The homotopy-equivalence comes from noticing the Faddell-Neuwirth fibration $C_1 (\mathbb R^2 \setminus \{0,1\}) \to C_3 \mathbb R^2 \to C_2 \mathbb R^2$ is trivial.  But why a relation of that specific type?  You can follow this line of reasoning to its conclusion and derive the relation from a close inspection of this model, say, using cellular cohomology. Because of the action of the symmetric group, there's basically no other relation possible (modulo a small sign issue). 
Another way to go about it would be to think of cohomology as dual to homology, but for that you need a compact manifold.  So you could compactify the manifold in the Fulton-Macpherson manner.  In this model, $\omega_{i,j}$ is dual to the subspace of the compactified configuration space where $j$ is directly above $i$.  Cup product corresponds to intersection product, so your relation boils down to saying that the homology class where $m$ is directly over $l$ and $l$ is directly over $k$ (or any cyclic permutation of that) is a boundary. You can write that as a boundary of a class -- the idea is to swing the bottom point of the configuration around to be the top point.  
Those are two things that come to mind.  I suspect someone like Dev Sinha or Fred Cohen have cuter ways of thinking of this. 
