For any manifold $M$, the unordered configuration space of $k$ points is obtained as a quotient of ordered configuration space of $k$ points by the group action of symmetric group on $k$ letters. Does it induce some relation between the cohomology algebras of the two spaces?



Relation between cohomology of ordered and unordered configuration spaces?
If $F\_n(M)$ denotes the ordered configuration space and $C\_n(M)$ the unordered configuration space, the quotient map gives a map
$$H^\*(C\_n(M)) \longrightarrow H^\*(F\_n(M)).$$
If one takes rational coefficients, then this induces an isomorphism
$$H^\*(C\_n(M);\mathbb{Q}) \longrightarrow H^\*(F\_n(M);\mathbb{Q})^{\Sigma\_n}$$
onto the symmetric group invariant subspace.


In positive characteristic I don't think there is any pleasant relation between them.


