  We express the rational cohomology of the unordered configuration space of a
compact oriented manifold as a representation of its mapping class group in
terms of a weight-decomposition of the rational cohomology of the mapping space
from the manifold to a sphere. We apply this to the case of a compact oriented
surface with one boundary component and explicitly compute the rational
cohomology of its unordered configuration space as a representation of its
mapping class group. In particular, this representation is not symplectic, but
has trivial action of the second Johnson filtration subgroup of the mapping
class group.
