  We define the dual of a set of generators of the fundamental group of an
oriented two-surface $S_{g,n}$ of genus $g$ with $n$ punctures and the
associated surface $S_{g,n}\setminus D$ with a disc $D$ removed. This dual is
another set of generators related to the original generators via an involution
and has the properties of a dual graph. In particular, it provides an algebraic
prescription for determining the intersection points of a curve representing a
general element of the fundamental group $\pi_1(S_{g,n}\setminus D)$ with the
representatives of the generators and the order in which these intersection
points occur on the generators.We apply this dual to the moduli space of flat
connections on $S_{g,n}$ and show that when expressed in terms both, the
holonomies along a set of generators and their duals, the Poisson structure on
the moduli space takes a particularly simple form. Using this description of
the Poisson structure, we derive explicit expressions for the Poisson brackets
of general Wilson loop observables associated to closed, embedded curves on the
surface and determine the associated flows on phase space. We demonstrate that
the observables constructed from the pairing in the Chern-Simons action
generate of infinitesimal Dehn twists and show that the mapping class group
acts by Poisson isomorphisms.
