  Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra
$g$, $\xi \colon P \to \Sigma$ a principal $G$-bundle, let $N(\xi)$ denote the
moduli space of central Yang-Mills connections on $\xi$, for suitably chosen
additional data, and let $\roman{Rep}_{\xi}(\Gamma,G)$ be the space of
representations of the universal central extension $\Gamma$ of the fundamental
group of $\Sigma$ in $G$ that corresponds to $\xi$. We construct smooth
structures on $N(\xi)$ and $\roman{Rep}_{\xi}(\Gamma,G)$ and show that the
assignment to a smooth connection $A$ of its holonomies with reference to
suitable closed paths yields a diffeomorphism from $N(\xi)$ onto
$\roman{Rep}_{\xi}(\Gamma,G)$; moreover we show that the derivative of the
latter at the non-singular points of $N(\xi)$ amounts to a certain twisted
integration mapping. Finally we examine the infinitesimal geometry of these
moduli spaces with reference to the smooth structures and, for illustration, we
show that, on the moduli space of flat $\roman{SU}(2)$-connections for a
surface of genus two which, as a space, is just complex projective 3-space, our
smooth structure looks rather different from the standard structure.
