  Let $Y$ be a CW-complex with a single 0-cell, $K$ its Kan group, a model for
the loop space of $Y$, and let $G$ be a compact, connected Lie group. We give
an explicit finite dimensional construction of generators of the equivariant
cohomology of the geometric realization of the cosimplicial manifold
$\roman{Hom}(K,G)$ and hence of the space $\roman{Map}^o(Y,BG)$ of based maps
from $Y$ to the classifying space $BG$. For a smooth manifold $Y$, this may be
viewed as a rigorous approach to lattice gauge theory, and we show that it then
yields, (i) when {$\roman{dim}(Y)=2$,} equivariant de Rham representatives of
generators of the equivariant cohomology of twisted representation spaces of
the fundamental group of a closed surface including generators for moduli
spaces of semi stable holomorphic vector bundles on complex curves so that, in
particular, the known structure of a stratified symplectic space results; (ii)
when {$\roman{dim}(Y)=3$,} equivariant cohomology generators including the
Chern-Simons function; (iii) when {$\roman{dim}(Y) = 4$,} the generators of the
relevant equivariant cohomology from which for example Donaldson polynomials
are obtained by evaluation against suitable fundamental classes corresponding
to moduli spaces of ASD connections.
