  We develop an equivariant Cerf theory for Morse functions on
finite-dimensional manifolds with group actions, and adapt the technique to the
infinite-dimensional setting to study the moduli space of perturbed flat
$SU(n)$-connections. As a consequence, we prove the existence of perturbative
$SU(n)$ Casson invariants on integer homology spheres for all $n\ge 3$, and
write down an explicit formula when $n=4$. This generalizes the previous works
of Boden and Herald.
