  The space of Fredholm operators of fixed index is stratified by submanifolds
according to the dimension of the kernel. Geometric considerations often lead
to questions about the intersections of concrete families of elliptic operators
with these submanifolds: are the intersections non-empty? are they smooth? what
are their codimensions? The purpose of this article is to develop tools to
address these questions in equivariant situations. An important motivation for
this work are transversality questions for multiple covers of $J$-holomorphic
maps. As an application, we use our framework to give a concise exposition of
Wendl's proof of the super-rigidity conjecture.
