On the index of $G$-spaces

With a $G$-space, where $G$ is a compact Lie group, one can associate an ideal in the cohomology ring of the classifying space for $G$. It is called the ideal-valued index of the $G$-space. A filtration of the ideal-valued index that arises in a natural way from the Leray spectral sequence is considered. Properties of the index with filtration are studied and numerical indices are introduced. These indices are convenient for estimates of the $G$-category and the study of the set of critical points of a $G$-invariant functional defined on a manifold.
A generalization of the Bourgin–Yang theorem for the index with filtration is proved. This result is used for estimates of the index of the space of partial coincidences for a map of a space with $p$-torus action in a Euclidean space.