On the index of \(G\)-spaces (Q2710722)

Let \(G\) be a compact Lie group. For a \(G\)-space \(X\), \textit{E. Fadell} and \textit{S. Husseini} [Ergodic Theory Dyn. Syst. 8, 73-85 (1988; Zbl 0657.55002)] have introduced the so-called ideal-valued index \(\text{Ind}^G X\) which is an ideal of the cohomology ring \(H^*(BG)\) of the classifying space \(BG\) of \(G\), where the cohomology groups are considered with coefficients in some field. By definition, \(\text{Ind}^G X=\text{Ker} (H^*_G(pt)\to H^*_G (X))\) where \(H^*(BG)= H^*_G(pt) \to H^*_G(X)\) is induced by the constant map \(X \to pt\). The ideal-valued index \(\text{Ind}^GX\) is (only) the first obstruction to the existence of a \(G\)-map \(f:X\to Y\) into a \(G\)-space \(Y\) (if \(f\) exists, then \(\text{Ind}^G Y\subset\text{Ind}^GX)\).NEWLINENEWLINENEWLINEIn the paper under review, the author considers an increasing filtration on \(\text{Ind}^GX\) arising in a natural way from the Leray spectral sequence, and studies properties of the ideal-valued index with filtration. In particular, he obtains a refinement of the generalized Bourgin-Yang theorem for the ideal-valued index with filtration. Moreover, the author introduces numerical indices and shows that they can be used to obtain estimates of the \(G\)-category of a \(G\)-space and the study of the set of critical points of a \(G\)-invariant functional defined on a manifold. In particular, the generalized Bourgin-Yang theorem is used for estimates of the index of the space of partial coincidences for a map of \(X\) into an Euclidean space, where \(X\) is a space with \(p\)-torus action.