2 problems

Let $G$ be a second countable locally compact Hausdorff group, let $X$ be a $G$-compact proper complete $G$-Riemannian manifold, and let $X'$ be another complete Riemannian $G$-com…

Let $X$ be a $\mathbb{Z}/k$ spin manifold with a $\mathbb{Z}/k$ circle action, and let $D$ be a Dirac operator commuting with the circle action. Write its equivariant topological i…