The \(\mathbb{Z}/2\) Fadell-Husseini index of the complex Grassmann manifolds \(G_n (\mathbb{C}^{2n})\) (Q6638358)

This paper aims to study a free \(\mathbb{Z}/2\) action on complex Grassmann manifolds \(G_{n}(\mathbb{C}^{2n})\) induced by taking orthogonal complements. The authors apply the cohomological ideal-valued index theory introduced by \textit{E. Fadell} and \textit{S. Husseini} [Ergodic Theory Dyn. Syst. 8, 73--85 (1988; Zbl 0657.55002)] to determine the associated \(\mathbb{Z}/2\) Fadell-Husseini index\ of \(G_{n}(\mathbb{C}^{2n})\). The heart of this paper is to compute the Fadell-Husseini index of \(G_{n}(\mathbb{C}^{2n})\) with respect to the above \(\mathbb{Z}/2\) action using an analogue of the pioneering technique of evaluating Chern classes of wreath square introduced by \textit{D. Baralić} et al. [Forum Math. 30, No. 6, 1539--1572 (2018; Zbl 1405.51015)], where the \(\mathbb{Z}/2\) Fadell-Husseini index of real Grassmann manifolds \(G_{n}(\mathbb{R}^{2n})\) was computed.