Twisted Milnor Hypersurface I

In this paper, we study {\bf twisted Milnor hypersurfaces} and compute their $\widehat{A}$-genus and Atiyah-Singer-Milnor $\alpha$-invariant. Our tool to compute the $\alpha$-invariant is Zhang’s analytic Rokhlin congruence formula. We also give some applications about group actions and metrics of positive scalar curvature on twisted Milnor hypersurfaces.