Uniqueness of compact ancient solutions to the higher-dimensional Ricci flow

In dimensions n >= 4, an ancient kappa-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is kappa-noncollapsed. In this paper, we study the classification of ancient kappa-solutions to n-dimensional Ricci flow on Sn, extending the result in [S. Brendle, P. Daskalopoulos and N. Sesum, Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Invent. Math. 226 (2021), no. 2, 579-651] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.