Strong stability and shifted stability for the cohomology of configuration spaces

Homological stability for unordered configuration spaces of connected manifolds was discovered by Th. Church and extended by O. Randal-Williams and B. Knudsen: $H_i(C_k(M);\mathbb{Q})$ is constant for $k\geq f(i)$. We characterize the manifolds satisfying strong stability: $H^*(C_k(M);\mathbb{Q})$ is constant for $k\gg 0$. We give few examples of manifolds whose top Betti numbers are stable after a shift of degree.