  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.
