Strong stability and shifted stability for the cohomology of configuration spaces

TL;DR

This paper investigates the conditions under which the rational cohomology of unordered configuration spaces stabilizes, introducing the concept of strong stability and providing examples where Betti numbers stabilize after a degree shift.

Contribution

It characterizes manifolds with strong stability in cohomology and offers new examples of manifolds with shifted stability of Betti numbers.

Findings

Homological stability is extended to cohomology with strong stability conditions.

Identifies manifolds where cohomology stabilizes after a degree shift.

Provides examples of manifolds with top Betti numbers stabilizing post-shift.

Abstract

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.