Homological stability in dimension five

Establish homological stability for the classifying spaces BDiff(W_g, ∂) of the relative diffeomorphism groups of the 5-manifolds W_g = (#^g(S^2 × S^3)) \setminus int(D) under stabilization.

Background

The paper defines stabilization maps for mapping class groups and proves stabilization of their first homology in the specific family W_g for g ≥ 4. It then places this result in the broader context of homological stability for classifying spaces of relative diffeomorphism groups. Full homological stability in dimension five is explicitly stated to remain unestablished.