Bi-invariant Grassmannians and Atiyah–Jones theorems

We study completions of instanton moduli spaces, corresponding to moduli spaces of framed torsion free sheaves on CP2. We compute their homology, by studying their lagrangian deformation retracts: the bi-invariant Grassmann manifolds of spaces fixed by two nilpotent commuting operators. orollary, we obtain sharp ranges for Atiyah–Jones homological stability: this doubles the previously known ranges for SU(2) and SU(3) instanton moduli spaces.