Homological stability under connected-sum stabilization

Establish whether, for a smooth closed \(2n\)-manifold \(M\), the homology groups \(H_k({M_g\langle n\rangle})\) of the spaces of homotopy self-equivalences of \(M_g=M\# g(S^n\times S^n)\) preserving the stabilized normal \(n\)-type admit a stability range for each fixed \(k>0\), depending on \(g\) and \(M\).

Background

The paper discusses stabilization of a closed even-dimensional manifold by connected sum with $S^n\times S^n$, writing $M_g=M\# g(S^n\times S^n)$. This operation induces stabilization homomorphisms between the corresponding spaces of homotopy self-equivalences preserving the stabilized normal $n$-type.

The braid relates these self-equivalence spaces to auxiliary groups associated with Postnikov sections of $M_g$. The authors ask whether the homology of the self-equivalence spaces themselves becomes stable in a range as the number $g$ of connected-sum factors increases.