Homotopy-type information carried by the two symmetric quotients

Characterize the extent to which the rational or complex homotopy type of a BG_n-manifold is determined by the commensurable symmetric quotients X_n=N_n/S_{n+1} and Y_{n,t}=M_{n,t}/S_{n+1}, and identify which higher homotopy operations are visible in their common invariant de Rham model versus those requiring the additional topology and local systems of the collision strata of Y_{n,t}.

Background

The paper uses the quotient X_n=N_n/S_{n+1} to construct the nonformality obstruction and the quotient Y_{n,t}=M_{n,t}/S_{n+1} to analyze the semismall resolution and compute low-degree Betti numbers. Although the two quotients have the same invariant de Rham model, they arise from different integral lattices and have different geometric roles.

The unresolved question asks how much of the rational or complex homotopy type of a BG_n-manifold can be recovered from these quotients. It also asks whether higher homotopy operations are already encoded by the common invariant model or instead depend on the additional topology and local-system data associated with collision strata in Y_{n,t}.