Group structures of topological and homotopy mapping class groups

Determine the group structures of the topological mapping class group MCG^Top(M_g) and the homotopy automorphism group haut(M_g) for M_g = #^g(S^2 × S^3).

Background

The paper determines the smooth mapping class group of M_g and proves splitting results for the corresponding topological mapping class and homotopy automorphism extensions. However, it does not determine the internal group structures of MCGTop(M_g) or haut(M_g), leaving these as unresolved problems.

References

The determination of the group structures of $\mathrm{MCG}{\mathrm{Top}(M_g)$ and $\mathrm{haut}(M_g)$ leaves open.

— Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology  (2608.26698 - Jin, 27 Aug 2026) in Remark following Theorem 6.2 in Section 6

It is unknown that which homotopy classes do these diffeomorphisms correspond to (except for $S2 \times S3$, see below).

— Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology  (2608.26698 - Jin, 27 Aug 2026) in Remark 8.5 in Section 8