Action on homotopy Torelli elements and the structure of the homotopy Torelli group

Determine the action of H_M on those elements of haut(M | dot{M}) that cannot be represented by diffeomorphisms, and thereby determine the group structure of the homotopy Torelli group I^h(M), including whether it is abelian.

Background

For simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology, the paper identifies the homotopy Torelli group as an extension involving haut(M | dot{M}) and H_M. The smooth calculations determine the corresponding diffeomorphism subgroup and show that the extension splits, but the action on homotopy equivalences not realizable by diffeomorphisms is not determined. Consequently, the full group structure of Ih(M) remains unresolved; the paper notes that the special case M = S2 × S3 is known to be nonabelian.

References

The action of $H_M$ on the elements of $\mathrm{haut}(M | \dot{M})$ that cannot be represented by diffeomorphisms is unknown, so the group structure of $\mathcal{I}h(M)$ (for example, whether it is abelian) is unknown.

— 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 8.4 in Section 8