Compatibility of stabilized cohomology classes

Establish that the cohomology classes h^{(m)} in H^p(GL_{m+n}(F);k) constructed from a bi-Grassmannian cocycle restrict along GL_{m+n-1}(F) to h^{(m-1)}, and consequently assemble into a class in H^p(GL_\infty(F);k).

Background

The extension criterion constructs, for a bi-Grassmannian cocycle h on the configuration complex C_*(n), a family of classes h{(m)} in the cohomology of GL_{m+n}(F), each restricting to the original class h=h{(0)} under the inclusion GL_n(F) to GL_{m+n}(F). The construction therefore provides stability relative to the initial group, but the paper does not establish compatibility between successive classes in the family.

The unresolved issue is whether h{(m)} restricts to h{(m-1)} under the standard inclusion GL_{m+n-1}(F) into GL_{m+n}(F). Such compatibility would allow the classes to define a single class in the stable cohomology Hp(GL_\infty(F);k).

References

We expect that h{(m)} restricts to h{(m-1)} and hence all assemble to a class in Hp(GL_\infty(F);{k}), but do not prove this.

— On the surjectivity conjectures of Dupont and Monod  (2609.03963 - Kupers et al., 3 Sep 2026) in Appendix, Section “Goncharov's extension criterion,” paragraph following the definition of stable classes