Integral cohomology of the indefinite Grassmannian

Determine the integral cohomology of the indefinite Grassmannian \(\Gr_{p,q}(F^{m+n})\), extending the coefficient-field cohomology calculations established in the paper.

Background

The paper computes the cohomology ring of the indefinite Grassmannian with coefficients in an arbitrary field and gives an explicit mod-2 presentation in the real case. It also explains that analogous descriptions over C\mathbb{C} and H\mathbb{H} involve Chern and symplectic Pontryagin classes with integral coefficients, for which the field-coefficient argument does not apply. The authors therefore identify the integral cohomology, including these more involved coefficient-ring calculations, as an unresolved problem.

References

Given that this is a first foray, it is unsurprising that there are many open questions (e.g., the integral cohomology left unaddressed Corollary~\ref{cor:cohom}) and future directions (e.g., the indefinite analogs of the symplectic twistor spaces in ), which we hope would generate exciting further works for interested readers and ourselves.

— The Grassmannian of indefinite subspaces  (2608.30249 - Wang et al., 31 Aug 2026) in Section Conclusion; see also Corollary~\ref{cor:cohom}