Topology of the open Grassmannian stratum

Determine the topology of the open Grassmannian stratum Gr(m,n)° consisting of points whose Pluecker coordinates are all nonzero, equivalently the topology of the complement U_{m,n} of the maximal-minor determinantal arrangement after removing the GL_m factor, for general m and n.

Background

The paper proves an isomorphism U_{m,n} ≅ GL_m(C) × Gr(m,n)° and, via Gale duality, identifies Gr(m,n)° with Gr(n-m,n)°. Closed topological formulas are obtained when m=2 or m=n-2, using an identification with a torus times the moduli space M_{0,n} of ordered points on the projective line.

For general m and n, however, the topology of Gr(m,n)° is not computed. Consequently, the homotopy groups, cohomology, and related topological invariants of U_{m,n} remain unresolved in general.

References

Beyond standard facts about $M_{,n}$, the proof needs no fibrations or spectral sequences. The fibration obstruction of Ex.~4.6 for $3\times7$ matrices plays no role for the slice, which exists for all $m\times n$; what is not known in general is the topology of $Gr(m,n)\circ$.

— Geometry of some determinantal arrangements  (2610.13291 - Janasz, 9 Oct 2026) in Remark 2.6, Section 3 (Topology of the complements)