There is no tame triangulation of the infinite real Grassmannian

We show that there is no triangulation of the infinite real Grassmannian G(k,R∞) nicely situated with respect to the coordinate axes. In terms of matroid theory, this says there is no triangulation of G(k,R∞) subdividing the matroid stratification. This is proved by an argument in projective geometry, considering a specific sequence of configurations of points in the plane. The Grassmannian G(k,Rn ) of k-planes in Rn is a smooth manifold, hence can be triangulated. Identify Rn as a subspace of Rn+1 , and let R∞ be the union (colimit) of the Rn ’s. The Grassmannian G(k,R∞ ) is infinite dimensional; it is unclear whether it can be triangulated for k ≥ 3. We are interested in triangulations which are nicely situated with respect to the coordinates axes. Such triangulations are of interest in combinatorics in the context of matroid theory; see Section 4. Definition 0.1. A triangulation of G(k,Rn ) or G(k,R∞ ) is tame if for every simplex σ, for every pair of k-planes V,W ∈ int σ, and for every vector v ∈ V , there is a vector w ∈ W so that for all of the standard basis vectors ei, v · ei = 0 ⇔ w · ei = 0 Using triangulation theorems from real algebraic geometry, it is not difficult to prove the following theorem (see [Mac93], [AD]). Theorem 0.2. For every k and n, there is a tame triangulation Tn,k. Furthermore for n ≤ n and k ≤ k, the triangulation Tn,k restricts to a subdivision of Tn′,k′ This theorem does not lead to a triangulation of G(k,R∞ ), because perhaps one would have to infinitely subdivide G(k,Rn ) ⊂ G(k,R∞ ). Main Theorem. There is no tame triangulation of G(3,R∞ ). ∗Partially supported by grants from the National Science Foundation.