arXiv:1703.09276 (2017)

arXiv

Abstract

We study the Grassmann manifold \(G_k\) of all \(k\)-dimensional subspaces of \({\mathbb R}^n\). The Cartan embedding \(G_k\subset O(n)\) realizes \(G_k\) as a subspace of \(Sl_n({\mathbb R})\) and we study the decomposition \(G_k=\coprod_w (BwB\cap G_k)\) inherited from the classical Bruhat decomposition. We prove that this defines a CW structure on \(G_k\) and determine the incidence numbers between cells.

Comments

This paper hasn’t been submitted (yet) because I was not happy with the way it ends: I had been hoping for a clean formula for the intersection numbers in the Grassmannian but I couldn’t find one. I haven’t given up on this, I’m just waiting for the right moment to continue.

There were two reasons for writing this paper, neither of which is actually mentioned in it:

  1. I wanted to prove that the mod-\(2\) cohomology of the \(PG^{\mathrm or}_{2k}({\mathbb R}^{4k})\) mentioned in the introduction coincides with the mod-\(2\) cohomology of the ordinary unoriented Grassmannian \(G_{2k}({\mathbb R}^{4k})\) (as vector spaces). The same is true for the (related) groups \(Ss^{\pm}(4k)\) and \(SO(4k)\) by the work of Baum and Browder. I was not able to show this.

  2. The space \(PG^{\mathrm or}_8({\mathbb R}^{16})\) is Rosenfeld’s projective line \({\mathbb P}^1({\mathbb O}\otimes{\mathbb O})\) over the octo-octonions. Understanding its cell structure might be a first step to modelling \({\mathbb P}^2({\mathbb O}\otimes{\mathbb O}) = E_8/Ss^+(16)\).

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