The mod-2 cohomology groups of low-dimensional unordered flag manifolds and Auerbach bases

Unordered flag manifolds are the manifolds of unordered $n$-tuple of mutually orthogonal lines in ${\mathbb{R}}^n$. In this paper, we develop some basic tools to compute the mod-$2$ cohomology groups of these spaces, and apply them for explicit computation for small $n$. We show that this computation improves the known estimate of the number of Auerbach bases of normed linear spaces of small dimensions.