The space of all triples of projective lines of distinct intersections in $\mathbb{RP}^n$

Abstract

We study the space of all triples of projective lines in $\mathbb{RP}^n$ such that any line in a triple intersects the two others at distinct points. We show that for $n=2$ and $3$ these spaces are homotopically equivalent to the real complete flag variety $Flag(\mathbb{R}^n)$ for $n=3$ and $4,$ respectively, and we explicitly calculate the integral homology of the corresponding spaces. We prove that for arbitrary $n$, this space is homotopy equivalent to $Flag(1,2,3,\mathbb{R}^{n+1}),$ the variety of all partial flags of signature $(0,1,2,3,n+1)$ in an $(n+1)$-dimensional vector space over $\mathbb{R}.$