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

TL;DR

This paper investigates the topological structure of the space of triples of projective lines in real projective space, revealing its homotopy equivalence to certain flag varieties and computing their integral homology.

Contribution

It establishes the homotopy equivalence between these spaces and flag varieties for all dimensions, providing explicit homology calculations for low dimensions.

Findings

Spaces are homotopically equivalent to real flag varieties for n=2,3,4.

Explicit integral homology calculations for these spaces.

General result: space is homotopy equivalent to a partial flag variety for all 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 $\mathbb{R}.$