On Line-Hyperplane Structures for Hitchin Representations

Let $n\ge 4$ and $\rho :{\pi }_{1}S\to PSL(n,\mathbb{R})$ be a Hitchin representation. We study the topology of a cocompact domain of discontinuity ${\mathrm{\Omega }}_{\rho }$ in the flag manifold ${\mathcal{F}}_{1,n-1}$ of line-hyperplane pairs in ${\mathbb{R}}^n$ defined by Guichard-Wienhard. In particular, we lift $\mathrm{\Omega }$ to its $({\mathbb{Z}}_2\times {\mathbb{Z}}_2)$-cover $\widehat{\mathrm{\Omega }}$ in the Stiefel manifold $V_2({\mathbb{R}}^n)$. The domain $\widehat{\mathrm{\Omega }}$ is highly connected and smoothly fibers over hyperbolic space ${\mathbb{H}}^2$ with unknown fiber $\widehat{\mathfrak{F}}$, a closed $(2n-5)$-manifold. We determine the homeomorphism type of $\widehat{\mathfrak{F}}$ as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of $\widehat{\mathfrak{F}}$ involves computing both its homology and a certain differential topology invariant needed to invoke Wall’s classification of highly connected odd-dimensional manifolds.