关于Hitchin表示的线-超平面结构
On Line-Hyperplane Structures for Hitchin Representations
AI总结：
本文研究Hitchin表示对应的不连续域提升到Stiefel流形后的纤维拓扑，确定了闭的$(2n-5)$维纤维的同胚及微分同胚类型，并利用Wall分类方法。
AI中文摘要：
设$n \geq 4$且$\rho: \pi_1S \rightarrow PSL(n,\mathbb{R})$为Hitchin表示。我们研究由Guichard-Wienhard定义的旗流形$\mathcal{F}_{1,n-1}$（即$\mathbb{R}^n$中线-超平面对）中的余紧致不连续域$\Omega_{\rho}$的拓扑。特别地，我们将$\Omega$提升到其在Stiefel流形$V_2(\mathbb{R}^n)$中的$(\mathbb{Z}_2\times \mathbb{Z}_2)$覆盖$\hat{\Omega}$。域$\hat{\Omega}$高度连通，并光滑纤维化于双曲空间$\mathbb{H}^2$上，其未知纤维$\hat{\mathfrak{F}}$是一个闭的$(2n-5)$维流形。我们确定了$\hat{\mathfrak{F}}$的同胚类型，以及其与同伦球做连通和意义下的微分同胚类型。$\hat{\mathfrak{F}}$的分类涉及计算其同调以及某个特定的微分拓扑不变量，该不变量用于调用Wall对高度连通奇数维流形的分类。
英文摘要：
Let $n \geq 4$ and $ρ: π_1S \rightarrow PSL(n,\mathbb{R})$ be a Hitchin representation. We study the topology of a cocompact domain of discontinuity $Ω_ρ$ 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 $Ω$ to its $(\mathbb{Z}_2\times \mathbb{Z}_2)$-cover $\hatΩ$ in the Stiefel manifold $V_2(\mathbb{R}^n)$. The domain $\hatΩ$ is highly connected and smoothly fibers over hyperbolic space $\mathbb{H}^2$ with unknown fiber $\hat{\mathfrak{F}}$, a closed $(2n-5)$-manifold. We determine the homeomorphism type of $\hat{\mathfrak{F}}$ as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of $\hat{\mathfrak{F}}$ involves computing both its homology and a certain differential topology invariant needed to invoke Wall's classification of highly connected odd-dimensional manifolds.
