曲面群到酉群的表示之拓扑分支

中文摘要 AI 辅助

我们研究紧致可定向曲面群到 $\mathrm U(p)$ 的表示，其边界单值化为椭圆-幂单型，即任何边界单值化都不具有特征值 $1$。我们证明相关的平坦厄米丛的符号完全决定了连通分支，并且每个分支都是道路连通的。对于亏格 $g\geq1$ 和 $n\geq1$ 个边界分支，共有 $np-1$ 个分支；对于 $g=0$ 和 $n\geq2$，共有 $(n-2)p+1$ 个。圆盘情形为空，而闭曲面表示空间是连通的。在取共轭商后，相同的分支分类仍然成立。我们的证明使用边界 $\rho$ 不变量和显式矩阵变形。对于三孔球面，表示分支具有复格拉斯曼流形的同伦类型，其共轭商是可缩的。

英文摘要

We study representations of compact oriented surface groups into $\mathrm U(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component is path-connected. For genus $g\geq1$ and $n\geq1$ boundary components, there are $np-1$ components; for $g=0$ and $n\geq2$, there are $(n-2)p+1$. The disk case is empty, while the closed-surface representation spaces are connected. The same component classification holds after taking the quotient by conjugation. Our proofs use the boundary rho invariant and explicit matrix deformations. For a three-holed sphere, the representation components have the homotopy types of complex Grassmannians, and their conjugation quotients are contractible.

发表机构

• Mathematical Science Research Center, Chongqing University of Technology（重庆理工大学数学科学研究中心）

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