Topological components of surface group representations into the unitary group
Abstract
We study representations of compact oriented surface groups into 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≥1 and n≥1 boundary components, there are np−1 components; for g=0 and n≥2, 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.
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BibTeX
@misc{wan2026topological,
title = {{Topological components of surface group representations into the unitary group}},
author = {Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30603},
eprint = {2609.30603},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30603},
}
RIS
TY - UNPB
TI - Topological components of surface group representations into the unitary group
AU - Xueyuan Wan
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30603
UR - https://arxiv.org/abs/2609.30603
AB - We study representations of compact oriented surface groups into 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≥1 and n≥1 boundary components, there are np−1 components; for g=0 and n≥2, 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.
ER -