{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30603",
  "author_total": 1,
  "authors": [
    {
      "name": "Xueyuan Wan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9k18bh9d9ckdb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30603",
  "fulltext_url": null,
  "id": "9k18bh9d9ckdb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30603",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214640180",
  "pdf_url": "https://arxiv.org/pdf/2609.30603",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9k18bh9d9ckdb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30603",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30603"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Topological components of surface group representations into the unitary group",
  "title_text": "Topological components of surface group representations into the unitary group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/9k18bh9d9ckdb",
  "venue": "arXiv",
  "versions": []
}