{
  "abstract": "If $M$ is a closed oriented manifold of even dimension, then its $n$th symmetric product $M(n)=M^n/S_n$ is a rational homology manifold for any $n \\geq 0$ and hence has a canonically defined $L$-class in $H^*(M(n);\\mathbb{Q})$. A complicated formula for these classes was given many years ago in the second author's thesis. We obtain a simpler result by studying the dual classes in homology, showing that their generating function (after embedding each $H_*(M(n); \\mathbb{Q})$ into the Hopf algebra $H_*(M(\\infty); \\mathbb{Q})$) is the product of an elementary factor depending only on the Euler characteristic of $M$ and the exponential of a series consisting of odd Adams twists of the Poincaré dual of the $L$-class of $M$. We also use this to obtain both a simpler proof and a simpler expression for the cohomological $L$-classes than the previous one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32793",
  "author_total": 2,
  "authors": [
    {
      "name": "Julia Semikina"
    },
    {
      "name": "Don Zagier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b6a99cmmg1am4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32793",
  "fulltext_url": null,
  "id": "b6a99cmmg1am4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32793",
  "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.AT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214803635",
  "pdf_url": "https://arxiv.org/pdf/2609.32793",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b6a99cmmg1am4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32793",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32793"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$L$-classes of symmetric products: a homological formula",
  "title_text": "L-classes of symmetric products: a homological formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/b6a99cmmg1am4",
  "venue": "arXiv",
  "versions": []
}