Pith Number

pith:5STATZAP

pith:2026:5STATZAPRHCXFIMFAUCXHTB33U

not attested not anchored not stored refs pending

$\mathbb{K}$-framings and $\mathbb{K}$-quadratic forms on surfaces

Nariya Kawazumi

K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.

arxiv:2604.27531 v2 · 2026-04-30 · math.GT · math.AT

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Claims

C1strongest claim

This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ.

C2weakest assumption

The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected).

C3one line summary

K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces.

Receipt and verification

First computed	2026-06-01T01:02:40.848072Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd

Aliases

arxiv: 2604.27531 · arxiv_version: 2604.27531v2 · doi: 10.48550/arxiv.2604.27531 · pith_short_12: 5STATZAPRHCX · pith_short_16: 5STATZAPRHCXFIMF · pith_short_8: 5STATZAP

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/5STATZAPRHCXFIMFAUCXHTB33U \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "810a236a128a37c70d5804461ad6f2f65003e2ce5f55ff7fabda0e2ec26c58f6",
    "cross_cats_sorted": [
      "math.AT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-04-30T07:34:00Z",
    "title_canon_sha256": "45e6b244e48e0861fa5d368995bba53808f264c178d4ea93805cb7670ed5a48f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.27531",
    "kind": "arxiv",
    "version": 2
  }
}