Pith Number

pith:G5LTEK65

pith:2025:G5LTEK65KPICEDW3TMOZN5KRQX

not attested not anchored not stored refs pending

The $\mathrm{L}^p$-index of the Hodge-Dirac operator on compact Riemannian manifolds

C\'edric Arhancet

The Hodge-Dirac operator on compact manifolds is bisectorial with bounded H^∞ calculus, yielding p-independent topological indices.

arxiv:2512.22517 v3 · 2025-12-27 · math.FA · math.DG · math.KT

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{G5LTEK65KPICEDW3TMOZN5KRQX}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We establish that this operator is bisectorial and admits a bounded H^∞ functional calculus, without curvature assumptions. This result enables us to prove that the triple (C(M), L^p(Ω^•(M)), D) constitutes a compact Banach spectral triple. We then investigate consistent pairings between the Banach K-homology and the K-theory of the algebra C(M), identifying the resulting Fredholm indices with classical topological invariants, and hence showing that they are independent of p. We recover the classical Euler characteristic and the Hirzebruch signature as L^p-indices.

C2weakest assumption

Relying on the compactness of M, we establish that this operator is bisectorial and admits a bounded H^∞ functional calculus, without curvature assumptions.

C3one line summary

L^p-indices of the Hodge-Dirac operator on compact Riemannian manifolds recover the Euler characteristic and Hirzebruch signature and are independent of p.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-26T02:04:01.713005Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3757322bdd53d0220edb9b1d96f55185efacc421d1f797dbdc0253438eddf77d

Aliases

arxiv: 2512.22517 · arxiv_version: 2512.22517v3 · doi: 10.48550/arxiv.2512.22517 · pith_short_12: G5LTEK65KPIC · pith_short_16: G5LTEK65KPICEDW3 · pith_short_8: G5LTEK65

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/G5LTEK65KPICEDW3TMOZN5KRQX \
  | 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: 3757322bdd53d0220edb9b1d96f55185efacc421d1f797dbdc0253438eddf77d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bc32716afbf459e883e32abdae920ef7f3959ffd5424408ba8c4b37b5c5852b5",
    "cross_cats_sorted": [
      "math.DG",
      "math.KT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2025-12-27T08:30:26Z",
    "title_canon_sha256": "456408b60096ff03516598f40603b51f4a443a619a9a7c509f8eaca886c5cf5a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2512.22517",
    "kind": "arxiv",
    "version": 3
  }
}