Pith Number

pith:CGQLYITK

pith:2026:CGQLYITKYCYSED6HZNHA3XE6MJ

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Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations

Jianzhong Pan, Jie Wu, Pengcheng Li

Stable cohomotopy groups in codimensions two and three admit complete algebraic characterizations for CW complexes, with bordism interpretations for oriented and string manifolds.

arxiv:2605.13239 v1 · 2026-05-13 · math.AT · math.GT

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\pithnumber{CGQLYITKYCYSED6HZNHA3XE6MJ}

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Record completeness

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Claims

C1strongest claim

For general CW complexes, we give a complete characterization of stable cohomotopy in codimension two and a characterization in codimension three up to a 3-primary parameter. Geometrically, we provide bordism-theoretic interpretations of these stable cohomotopy groups for oriented manifolds in codimension two and string manifolds in codimension three.

C2weakest assumption

The characterizations assume the spaces are CW complexes or the manifolds are oriented/string; the algebraic descriptions and bordism equivalences may fail or require additional corrections outside these categories.

C3one line summary

Stable cohomotopy in codimensions 2 and 3 receives complete algebraic characterizations for CW complexes and bordism interpretations for manifolds, yielding necessary and sufficient conditions for nowhere-vanishing vector bundle sections.

References

45 extracted · 45 resolved · 2 Pith anchors

[1] Adams,On the non-existence of elements of Hopf invariant one, Ann 1960

[2] S. Amelotte, T. Cutler, and T. So,Suspension splittings of 5-dimensional Poincar´ e duality complexes and their applications, Algebr. Geom. Topol.26 (2026), no. 1, 283–319. MR 5018431 2026

[3] D.W. Anderson, E.H. Brown, Jr., and F.P. Peterson,The structure of the spin cobordism ring, Ann. of Math. (2)86(1967), 271–298. MR 219077 1967

[4] M. Ando, M.J. Hopkins, and C. Rezk,Multiplicative orientations of ko-theory and of the spectrum of topological modular forms, preprint (2010) 2010

[5] Atiyah,Bordism and cobordism, Proc 1961

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:49.520492Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

11a0bc226ac0b1220fc7cb4e0ddc9e62703e6cd778a501bb503f7f5b4530fe32

Aliases

arxiv: 2605.13239 · arxiv_version: 2605.13239v1 · doi: 10.48550/arxiv.2605.13239 · pith_short_12: CGQLYITKYCYS · pith_short_16: CGQLYITKYCYSED6H · pith_short_8: CGQLYITK

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CGQLYITKYCYSED6HZNHA3XE6MJ \
  | 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: 11a0bc226ac0b1220fc7cb4e0ddc9e62703e6cd778a501bb503f7f5b4530fe32

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b500a5aa0fb3400ae5bac48bb6a732b0b93632db2574c63ff7982934f6f9103b",
    "cross_cats_sorted": [
      "math.GT"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.AT",
    "submitted_at": "2026-05-13T09:25:36Z",
    "title_canon_sha256": "827c98c6236dfb156826f903ca0bebb28b72ed85107265b788b3e671fee3290d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13239",
    "kind": "arxiv",
    "version": 1
  }
}