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

Recognition: 2 theorem links

· Lean Theorem

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

Pengcheng Li , Jianzhong Pan , Jie Wu

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:03 UTC · model grok-4.3

classification 🧮 math.AT math.GT

keywords stable cohomotopycodimensionbordismCW complexesoriented manifoldsstring manifoldsvector bundles

0 comments

The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides algebraic characterizations of stable cohomotopy groups for general CW complexes, complete in codimension two and holding up to a 3-primary parameter in codimension three. Geometrically, these groups receive bordism-theoretic interpretations, corresponding to oriented bordism in codimension two and string bordism in codimension three. As an application, the work derives necessary and sufficient conditions for the existence of nowhere-vanishing sections of vector bundles, extending earlier codimension-one results.

Core claim

For any CW complex, stable cohomotopy in codimension two is given by a full algebraic characterization, while in codimension three the characterization holds except possibly for 3-primary components; geometrically, the stable cohomotopy groups of oriented manifolds in codimension two are identified with their oriented bordism groups, and those of string manifolds in codimension three with their string bordism groups.

What carries the argument

Algebraic characterizations of stable cohomotopy groups together with their bordism-theoretic interpretations for oriented and string manifolds.

If this is right

Necessary and sufficient conditions for nowhere-vanishing sections of vector bundles over manifolds.
Direct computation of certain stable homotopy groups via algebraic or bordism data.
Geometric realization of algebraic homotopy invariants in low codimensions.
Extension of codimension-one section existence criteria to codimensions two and three.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The characterizations may simplify explicit calculations of stable homotopy groups for concrete CW complexes such as spheres or projective spaces.
Bordism interpretations could link to other geometric invariants like characteristic classes or index theorems in manifold topology.
The 3-primary caveat in codimension three suggests possible refinements using additional 3-local tools or higher chromatic data.

Load-bearing premise

The algebraic characterizations and bordism equivalences are stated for CW complexes and for oriented or string manifolds.

What would settle it

An explicit computation of stable cohomotopy for a non-CW complex in codimension two that fails to match the claimed algebraic description.

Figures

Figures reproduced from arXiv: 2605.13239 by Jianzhong Pan, Jie Wu, Pengcheng Li.

**Figure 1.** Figure 1: The Whitehead tower of BOn, n ≥ k. Lemma 5.1. Let n ≥ k + 2 and let M be a closed smooth (n + k)-manifold with a tangential O⟨m⟩-structure. The canonical homomorphism Ω fr k (M) → Ω O⟨m⟩ k (M) is an isomorphism for k ≤ m − 2. In particular, there are canonical group isomorphisms Ω fr k (M) ∼=    Ω Spin k (M) for k ≤ 2, Ω String k (M) for k ≤ 6, Ω Fivebrane k (M) for k ≤ 7. Proof. Recall that there ho… view at source ↗

**Figure 2.** Figure 2: Partial Moore-Postnikov tower of Bιn−1. For each i ≥ 1, the composition p1,i = p1 ◦ p2 ◦ · · · ◦ pi : Yi → BSOn is (n+i−1)-anticonnected, which means that the induced homomorphism (p1,i)∗ : πj (Yi) → πj (BSOn) is an isomorphism for j ≥ n+i and is a monomorphism for j = n+i−1; the map fi : BSOn−1 → Yi is (n + i − 1)-connected, implying that the induced map (fi)∗ : [skn+i−1(X), BSOn−1] → [skn+i−1(X), Yi ] (7… view at source ↗

read the original abstract

This paper investigates stable cohomotopy groups in codimensions two and three from complementary algebraic and geometric viewpoints. 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. As an application, we derive necessary and sufficient conditions for the existence of nowhere-vanishing sections of vector bundles, extending the foundational codimension-one results of Konstantis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives algebraic characterizations of stable cohomotopy in codimensions two and three, with bordism interpretations for oriented and string manifolds, extending Konstantis' codim-one work.

read the letter

The main advance is a complete algebraic characterization of stable cohomotopy in codimension two for CW complexes, plus a characterization in codimension three that holds up to the 3-primary part. On the geometric side it supplies bordism-theoretic interpretations for oriented manifolds in codim two and string manifolds in codim three, then applies the results to give necessary and sufficient conditions for nowhere-vanishing sections of vector bundles. This is a direct, scoped extension of the codim-one results, and the dual algebraic-geometric framing is the part that looks most useful on first reading. The algebraic descriptions appear to rest on standard homotopy-theoretic tools, while the bordism side ties the groups to concrete manifold invariants without obvious circularity. If the proofs hold, the section-existence criteria could simplify some obstruction-theory calculations in low dimensions. The soft spots are limited and mostly declared in the abstract. The codim-three case is explicitly incomplete at the 3-primary level, so it is not a full characterization there. The statements are restricted to CW complexes and to oriented or string manifolds, which is reasonable but means the results do not apply verbatim outside those classes. Without the full derivations in front of me it is impossible to check for calculation slips or missing edge cases, but the setup itself shows no internal contradictions or post-hoc fitting. This is for algebraic and geometric topologists who work with stable homotopy groups or with section and framing problems on manifolds. A reader who needs explicit low-codimension tools or who wants to compute these groups on concrete spaces would get immediate value. It deserves a serious referee. The claims are clearly scoped, build on cited prior work, and address a concrete gap, so the paper should go to review rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper investigates stable cohomotopy groups in codimensions two and three from algebraic and geometric viewpoints. For general CW complexes, it gives a complete characterization of stable cohomotopy in codimension two and a characterization in codimension three up to a 3-primary parameter. Geometrically, it provides bordism-theoretic interpretations of these groups for oriented manifolds in codimension two and string manifolds in codimension three. As an application, it derives necessary and sufficient conditions for the existence of nowhere-vanishing sections of vector bundles, extending Konstantis' codimension-one results.

Significance. If the characterizations and bordism interpretations hold, the work would provide useful explicit descriptions of stable cohomotopy groups in low codimensions, which are often hard to compute directly, along with geometric realizations that could aid computations and applications in bundle theory. The extension of Konstantis' results to higher codimensions is a clear strength, offering necessary and sufficient conditions that may prove practical for differential topology.

minor comments (2)

[Abstract] The abstract states the codimension-three characterization holds 'up to a 3-primary parameter' without indicating whether this parameter is a specific homomorphism, a torsion subgroup, or an extension class; a short clarification in the introduction or statement of the main theorem would help readers assess the precision of the result.
[Bordism section] In the bordism interpretations, ensure that the relevant bordism groups (e.g., for string manifolds) are equipped with explicit references to standard definitions or prior computations, such as those appearing in works on spin or string bordism, to make the equivalence statements self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point at this stage. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents algebraic characterizations of stable cohomotopy in codimensions two (complete) and three (up to 3-primary parameter) for general CW complexes, together with bordism interpretations for oriented manifolds (codim 2) and string manifolds (codim 3). These are framed as independent results that extend Konstantis' codimension-one work. No quoted equations, definitions, or steps in the provided abstract or program description reduce a claimed prediction or characterization to a fitted input, self-definition, or self-citation chain by construction. The derivations are scoped to CW complexes and oriented/string manifolds without internal reduction to the target quantities themselves. This is the expected non-finding for a paper whose central claims remain externally falsifiable and non-tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard axioms of stable homotopy theory and bordism theory for CW complexes and manifolds; no free parameters or new entities are indicated in the abstract.

axioms (1)

standard math Standard axioms and long exact sequences of stable homotopy groups and bordism groups
Invoked to obtain algebraic characterizations for CW complexes and bordism interpretations for manifolds.

pith-pipeline@v0.9.0 · 5401 in / 1210 out tokens · 40753 ms · 2026-05-14T19:03:42.099290+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

complete characterization of stable cohomotopy in codimension two... via Postnikov tower... short exact sequence 0→Z/2^{1-ε}⊕QH_{n+1}(M;Sq2Z)→πn(M)→ker(Sq2Z:n)→0
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

bordism-theoretic interpretations... Ωfr2(M) and Ωfr3(M) for oriented/string manifolds

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · 2 internal anchors

[1]

Adams,On the non-existence of elements of Hopf invariant one, Ann

J.F. Adams,On the non-existence of elements of Hopf invariant one, Ann. of Math. (2)72(1960), 20–104. MR 141119

work page 1960
[2]

Amelotte, T

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

work page 2026
[3]

Anderson, E.H

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

work page 1967
[4]

Ando, M.J

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

work page 2010
[5]

Atiyah,Bordism and cobordism, Proc

M.F. Atiyah,Bordism and cobordism, Proc. Cambridge Philos. Soc.57(1961), 200–208. MR 126856

work page 1961
[6]

Atiyah, R

M.F. Atiyah, R. Bott, and A. Shapiro,Clifford modules, Topology3(1964), no. suppl, suppl. 1, 3–38. MR 167985

work page 1964
[7]

Bauer,A stable cohomotopy refinement of Seiberg-Witten invariants

S. Bauer,A stable cohomotopy refinement of Seiberg-Witten invariants. II, In- vent. Math.155(2004), no. 1, 21–40. MR 2025299

work page 2004
[8]

Bauer and M

S. Bauer and M. Furuta,A stable cohomotopy refinement of Seiberg-Witten invariants. I, Invent. Math.155(2004), no. 1, 1–19. MR 2025298

work page 2004
[9]

Borsuk,Sur la d´ ecomposition des poly` edres n-dimensionnels en poly` edres contractiles en soi, Compositio Math.3(1936), 431–434

K. Borsuk,Sur la d´ ecomposition des poly` edres n-dimensionnels en poly` edres contractiles en soi, Compositio Math.3(1936), 431–434. MR 1556956

work page 1936
[10]

Bott and L.W

R. Bott and L.W. Tu,Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304

work page 1982
[11]

The Pontrjagin Dual of 3-Dimensional Spin Bordism

G. Brumfiel and J. Morgan,The pontrjagin dual of3-dimensional spin bordism, arXiv preprint: 1612.02860 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
[12]

,The pontrjagin dual of4-dimensional spin bordism, arXiv preprint: 1803.08147 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
[13]

Crowley and C.M

D. Crowley and C.M. Escher,A classification ofS 3-bundles overS 4, Differential Geom. Appl.18(2003), no. 3, 363–380. MR 1975035

work page 2003
[14]

Dold,Erzeugende der Thomschen AlgebraN, Math

A. Dold,Erzeugende der Thomschen AlgebraN, Math. Z.65(1956), 25–35. MR 79269 46 P. LI, J. PAN, AND J. WU

work page 1956
[15]

Douglas, J

C.L. Douglas, J. Francis, A.G. Henriques, and M.A. Hill,Topological modular forms, vol. 201, American Mathematical Soc., 2014

work page 2014
[16]

Giambalvo,On⟨8⟩-cobordism, Illinois J

V. Giambalvo,On⟨8⟩-cobordism, Illinois J. Math.15(1971), 533–541. MR 287553

work page 1971
[17]

Harper,Secondary cohomology operations, Graduate Studies in Mathemat- ics, vol

J.R. Harper,Secondary cohomology operations, Graduate Studies in Mathemat- ics, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1913285

work page 2002
[18]

Hatcher,Algebraic topology, Cambridge University Press, Cambridge, 2002

A. Hatcher,Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354

work page 2002
[19]

Hill,The string bordism ofBE 8 andBE 8 ×BE 8 through dimension14, Illinois Journal of Mathematics53(2009), no

M.A. Hill,The string bordism ofBE 8 andBE 8 ×BE 8 through dimension14, Illinois Journal of Mathematics53(2009), no. 1, 183 – 196

work page 2009
[20]

Hopkins,Algebraic topology and modular forms, Proceedings of the Inter- national Congress of Mathematicians, Vol

M.J. Hopkins,Algebraic topology and modular forms, Proceedings of the Inter- national Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 291–317. MR 1989190

work page 2002
[21]

Hovey,v n-elements in ring spectra and applications to bordism theory, Duke Math

M.A. Hovey,v n-elements in ring spectra and applications to bordism theory, Duke Math. J.88(1997), no. 2, 327–356. MR 1455523

work page 1997
[22]

Jung and T.O

M. Jung and T.O. Rot,A geometric computation of cohomotopy groups in code- gree one, Algebr. Geom. Topol.25(2025), no. 6, 3603–3626. MR 4972721

work page 2025
[23]

Khare,On Dold manifolds, Topology Appl.33(1989), no

S.S. Khare,On Dold manifolds, Topology Appl.33(1989), no. 3, 297–307. MR 1026930

work page 1989
[24]

Kirby, P

R. Kirby, P. Melvin, and P. Teichner,Cohomotopy sets of 4-manifolds, Pro- ceedings of the Freedman Fest, Geom. Topol. Monogr., vol. 18, Geom. Topol. Publ., Coventry, 2012, pp. 161–190. MR 3084237

work page 2012
[25]

Konstantis,A counting invariant for maps into spheres and for zero loci of sections of vector bundles, Abh

P. Konstantis,A counting invariant for maps into spheres and for zero loci of sections of vector bundles, Abh. Math. Semin. Univ. Hambg.90(2020), no. 2, 183–199. MR 4217950

work page 2020
[26]

Kosinski,Differential manifolds, Pure and Applied Mathematics, vol

A.A. Kosinski,Differential manifolds, Pure and Applied Mathematics, vol. 138, Academic Press, Inc., Boston, MA, 1993. MR 1190010

work page 1993
[27]

Larmore and E

L.L. Larmore and E. Thomas,Group extensions and principal fibrations, Math. Scand.30(1972), 227–248. MR 328935

work page 1972
[28]

Lashof,Poincar´ e duality and cobordism, Trans

R. Lashof,Poincar´ e duality and cobordism, Trans. Amer. Math. Soc.109(1963), 257–277. MR 156357

work page 1963
[29]

Lawson and N

T. Lawson and N. Naumann,Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2, Int. Math. Res. Not. IMRN (2014), no. 10, 2773–2813. MR 3214285

work page 2014
[30]

P. Li, J. Pan, and J. Wu,Cohomotopy sets of(n−1)-connected(2n+ 2)- manifolds for smalln, Trans. London Math. Soc.12(2025), no. 1, Paper No. e70010. MR 4954692

work page 2025
[31]

Li and Z

P. Li and Z. Zhu,The homotopy decomposition of the suspension of a non- simply-connected five-manifold, Proc. Roy. Soc. Edinburgh Sect. A156(2026), no. 1, 93–121. MR 5021277

work page 2026
[32]

Madsen and R.J

I. Madsen and R.J. Milgram,The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, No. 92, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1979. MR 548575

work page 1979
[33]

Milnor and J.D

J.W. Milnor and J.D. Stasheff,Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974. MR 440554

work page 1974
[34]

Mosher and M.C

R.E. Mosher and M.C. Tangora,Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, New York-London, 1968. COHOMOTOPY GROUPS IN CODIMENSIONS TWO AND THREE 47 MR 0226634

work page 1968
[35]

Mukherjee,Differential topology, second ed., Hindustan Book Agency, New Delhi; Birkh¨ auser/Springer, Cham, 2015

A. Mukherjee,Differential topology, second ed., Hindustan Book Agency, New Delhi; Birkh¨ auser/Springer, Cham, 2015. MR 3379695

work page 2015
[36]

Novikov,Homotopy properties of Thom complexes, Topological Library

S.P. Novikov,Homotopy properties of Thom complexes, Topological Library. Part 1: Cobordisms and Their Applications (S.P. Novikov and I.A. Taimanov, eds.), Series on Knots and Everything, vol. 39, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 211–250. MR 2334179

work page 2007
[37]

Peterson,Generalized cohomotopy groups, Amer

F.P. Peterson,Generalized cohomotopy groups, Amer. J. Math.78(1956), 259–

work page 1956
[38]

Rohlin,A three-dimensional manifold is the boundary of a four- dimensional one, Doklady Akad

V.A. Rohlin,A three-dimensional manifold is the boundary of a four- dimensional one, Doklady Akad. Nauk SSSR (N.S.)81(1951), 355–357. MR 48808

work page 1951
[39]

Rudyak,On Thom spectra, orientability, and cobordism, Springer Mono- graphs in Mathematics, Springer-Verlag, Berlin, 1998, With a foreword by Haynes Miller

Y.B. Rudyak,On Thom spectra, orientability, and cobordism, Springer Mono- graphs in Mathematics, Springer-Verlag, Berlin, 1998, With a foreword by Haynes Miller. MR 1627486

work page 1998
[40]

Sati and U

H. Sati and U. Schreiber,Twisted cohomotopy implies M5-brane anomaly can- cellation, Lett. Math. Phys.111(2021), no. 5, Paper No. 120, 25. MR 4318438

work page 2021
[41]

Steenrod,Products of cocycles and extensions of mappings, Ann

N.E. Steenrod,Products of cocycles and extensions of mappings, Ann. of Math. (2)48(1947), 290–320. MR 22071

work page 1947
[42]

Stong,Notes on cobordism theory, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968, Mathematical notes

R.E. Stong,Notes on cobordism theory, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968, Mathematical notes. MR 248858

work page 1968
[43]

Taylor,The principal fibration sequence and the second cohomotopy set, Proceedings of the Freedman Fest, Geom

L.R. Taylor,The principal fibration sequence and the second cohomotopy set, Proceedings of the Freedman Fest, Geom. Topol. Monogr., vol. 18, Geom. Topol. Publ., Coventry, 2012, pp. 235–251. MR 3084240

work page 2012
[44]

Thom,Quelques propri´ et´ es globales des vari´ et´ es diff´ erentiables, Comment

R. Thom,Quelques propri´ et´ es globales des vari´ et´ es diff´ erentiables, Comment. Math. Helv.28(1954), 17–86. MR 61823

work page 1954
[45]

Wu,On Pontrjagin classes

W.-T. Wu,On Pontrjagin classes. II, Acta Math. Sinica4(1954), 171–199. MR 77933 Department of Mathematics, Great Bay University,523000, Dongguan, China Email address:lipcaty@outlook.com Institute of Mathematics, Chinese Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences,100190, Beijing, China Email address:pjz@amss.ac.cn...

work page 1954