Realification of stably trivial vector bundles

Guy Boyde; Niall Taggart

arxiv: 2606.26837 · v1 · pith:5YBGAUYVnew · submitted 2026-06-25 · 🧮 math.AT

Realification of stably trivial vector bundles

Guy Boyde , Niall Taggart This is my paper

Pith reviewed 2026-06-26 01:55 UTC · model grok-4.3

classification 🧮 math.AT

keywords stably trivial bundlesrealificationstabilisationcomplex projective spacesspheresgroup homomorphismsWeiss calculusstable homotopy theory

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The pith

Stably trivial complex bundles over projective spaces and spheres carry a group structure making realification and stabilisation homomorphisms.

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

The paper shows that when the corank is small enough, the stably trivial complex vector bundles over complex projective spaces and spheres admit a natural group structure. Under this structure both realification to the underlying real bundle and stabilisation by a trivial line bundle become group homomorphisms. The authors then compute the resulting maps in a range by translating the question, via Weiss calculus, into stable homotopy theory and building on prior enumerations of the bundles themselves.

Core claim

The set of stably trivial complex vector bundles over CP^n and spheres has a natural group structure when the corank is small enough; with respect to this group structure the realification map and the stabilisation map are both group homomorphisms. These homomorphisms are computed explicitly in a range by reducing the problem to stable homotopy groups via Weiss calculus.

What carries the argument

The natural group structure on the set of stably trivial complex vector bundles of sufficiently small corank, under which realification and stabilisation become homomorphisms.

If this is right

Realification induces well-defined maps between the computed groups of complex and real stably trivial bundles.
Stabilisation maps between different coranks are homomorphisms and can be tracked through the stable homotopy computations.
The image of each homomorphism can be read off from known stable homotopy groups in the relevant range.
These computations give concrete information on which real bundles arise from complex ones over the given base spaces.

Where Pith is reading between the lines

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

The same group structure may allow similar statements for other base spaces once their stably trivial bundles are enumerated.
The reduction to stable homotopy suggests that further computations could be obtained by applying additional calculus functors or spectral sequences.
If the group structure extends to larger coranks, the homomorphism property would give relations in real K-theory of these spaces.

Load-bearing premise

There exists a natural group structure on the set of these bundles when the corank is small enough.

What would settle it

An explicit pair of stably trivial complex bundles over CP^3 or S^6 whose sum fails to be stably trivial, or whose realifications fail to satisfy the homomorphism property under the proposed operation.

Figures

Figures reproduced from arXiv: 2606.26837 by Guy Boyde, Niall Taggart.

**Figure 1.** Figure 1: The action of realification and stabilisation on stably trivial bundles over complex projective space for small corank after localisation at 2. For compactness we write Vect0 k,𝑑 := Vect0 k,𝑑 (CPℓ )(2) , and use the convention that any map that is not written is zero [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: The action of realification and stabilisation on stably trivial bundles over even spheres 𝑆 2𝑛 for 𝑛 ≥ 8 after localisation at 2. For compactness we write Vect0 k,𝑑 := Vect0 k,𝑑 (𝑆 2𝑛 )(2) , and use the convention that any map that is not written is zero [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: The action of realification and stabilisation on stably trivial bundles over odd spheres 𝑆 2𝑛+1 for 𝑛 ≥ 8 after localisation at 2. For compactness we write Vect0 k,𝑑 := Vect0 k,𝑑 (𝑆 2𝑛+1 )(2) , and use the convention that any map that is not written is zero. As with above, we may make several deductions about vector bundles over spheres from Figs. 2 and 3. Note first that (since we know for𝑚 ≥ 2𝑑 that 𝜋𝑚 (… view at source ↗

**Figure 4.** Figure 4: The realification and truncation maps on even spheres in low codimensions after localisation at 2 (proven as Lemma 4.14). For compactness we omit zero maps, and abuse notation by writing 𝜋𝑖(𝑋) = 𝜋𝑖(𝑋)(2) . ( Z(2) ⊕ Z/2 𝑛 odd Z(2) 𝑛 even Z(2) 0 𝜋2𝑛+1 (ΣCP∞ 𝑛−1 ) 𝜋2𝑛+1 (ΣCP∞ 𝑛 ) 𝜋2𝑛+1 (ΣCP∞ 𝑛+1 ) 𝜋2𝑛+1 (RP∞ 2𝑛−2 ) 𝜋2𝑛+1 (RP∞ 2𝑛−1 ) 𝜋2𝑛+1 (RP∞ 2𝑛 ) 𝜋2𝑛+1 (RP∞ 2𝑛+1 ) 𝜋2𝑛+1 (RP∞ 2𝑛+2 )    (Z/8) 2 𝑛 ≡ 1… view at source ↗

**Figure 5.** Figure 5: The realification and truncation maps on odd spheres in low codimensions after localisation at 2 (proven as Lemma 4.15). For compactness we omit zero maps, and abuse notation by writing 𝜋𝑖(𝑋) = 𝜋𝑖(𝑋)(2) . 4.1. The Adams charts. We have the following Adams charts for certain finite skeleta of the stable representing object Σ dimR (k)−1kP∞ 𝑑 Figs. 6 to 11. In the complex case these charts were computed by ha… view at source ↗

**Figure 6.** Figure 6: Adams 𝐸2-pages for ΣCP4𝑘+2 4𝑘 (left) and ΣCP4𝑘+3 4𝑘+1 (right). The possible differentials which may affect the unshaded region are indicated by a dotted blue arrow. 8𝑘 + 4 8𝑘 + 8 8𝑘 + 12 0 2 4 6 0 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 0 1 2 8𝑘 + 6 8𝑘 + 10 8𝑘 + 14 0 2 4 6 0 0 1 2 3 4 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 0 1 2 3 4 0 1 2 3 0 1 2 0 1 2 … view at source ↗

**Figure 7.** Figure 7: Adams 𝐸2-pages for ΣCP4𝑘+4 4𝑘+2 (left) and ΣCP4𝑘+5 4𝑘+3 (right). 8𝑘 8𝑘 + 4 8𝑘 + 8 0 2 4 6 0 1 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 0 1 2 8𝑘 8𝑘 + 4 8𝑘 + 8 0 2 4 6 0 1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3… view at source ↗

**Figure 8.** Figure 8: Adams 𝐸2-pages for RP8𝑘+5 8𝑘 (left) and RP8𝑘+5 8𝑘+1 (right). The possible differentials which may affect the unshaded region are indicated by a dotted blue arrow. Some explanation on interpreting these charts is in order. First, the mod-2 cohomology of RP∞ is polynomial on a single generator 𝑥 in degree 1. There is only one non-zero class in each degree, so the Steenrod square Sq𝑖 must act by Sq𝑖 (𝑥 𝑑 ) … view at source ↗

**Figure 9.** Figure 9: Adams 𝐸2-pages for RP8𝑘+7 8𝑘+2 (left) and RP8𝑘+7 8𝑘+3 (right). The possible differentials which may affect the unshaded region are indicated by a dotted blue arrow. 8𝑘 + 4 8𝑘 + 8 8𝑘 + 12 0 2 4 6 0 1 2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 0 1 2 3 4 0 1 2 0 1 8𝑘 + 4 8𝑘 + 8 8𝑘 + 12 0 2 4 6 0 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 … view at source ↗

**Figure 10.** Figure 10: Adams 𝐸2-pages for RP8𝑘+9 8𝑘+4 (left) and RP8𝑘+9 8𝑘+5 (right). The possible differentials which may affect the unshaded region are indicated by a dotted blue arrow. 8𝑘 + 6 8𝑘 + 10 8𝑘 + 14 0 2 4 6 0 1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 0 1 2 3 0 1 2 0 1 0 1 0 1 8𝑘 + 6 8𝑘 + 10 8𝑘 + 14 0 2 4 6 0 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 0 1 2 3 4 0 1 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p02… view at source ↗

**Figure 11.** Figure 11: Adams 𝐸2-pages for RP8𝑘+11 8𝑘+6 (left) and RP8𝑘+11 8𝑘+7 (right). up to a degree shift, the cohomology of RP𝑑+4 𝑑 or RP𝑑+5 𝑑 (as a module over the Steenrod algebra) depends only on the value of 𝑑 modulo 8. In Figs. 8 to 11, we have given the eight possible Adams charts, writing 𝑑 = 8𝑘 + 𝜀 for 0 ≤ 𝜀 ≤ 7. Similar considerations in the complex case produce Figs. 6 and 7. Many more complicated periodicity phen… view at source ↗

**Figure 12.** Figure 12: Adams 𝐸2-pages for the mod-2 Moore spectrum 𝑆 0 /2 (left), the space 𝐶𝑝 𝑓 appearing in the proof of Lemma 4.21 (middle), and the cofibre of 𝜂 (right). The following is immediate from the 𝐸2-page for the mod-2 Moore spectrum 𝑆 0 /2, see [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗

**Figure 13.** Figure 13: Adams 𝐸2-pages for RP∞ 8𝑘 (left) and RP∞ 8𝑘+4 (right). In both cases it is immediate that the codimension 5 homotopy group is zero. Proposition 4.23. The map 𝑟∗ : Z(2) 𝜋4𝑚+1 (ΣCP∞ 2𝑚−1 )(2) −→ 𝜋4𝑚+1 (RP∞ 4𝑚−2 )(2) Z/4 is surjective when 𝑚 is even, zero when 𝑚 ≡ 1 (mod 4), and multiplication by 2 when 𝑚 ≡ 3 (mod 4). Proof. We work 2-locally. Consider the diagram ΣCP∞ 2𝑚−2 ΣCP∞ 2𝑚−1 Σ 2CP2𝑚−2 2𝑚−2 ≃ 𝑆 … view at source ↗

read the original abstract

The set of stably trivial complex vector bundles over complex projective spaces and spheres has a natural group structure when the corank is small enough. With respect to this group structure, the operations of taking the underlying real vector bundle (realification) and of adding a trivial line bundle (stabilisation), are group homomorphisms. Building on Hu's recent enumerations of stably trivial complex bundles, we compute these homomorphisms in a range by using Weiss calculus to translate the problem to stable homotopy theory.

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

The paper computes the realification and stabilisation homomorphisms on groups of stably trivial complex bundles over CP^n and spheres in a range, by translating via Weiss calculus to stable homotopy.

read the letter

The main takeaway is that the authors compute the homomorphisms for realification and stabilisation on the groups of stably trivial complex vector bundles over complex projective spaces and spheres, in a range, after establishing that these operations are homomorphisms with respect to the natural group structure for small corank.

What is new is the explicit computation of those maps. Hu's prior enumerations listed the bundles, but this work determines how the two operations act on the groups. The translation to stable homotopy via Weiss calculus is the step that makes the calculations feasible.

The paper does well in carrying out that translation and producing concrete data on the maps. Weiss calculus is an appropriate tool here, and the approach builds directly on existing enumerations without introducing new circularity.

The soft spots are limited. The abstract states the results without showing derivation steps or the precise range, so the actual calculations need to be checked in the full text for clarity and verification. The group structure and homomorphism claims rest on standard constructions in the area, and nothing in the description suggests they fail. The range is necessarily limited, which is normal for these computations.

This is for algebraic topologists who need specific information about these bundle groups and their maps. A reader working on stable homotopy or vector bundle classifications would get direct value from the numbers. It is not broad enough to reorganize the field, but the data is the kind that gets used.

I would bring this to a reading group focused on computational topology. I would not cite it unless I needed those particular homomorphisms. It deserves peer review because it supplies verifiable computations built on prior work.

Referee Report

0 major / 3 minor

Summary. The manuscript defines a natural group structure on the set of stably trivial complex vector bundles over complex projective spaces and spheres when the corank is small. It shows that realification (passage to the underlying real bundle) and stabilization (addition of a trivial complex line) are group homomorphisms with respect to this structure. Building on Hu's enumerations, the paper uses Weiss calculus to translate the problem into stable homotopy theory and computes the resulting homomorphisms in a range.

Significance. If the claims hold, the work supplies explicit homomorphisms between groups of stably trivial bundles and stable homotopy groups, clarifying the interaction between realification and stabilization in this setting. The reduction via Weiss calculus is a methodological strength that converts an enumerative problem into one amenable to existing stable-homotopy computations.

minor comments (3)

[Abstract and §1] The precise range in which the computations are valid (e.g., the values of n and corank) is stated only informally in the abstract and introduction; an explicit statement, perhaps as a table or theorem label, would improve readability.
[§2] Notation for the group law on the set of bundles is introduced without a dedicated display equation; adding a numbered definition would make subsequent references to the operation clearer.
[§3] The manuscript cites Hu's enumerations but does not include a short self-contained summary of the input data used; a one-paragraph recap would help readers verify the starting point of the Weiss-calculus translation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the methodological contribution via Weiss calculus, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on Hu's external enumerations of stably trivial bundles and standard applications of Weiss calculus to translate into stable homotopy theory. The group structure on the set of bundles (for small corank) and the homomorphism properties of realification and stabilisation are presented as following from the natural constructions in the area, without any reduction of the central claims to fitted parameters, self-definitions, or load-bearing self-citations. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable. The natural group structure on stably trivial bundles is presupposed without derivation details.

pith-pipeline@v0.9.1-grok · 5590 in / 1035 out tokens · 20282 ms · 2026-06-26T01:55:58.495216+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 2 internal anchors

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Vector fields on spheres

[Ada62] J. F. Adams. “Vector fields on spheres”.Ann. of Math. (2)75 (1962), pp. 603–632.doi:10.2307/ 1970213. [Aro02] G. Arone. “The Weiss derivatives of BO(-) and BU(-)”.Topology41.3 (2002), pp. 451–481.doi: 10.1016/S0040-9383(01)00008-8. [Ati61] M. F. Atiyah. “Thom complexes”.Proc. London Math. Soc. (3)11 (1961), pp. 291–310.doi:10. 1112/plms/s3-11.1.29...

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arXiv:2303 . 00736 [math.AT]. [DM89] D. M. Davis and M. Mahowald. “TheSO(𝑛)-of-origin”.Forum Math.1.3 (1989), pp. 239–250. doi:10.1515/form.1989.1.239. [Fuj67] M. Fujii. “𝐾 𝑂-groups of projective spaces”.Osaka Math. J.4.1 (1967), pp. 141–149.doi:10 . 18910/6753. 50 REFERENCES [GM23] D. Gepner and L. Meier. “On equivariant topological modular forms”.Compos...

work page doi:10.1515/form.1989.1.239 1989

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Metastable complex vector bundles over complex projective spaces

[Hu23] Y. Hu. “Metastable complex vector bundles over complex projective spaces”.Trans. Am. Math. Soc.376.11 (2023), pp. 7783–7814.doi:10.1090/tran/8955. [Joh95] B. Johnson. “The derivatives of homotopy theory”.Trans. Am. Math. Soc.347.4 (1995), pp. 1295– 1321.doi:10.2307/2154811. [Kac78] H. Kachi. “Homotopy groups of symmetric spacesΓ 𝑛”.J. Fac. Sci., Sh...

work page doi:10.1090/tran/8955 2023

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Diffeomorphisms of discs and the second Weiss derivative of BTop(-)

arXiv:2109.03500[math.AT]. [Lew00] L. G. Lewis.Splitting theorems for certain equivariant spectra. Vol

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

On conjectures of Mahowald, Segal and Sullivan

Mem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2000.doi:10.1090/memo/0686. [Lin80] W. Lin. “On conjectures of Mahowald, Segal and Sullivan”.Math. Proc. Camb. Philos. Soc.87 (1980), pp. 449–458.doi:10.1017/S0305004100056887. [LM24] W. Lück and T. Macko.Surgery theory. Foundations. With contributions by Diarmuid Crowley. Vol

work page doi:10.1090/memo/0686 2000

[7] [7]

A short proof of the James periodicity of𝜋 𝑘+𝑝 (𝑉𝑘+𝑚,𝑚)

Grundlehren Math. Wiss. Cham: Springer, 2024.doi:10.1007/978-3-031-56334-8. [Mah65] M. Mahowald. “A short proof of the James periodicity of𝜋 𝑘+𝑝 (𝑉𝑘+𝑚,𝑚)”.Proc. Amer. Math. Soc. 16 (1965), p. 512.doi:10.2307/2034684. [Mas61] W. S. Massey. “Obstructions to the existence of almost complex structures”.Bull. Am. Math. Soc.67 (1961), pp. 559–564.doi:10.1090/S0...

work page doi:10.1007/978-3-031-56334-8 2024

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A rational model for the fiberwise THH transfer I: Sullivan algebras

arXiv:2604.02516[math.AT]. [Opi24] M. P. Opie.Enumerating complex rank𝑛vector bundles onC𝑃 𝑛+1

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

Rational Pontryagin classes and functor calculus

arXiv:2410 . 23520 [math.AT]. [RW16] R. Reis and M. Weiss. “Rational Pontryagin classes and functor calculus”.J. Eur. Math. Soc. (JEMS)18.8 (2016), pp. 1769–1811.doi:10.4171/JEMS/629. [San64] B. J. Sanderson. “Immersions and embeddings of projective spaces”.Proc. London Math. Soc. (3)14 (1964), pp. 137–153.doi:10.1112/plms/s3-14.1.137. [Sch18] S. Schwede....

work page doi:10.4171/jems/629 2016

[10] [10]

A classification theorem for fibre spaces

New Math. Monogr. Cambridge: Cambridge Uni- versity Press, 2018.doi:10.1017/9781108349161. [Sta63] J. D. Stasheff. “A classification theorem for fibre spaces”.Topology2 (1963), pp. 239–246.doi: 10.1016/0040-9383(63)90006-5. [Tag21] N. Taggart. “Comparing the orthogonal and unitary functor calculi”.Homology Homotopy Appl.23.2 (2021), pp. 227–256.doi:10.431...

work page doi:10.1017/9781108349161 2018