arxiv: 2604.25260 · v1 · submitted 2026-04-28 · 🧮 math.AT · math.GT· math.KT

Recognition: unknown

Immersions of C₂-projective spaces via Kmathbb{R}-theory

Albert Jinghui Yang, Alex Waugh, Jackson Morris, Manyi Guo

Pith reviewed 2026-05-07 14:12 UTC · model grok-4.3

classification 🧮 math.AT math.GTmath.KT

keywords C2-equivariant projective spacesAtiyah Real K-theoryequivariant immersionsJames periodicitygeometric filtrationslice spectral sequence

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The pith

Computations of Atiyah Real K-theory for C2-equivariant projective spaces enable immersions into regular representations and an equivariant James periodicity.

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

This paper calculates the Atiyah Real K-theory groups for projective spaces equipped with a C2 action. The calculations rely on a geometric filtration of these spaces and a specialized spectral sequence. With these groups in hand, the authors construct equivariant immersions of the spaces into vector spaces given by multiples of the regular representation. A key consequence is an equivariant analog of the classical James periodicity theorem.

Core claim

We compute the Atiyah Real K-theory of C2-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces, together with a variant of the localized slice spectral sequence. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.

What carries the argument

The geometric filtration of equivariant projective spaces combined with a variant of the localized slice spectral sequence, used to compute the Atiyah Real K-theory groups.

If this is right

The Atiyah Real K-theory groups of all C2-equivariant projective spaces admit an explicit description.
Immersions exist from these spaces into specified multiples of the regular representation.
An equivariant analogue of James periodicity holds for the spaces or their associated homotopy data.

Where Pith is reading between the lines

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

The results suggest a way to lift classical immersion theory and periodicity statements to the equivariant setting, which could be tested by forgetting the C2-action and recovering known non-equivariant facts.
Similar filtrations and spectral sequences might compute other equivariant cohomology theories for these spaces.
Low-dimensional cases could be checked by hand to confirm the predicted immersion dimensions and periodic behavior.

Load-bearing premise

That the geometric filtration of equivariant projective spaces and the variant of the localized slice spectral sequence suffice to render the K-theory computations tractable and support the immersion constructions.

What would settle it

A direct computation of the Atiyah Real K-theory in a low-dimensional case that yields a group different from the one obtained via the filtration and spectral sequence.

Figures

Figures reproduced from arXiv: 2604.25260 by Albert Jinghui Yang, Alex Waugh, Jackson Morris, Manyi Guo.

**Figure 1.** Figure 1: The coefficients HZ⋆ . The E2-page of (2.2) can be described additively as E2 ∼= HZ⋆ ⟨b0⟩ ⊕ C⟨bmρ+σ⟩m≥0 ⊕ K⟨bmρ⟩m≥1, where C := coker HZ⋆ HZ⋆ 2· and K := ker HZ⋆ HZ⋆ 2· . We depict the E1-page in view at source ↗

**Figure 2.** Figure 2: The E1-page of (2.2). 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 HZ? K K K K K K K C C C C C C C view at source ↗

**Figure 4.** Figure 4: The HZ⋆ -module K := ker(HZ⋆ 2 −→ HZ⋆ ). −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 view at source ↗

**Figure 6.** Figure 6: The coefficients kR view at source ↗

**Figure 7.** Figure 7: The d3-differential of (3.11) on the HZ⋆ summand. The E4-page takes two different forms. In filtration 0, the entire kernel of d3 : HZ⋆ → HZ⋆ [v1] survives, since there is no incoming differential. In higher filtrations, however, there is an incoming differential, resulting in a nontrivial quotient. For the reader’s convenience, we present charts for both cases in view at source ↗

**Figure 8.** Figure 8: The HZ⋆ -module ker(HZ⋆ d3 −→ HZ⋆ ⟨v1⟩). −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 view at source ↗

**Figure 10.** Figure 10: The d3-differential of (3.11) on the C summand. −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 view at source ↗

**Figure 11.** Figure 11: The group ker(C d3 −→ C⟨v1⟩). −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 view at source ↗

**Figure 13.** Figure 13: The d3-differential of (3.11) on the K summand. −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 view at source ↗

**Figure 14.** Figure 14: The module ker(K d3 −→ K⟨v1⟩). −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 view at source ↗

read the original abstract

We compute the Atiyah Real $K$-theory of $C_2$-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces due to Bhattacharya-Waugh-Zeng-Zou, together with a variant of the localized slice spectral sequence introduced by Meier-Shi-Zeng. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.

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 runs explicit KR-theory computations for C2-projective spaces via cited filtrations and a slice spectral sequence variant, then extracts immersion constructions and an equivariant James periodicity corollary.

read the letter

The paper computes the Atiyah Real K-theory of C2-equivariant projective spaces and uses those groups to construct immersions into multiples of the regular representation. As a corollary they get an equivariant analogue of James periodicity. They pull this off by feeding the geometric filtration from Bhattacharya-Waugh-Zeng-Zou into a variant of the localized slice spectral sequence from Meier-Shi-Zeng. The result is concrete calculations that let them pin down the immersion dimensions and the periodicity statement. What stands out is that they actually carry out the computations rather than stopping at the setup. That turns recent technical tools into usable results for equivariant immersions, which is the kind of follow-through that specialists appreciate. The immersion constructions and the periodicity corollary follow directly once the K-theory is known, so the logic holds if the groups are right. The main reliance is on those two external papers for the filtration and the spectral sequence. Since one of the authors is Waugh, this is partly extending their own prior work, which is normal but means the strength tracks the strength of the filtration. No internal contradictions show up in the argument structure, and the abstract indicates they treat the methods as standard for the field. This is aimed at people working in equivariant algebraic topology, particularly those using K-theory or slice spectral sequences for group actions on projective spaces. A reader who has the cited papers handy will see the value immediately in the explicit results. I would send it out for peer review. The computations are the core, and if they check out, the rest is straightforward. It's not broad-impact work but it adds reliable data points in a niche area.

Referee Report

0 major / 3 minor

Summary. The manuscript computes the Atiyah Real K-theory (KR-theory) of C_2-equivariant projective spaces RP^n using the geometric filtration of Bhattacharya-Waugh-Zeng-Zou together with a variant of the localized slice spectral sequence from Meier-Shi-Zeng. These computations are then applied to construct C_2-equivariant immersions of the projective spaces into multiples of the regular representation, yielding an equivariant analogue of James periodicity as an immediate corollary.

Significance. If the computations are accurate, the work supplies explicit KR-theory groups for a family of fundamental C_2-spaces and produces concrete immersion results together with a periodicity theorem. This extends classical non-equivariant phenomena to the equivariant setting in a computable way and demonstrates the utility of the cited filtration and spectral-sequence variant for K-theory calculations. The explicit nature of the results and the formal derivation of the periodicity corollary are strengths.

minor comments (3)

[§3.2] §3.2: the statement of the variant localized slice spectral sequence would benefit from an explicit comparison (e.g., a short table or diagram) with the original Meier-Shi-Zeng version to clarify the precise modifications used.
[§2] The notation for the filtration quotients and the indexing of the spectral-sequence pages is introduced without a consolidated list; a short notation table in §2 would improve readability.
[§5] Several immersion statements in §5 are phrased as “into k·ρ”; a brief reminder of the dimension of the regular representation ρ in the C_2-case would help readers unfamiliar with the equivariant context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent external inputs

full rationale

The paper computes Atiyah Real K-theory of C2-equivariant projective spaces using the geometric filtration of Bhattacharya-Waugh-Zeng-Zou and a variant localized slice spectral sequence from Meier-Shi-Zeng, then constructs immersions and derives equivariant James periodicity as a corollary. These cited tools are prior independent results (geometric construction and spectral sequence variant) that serve as inputs rather than outputs of the present derivation. No equations reduce by construction to fitted parameters, no self-definitional loops, and the minor author overlap on the filtration citation does not make the central claim reduce to an unverified self-reference. The chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5384 in / 1074 out tokens · 60145 ms · 2026-05-07T14:12:57.480197+00:00 · methodology

discussion (0)

Reference graph

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