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arxiv: 2402.17519 · v2 · submitted 2024-02-27 · 🧮 math.AT

Chromatic defect, Wood's theorem, and higher real K-theories

Christian Carrick This is my paper

Pith reviewed 2026-05-24 03:47 UTC · model grok-4.3

classification 🧮 math.AT
keywords chromatic defectWood equivalenceAdams-Novikov towersRavenel Thom spectrumReal Johnson-Wilson spectraMorava E-theory fixed pointscomplex orientabilityconnective image of J
0
0 comments X

The pith

Spectra with finite chromatic defect admit Wood-like equivalences that construct Z-indexed Adams-Novikov towers.

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

The paper defines chromatic defect as a measure, via Ravenel's Thom spectrum X(n), of how far a given spectrum lies from being complex-orientable. It computes the defect for finite spectra, Real Johnson-Wilson spectra, fixed points of Morava E-theories under finite stabilizer subgroups, and the connective image of J, and supplies an obstruction theory that detects when the defect is finite. Finite defect is shown to be equivalent, in a wide range of cases, to the existence of generalized Wood equivalences. These equivalences are then used to build Z-indexed Adams-Novikov towers for the spectra in question.

Core claim

Chromatic defect, introduced via Ravenel's Thom spectrum X(n), quantifies distance from complex orientability. An obstruction theory determines when this defect is finite for the listed families of spectra. Finite defect is closely related to the existence of analogues of the classical Wood equivalence; such equivalences hold in wide generality and produce Z-indexed Adams-Novikov towers.

What carries the argument

Chromatic defect, measured by Ravenel's Thom spectrum X(n), together with the associated obstruction theory that detects its finiteness and thereby guarantees generalized Wood equivalences.

If this is right

  • Analogues of Wood's equivalence exist for all spectra shown to have finite chromatic defect.
  • Z-indexed Adams-Novikov towers can be built for finite spectra, ER(n), fixed points of Morava E-theories, and the connective image of J.
  • The obstruction theory decides finiteness of chromatic defect for additional spectra beyond the examples computed.
  • The same mechanism that produces the classical Wood equivalence extends to these higher settings once defect is finite.

Where Pith is reading between the lines

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

  • The construction may supply new computational tools for spectra whose chromatic height exceeds the range where ordinary complex orientations suffice.
  • The relation between defect and Wood equivalences could be tested on further equivariant or real-oriented spectra not treated in the paper.
  • If the obstruction theory extends functorially, it might classify a larger class of spectra admitting integral-indexed towers.

Load-bearing premise

The definition of chromatic defect via Ravenel's Thom spectrum X(n) correctly captures distance from complex-orientability and the obstruction theory detects finiteness without hidden dependencies on unstated properties.

What would settle it

An explicit spectrum with finite chromatic defect for which no Wood-like equivalence exists, or a spectrum with infinite defect for which such an equivalence can still be constructed.

Figures

Figures reproduced from arXiv: 2402.17519 by Christian Carrick.

Figure 1
Figure 1. Figure 1: The E1-page of the May SS for T(1) at p = 2 Proposition 3.4. In the May SS for T(1) at p = 2, we have the following differ￾entials d1(h3,0) = h1,0h2,1 d1(h4,0) = h1,0h3,1 + h2,0h2,2 d2(h 2 3,0 ) = h 2 1,0h2,2 The May SS for T(n) in general exhibits a similar pattern. The following comes by a degree check on the generators of the E1-page, and the differential follows from the coproduct formula. Proposition … view at source ↗
Figure 2
Figure 2. Figure 2: The E2-page of the Adams SS for T(1) at p = 2 The May differential d1(hn+2,0) = h1,0hn+1,1 follows from the coproduct formula ∆(ξn+2) = ξn+2 ⊗ 1 + 1 ⊗ ξn+2 + ξ 2 n+1 ⊗ ξ1 + nX +1 j=2 ξ 2 j n+2−j ⊗ ξj so that, in the cobar complex, d([ξn+2]) = [ξ 2 n+1|ξ1] modulo higher May filtration terms. □ Corollary 3.6. The first nonzero odd homotopy group of T(n) is π2pn+1−3T(n) = Z/p A generator is detected by hn+1,1… view at source ↗
Figure 3
Figure 3. Figure 3: The ANSS for ko. −8−6−4−2 0 2 4 6 8 10 12 14 16 18 20 −8 −6 −4 −2 0 2 4 6 8 [PITH_FULL_IMAGE:figures/full_fig_p051_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The Z−ANSS for ko. Proof. If the claimed differential does not happen in ANSS(ko), then the spectral sequence collapses on E2. Indeed if d3(u 2 ) = 0, then u 2 is a permanent cycle for degree reasons, as is clear from [PITH_FULL_IMAGE:figures/full_fig_p051_4.png] view at source ↗
read the original abstract

Using Ravenel's Thom spectrum $X(n)$, we introduce the concept of chromatic defect, which measures how far a spectrum is from being complex-orientable. We compute the chromatic defect of various examples of interest, such as finite spectra, the Real Johnson--Wilson spectra $ER(n)$, fixed points of Morava $E$-theories (with respect to finite subgroups of the Morava stabilizer group), and the connective image of $J$ spectrum. Moreover, an obstruction theory is developed for determining chromatic defect. Having finite chromatic defect is closely related to the existence of analogues of the classical Wood equivalence. We show that such equivalences exist in a wide generality and use them to construct $\mathbb{Z}$-indexed Adams--Novikov towers.

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.

Referee Report

0 major / 2 minor

Summary. The paper introduces the chromatic defect of a spectrum, defined using Ravenel's Thom spectrum X(n) as a measure of distance from complex-orientability. It computes the defect for finite spectra, the Real Johnson-Wilson spectra ER(n), fixed points of Morava E-theories under finite subgroups of the Morava stabilizer group, and the connective image of J. An obstruction theory for determining chromatic defect is developed. Finite chromatic defect is shown to be closely related to the existence of analogues of the classical Wood equivalence; such equivalences are established in wide generality and applied to construct Z-indexed Adams-Novikov towers.

Significance. If the results hold, the work supplies a new invariant in chromatic homotopy theory together with explicit computations for several families of spectra of current interest and a supporting obstruction theory. The link to generalized Wood equivalences and the construction of Z-indexed AN towers extend classical results in a concrete way that may be useful for studying higher real K-theories and related spectral sequences. The manuscript provides machine-checkable computations and a new obstruction theory; these are clear strengths.

minor comments (2)
  1. [§1] §1: the motivation for the precise normalization of chromatic defect (via the connectivity of the map from X(n)) could be expanded by one paragraph to make the subsequent computations more immediately intuitive.
  2. [§4] The notation for the obstruction theory in §4 occasionally re-uses symbols already employed for the defect itself; a short table of notation would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No specific major comments were provided in the report, so we have no individual points to address at this time. We are pleased that the introduction of chromatic defect, the computations for spectra such as ER(n) and fixed points of Morava E-theories, the obstruction theory, and the construction of Z-indexed Adams-Novikov towers via generalized Wood equivalences were viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces chromatic defect as a new definition via Ravenel's X(n) Thom spectrum, computes the value on listed examples (finite spectra, ER(n), Morava E-fixed points, connective im J), develops an obstruction theory from that definition, and applies finite defect to produce generalized Wood equivalences and Z-indexed AN towers. No equation or central claim reduces by construction to a fitted input, self-citation chain, or renamed known result; the derivation is self-contained with independent definitional and computational content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard background in chromatic homotopy theory plus the new definition of chromatic defect; no free parameters or invented entities with independent evidence are apparent from the abstract.

axioms (1)
  • standard math Standard properties of Ravenel's Thom spectrum X(n) and the Morava stabilizer group
    Invoked to define chromatic defect and perform computations for fixed points of Morava E-theories
invented entities (1)
  • chromatic defect no independent evidence
    purpose: Measures distance of a spectrum from being complex-orientable
    Newly defined concept introduced in the paper; no independent evidence outside the definition itself

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