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

Christian Carrick

TL;DR

The paper introduces chromatic defect as a quantitative measure of how far a spectrum is from being complex-orientable, using Ravenel's X(n) filtration and a flexible orientability notion that extends beyond ring spectra. It develops obstruction theory via the elements $\chi_{m+1}$ and links finite chromatic defect to Wood-type phenomena, providing both algebraic criteria (via $\mathcal{A}(n)_*$ and $\mathcal{E}(n)_*$ Ext) and algebro-geometric criteria (Hopkins' stacks) for descent along Ravenel's filtration. The authors compute chromatic defect for key families such as the Real Johnson–Wilson theories $\mathrm{ER(n)}$, the Hopkins–Miller higher real K-theories $\mathrm{EO}_n(G)$, and related fp spectra, establishing explicit defect formulas like $\Phi(\mathrm{ER(n)})=2^n$ and $\Phi(\mathrm{EO}_n(G))=p^{N(G)}$. They also develop a $\mathbb{Z}$-indexed Adams–Novikov framework that parallels Tate cohomology and apply it to KO, TMF, and related cases, clarifying how Wood-type structure governs the ANSS and enabling more tractable computations. Overall, the work connects chromatic defects, Wood-type phenomena, and stack-theoretic descent to yield concrete tools for understanding when and how stable homotopy data can be recovered from structured, finite-stage decompositions, with implications for fixed-point theories and higher real K-theories."

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.

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

TL;DR

The paper introduces chromatic defect as a quantitative measure of how far a spectrum is from being complex-orientable, using Ravenel's X(n) filtration and a flexible orientability notion that extends beyond ring spectra. It develops obstruction theory via the elements and links finite chromatic defect to Wood-type phenomena, providing both algebraic criteria (via and Ext) and algebro-geometric criteria (Hopkins' stacks) for descent along Ravenel's filtration. The authors compute chromatic defect for key families such as the Real Johnson–Wilson theories , the Hopkins–Miller higher real K-theories , and related fp spectra, establishing explicit defect formulas like and . They also develop a -indexed Adams–Novikov framework that parallels Tate cohomology and apply it to KO, TMF, and related cases, clarifying how Wood-type structure governs the ANSS and enabling more tractable computations. Overall, the work connects chromatic defects, Wood-type phenomena, and stack-theoretic descent to yield concrete tools for understanding when and how stable homotopy data can be recovered from structured, finite-stage decompositions, with implications for fixed-point theories and higher real K-theories."

Abstract

Using Ravenel's Thom spectrum , 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 , fixed points of Morava -theories (with respect to finite subgroups of the Morava stabilizer group), and the connective image of 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 -indexed Adams--Novikov towers.
Paper Structure (26 sections, 82 theorems, 185 equations, 4 figures)

This paper contains 26 sections, 82 theorems, 185 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results and outline of the paper
  3. Some numerology and questions
  4. Acknowledgments
  5. Orientability and Ravenel's X(n)
  6. Complex-orientability
  7. X(n)-orientations and chromatic defect
  8. Wood-types
  9. The T(n)'s and Fp-homology
  10. The Adams spectral sequence of T(n)
  11. The chromatic defect of finite spectra
  12. P(n)-free complexes
  13. Chromatic defect and Wood equivalences for fp spectra
  14. Chromatic defect for fp spectra
  15. Wood-type fp spectra
  16. ...and 11 more sections

Key Result

Theorem 1.3

A spectrum $E$ is complex-orientable if and only if it is a weak $\mathrm{MU}$-module; that is, the unit map $E\to \mathrm{MU}\otimes E$ admits a retraction.

Figures (4)

  • Figure 1: The $E_1$-page of the May SS for $\mathrm{T(1)}$ at $p=2$
  • Figure 2: The $E_2$-page of the Adams SS for $\mathrm{T(1)}$ at $p=2$
  • Figure 3: The ANSS for $\mathrm{ko}$.
  • Figure 4: The ${\mathbb Z}\mathrm{-ANSS}$ for $\mathrm{ko}$.

Theorems & Definitions (188)

  • Definition 1.1: \ref{['def:comporient']}
  • Remark 1.2
  • Theorem 1.3: \ref{['thm:weakMU']}
  • Definition 1.4: \ref{['def:wood']}, \ref{['def:defect']}
  • Definition 1.5: \ref{['def:defect']}
  • Theorem 1.6: \ref{['cor:woodimpliesfinitedefect']}
  • Theorem 1.7: \ref{['prop:chi']}
  • Corollary 1.8: \ref{['cor:finitespectradefect']}
  • Theorem 1.9: \ref{['cor:fpeven']}
  • Theorem 1.11: \ref{['thm:fpwoodsplittings']}
  • ...and 178 more