Chromatic splitting: filtrations and counterexamples

1 · The essentials

These manuscripts examine whether a complicated mathematical object can be separated into a predicted collection of simpler pieces. The objects come from topology, and the questions depend on a prime number and a level called height. The main manuscript shows that, when the prime is greater than the height plus one, the predicted pieces can be arranged into a step-by-step construction. At height n, that construction has $2^{n}$ stages. But building an object from pieces does not mean those pieces separate cleanly: the connections between them can matter. Companion manuscripts show that proposed splitting formulas fail in several specified ranges. At prime three and height three, even a broader kind of finite construction fails in the setting considered. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of assembling a cabinet from a specified set of panels: having the right panels does not mean the finished cabinet is a loose stack of them. The filtration records an assembly in stages, while the attachment information records connections that can prevent the desired separation.

Why it matters

The family distinguishes having the predicted pieces from having the predicted splitting. It retains a precise staged construction when $p > n + 1$ , while giving counterexamples to strong and weak splitting in stated ranges. The prime-three result also places a limit on finite assembly, specifically in the module category named in that manuscript.

2 · Going further

The main object is the chromatic overlap $L_{n - 1} L_{K (n)} S_{p}^{\land}$ . The question is whether the classical chromatic-splitting pieces describe this object as a splitting, or only as layers joined together. For every $n \geq 1$ and prime $p > n + 1$ , the main manuscript constructs an ordered filtration with $2^{n}$ stages and the classical chromatic-splitting cofibers. The map from the first stage to the overlap is the canonical localization unit.

The height-three companion makes this construction explicit for every prime $p \geq 5$ : it gives eight stages for $L_{2} L_{K (3)} S_{p}^{\land}$ . Two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem implies that the first height-one attachment is nonzero. Thus the abstracts describe a construction that retains the predicted layers while tracking how they are attached; they do not provide the full argument for the general filtration.

The rational-obstruction manuscript shows that the canonical map $L_{0} L_{K (3)} S \to L_{0} L_{K (2)} L_{K (3)} S$ is nonzero on $π_{- 3}$ for every prime $p \geq 5$ . It concludes that the height-three strong splitting formula fails even as an equivalence of underlying $E (2)$ -local spectra, without specified summand maps. The weak-splitting manuscript instead uses the nonzero kernel class $β_{1}^{(p - 1)^{2}}$ to show that the canonical weak splitting map for the derived $p$ -completed sphere has no homotopy retraction at height $p$ for $p \geq 5$ , and at height $p + 1$ for $p \geq 7$ .

At prime three and height three, another companion shows that the overlap cannot be assembled from the rational, height-one, and height-two local spheres using finitely many sums, shifts, cofibers, and retracts. This statement is specifically about $E (2)$ -local modules over the derived 3-complete sphere. It addresses finite assembly rather than merely the failure of a particular splitting formula.

Useful notions

Prime numbers and height parameters
Spectra and the sphere spectrum
Localization and completion
Filtrations and cofibers
Homotopy groups and retractions

What was known before / what is new

Before

The source refers to classical chromatic-splitting cofibers and a height-three chromatic-splitting pattern, which supply the predicted layers. It also distinguishes strong and weak chromatic splitting and an ordinary finite-assembly question. The supplied abstracts do not give a historical account or a list of earlier established cases.

New

The main manuscript shows that the predicted cofibers form a $2^{n}$ -stage ordered filtration whenever $n \geq 1$ and $p > n + 1$ , with the prescribed canonical first-stage map. The height-three companion supplies explicit attachments, while the rational-obstruction companion shows failure of strong splitting for $p \geq 5$ . The other companions show failure of finite assembly in the specified prime-three module category and failure of weak splitting at the stated heights $p$ and $p + 1$ .

Limits and assumptions

The general filtration requires $n \geq 1$ and $p > n + 1$ ; it is not a splitting assertion. The explicit height-three filtration and strong-splitting counterexample require $p \geq 5$ . The finite-assembly obstruction is restricted to prime three, height three, and $E (2)$ -local modules over the derived 3-complete sphere. The weak-splitting counterexamples concern the derived $p$ -completed sphere at height $p$ for $p \geq 5$ and height $p + 1$ for $p \geq 7$ . The source does not settle the remaining primes, heights, or categorical settings. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Chromatic overlap: The object $L_{n - 1} L_{K (n)} S_{p}^{\land}$ studied by the main filtration result.
Filtration: An ordered, staged construction whose successive layers are the specified cofibers.
Attachments: The connecting data describing how the layers in the filtration are joined.
Finite assembly: Construction from the stated local spheres using finitely many sums, shifts, cofibers, and retracts in the specified category.
Homotopy retraction: A map back that undoes the original map up to homotopy.

Keywords : Chromatic splitting · Filtrations · Counterexamples

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

27/09/2026

To be verified

For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height $p + 1$ for every prime p ≥ 7. The classical sphere product $β_{1}^{(p - 1)^{2}}$ gives a nonzero kernel class in both ranges.

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