Browse › Topology › 318 Chromatic splitting: filtrations and counterexamples
Counterexamples to weak chromatic splitting: sphere kernels and descent exponents
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Abstract
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 $\beta_1^{(p-1)^2}$ gives a nonzero kernel class in both ranges.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- Claude (proof), Agnès Beaudry and Tomer Schlank (human write-up), Weak chromatic splitting fails at n = p for every odd prime p (hexagon:2610.00138, v1 7 October 2026; announced 2026-09-21). Theorem 1.1 uses the same $\beta_1^{(p-1)^2}$ kernel for $p\geq5$ and $p\leq n<\sqrt{2}(p-1)$, covering both height ranges here and additionally treating $(n,p)=(3,3)$; it does not mention the OpenAI papers. [AI-generated per source]
Digests and expositions
- Latent Seminar, Chromatic splitting: filtrations, attachments and sphere kernels (2026-10-07). Explains the named sphere kernels and separate descent-exponent argument, without independently verifying the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 318
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026,
author = {{OpenAI}},
title = {{Counterexamples to weak chromatic splitting: sphere kernels and descent exponents}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf}{OAI:Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026}},
year = {2026}
}Family 318: Chromatic splitting: filtrations and counterexamples
Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and $p+1$ (p ≥ 7). Nevertheless, for n ≥ 1 and $p\gt n+1$, the overlap $L_{n-1}L_{K(n)}S_p^\wedge$ admits a $2^n$-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of $E(2)$-local modules over the derived completed sphere.
Other papers in this family:
- Filtered chromatic splitting at generic primes · September 25, 2026
- The height-three chromatic overlap: an explicit filtration and its attachments · September 27, 2026
- A rational obstruction to strong chromatic splitting at height three · September 25, 2026
- Failure of finite assembly for a chromatic overlap at the prime three · September 25, 2026