Weak chromatic splitting at nongeneric heights and primes
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Does the canonical chromatic localization map Lₙ₋₁X→Lₙ₋₁L_K₍ₙ₎X admit a homotopy retraction for the derived p-completed sphere X? The weak chromatic-splitting prediction includes completions of finite spectra.
Why it matters and what counts as progress
Why it matters: Failure of this weak retraction is stronger than failure of a more detailed summand formula.
What counts as progress: A retraction theorem or a nonzero kernel class obstructing it, with height and prime restrictions explicit, including the proposed n=p and n=p+1 families.
How the problem developed
- 25 Jan 2019 · Conjecture [1]
Tobias Barthel and Agnès Beaudry Surveyed the weak chromatic splitting prediction and the then-known low-height range.
- 27 Sept 2026 · Reported claim [2]
OpenAI Reported failure of retraction for the canonical map of derived p-completed spheres at n=p for p≥5 and n=p+1 for p≥7, with the stated beta-family description of the kernel.
AI contribution
Counterexamples to weak chromatic splitting: sphere kernels and descent exponents
For the derived p-completed sphere X, the canonical map L_{n−1}X→L_{n−1}L_{K(n)}X is claimed to have no homotopy retraction at n=p for primes p≥5 and at n=p+1 for primes p≥7. The proposed kernel is the image of β₁^((p−1)²); at p=5 it lies in π₆₀₈L₄X.
Sources
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Publication dates and editorial changes.
OpenAI reports higher-height failures of weak chromatic splitting