OpenAI research collection · 5 manuscripts · Source checked 2026-10-19
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> 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.
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For every n ≥ 1 and prime p> n+1, we construct a 2^n-stage ordered filtration of L_{n-1}L_{K(n)}S_p^\wedge with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.
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For every prime p ≥ 5, we construct an explicit eight-stage filtration of L_2L_{K(3)}\mathbb S_p^\wedge by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local…
For every prime p ≥ 5, we construct an explicit eight-stage filtration of L_2L_{K(3)}\mathbb S_p^\wedge by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.
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For every prime p ≥ 5, the canonical map L_0L_{K(3)}S→ L_0L_{K(2)}L_{K(3)}S for the sphere spectrum S is nonzero on π_(−3). Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying E(2)-local spectra without specified summand maps.
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At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of E(2)-local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly…
At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of E(2)-local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.
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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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