{"id":"openai-math/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents","pdf":"https://github.com/openai/math/blob/main/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026","tags":["algebraic-topology","chromatic-homotopy","sphere-spectrum"],"score":0,"title":"Counterexamples to weak chromatic splitting: sphere kernels and descent exponents","domain":"Topology","family":"318","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.GT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"formalized":false,"family_title":"Chromatic splitting: filtrations and counterexamples"}