∑math20

Topology · Result 314

Cyclic length and chromatic fixed-point loss

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Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.

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Cyclic length and chromatic fixed-point loss

OpenAI · September 24, 2026

For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.

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@misc{OAI:Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026,
  author = {{OpenAI}},
  title = {{Cyclic length and chromatic fixed-point loss}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf}{OAI:Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026}},
  year = {2026}
}

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