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Filtered products and boundary-preserving compression in complex cobordism
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Abstract
We prove that, in sufficiently large smash powers, an $MU$-null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space. When the torus slot bundle embeds continuously into a trivial bundle of rank less than twice the source rank, the compression places a positive proportion of slots on their boundaries in arbitrarily large powers and fixes every original boundary slot exactly. An example at equality shows that the strict rank inequality cannot be removed.
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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.
Follow-up works
- George A. Elliott and Zhuang Niu, Mean dimension and radius of comparison (arXiv:2610.11143v1, 2026-10-08). This follow-up, whose authors disclose ChatGPT use, takes the torus-compression theorem (Theorem 1.3) as its main technical input and adapts the companion's arguments to diagonal AH algebras and free minimal amenable actions with the uniform Rokhlin property, with Cuntz comparison of open sets additionally assumed for dynamical equality. [family-level]
Digests and expositions
- AI Proved What?, Exactly one half? The Phillips–Toms conjecture (Episode 302, October 2026). This AI-assisted video and transcript explain both papers' claims, the role of cobordism compression in reducing cover order, and the Elliott–Niu follow-up. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 302
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026,
author = {{OpenAI}},
title = {{Filtered products and boundary-preserving compression in complex cobordism}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf}{OAI:Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026}},
year = {2026}
}Family 302: Radius of comparison equals half the mean dimension
For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of $C(X)\rtimes_h\mathbb Z$ equals $\tfrac12\mathrm{mdim}(X,h)$, including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.
Other papers in this family:
- Radius of comparison equals half the mean dimension · September 25, 2026