Cyclic length and chromatic fixed-point loss

Claims the exact chromatic loss between fixed points of p-group actions is a count of cyclic steps.

1 paper September 24, 2026 Explained by AI

Smith theory says a group of prime-power order acting on a space carries certain properties over to its fixed points. Chromatic versions ask how many heights of information are lost on the way. The answer was known for abelian groups and some special cases. This paper claims the exact answer for every finite p-group: the shortest chain of cyclic steps from the subgroup H up to G. It confirms an equality proposed by Kuhn and Lloyd.

First step

1952

Edwin Floyd, periodic maps and Euler characteristics, a quantitative form of Smith theory

Significance

Major

It settles the equality Kuhn and Lloyd proposed for all finite p-groups, which pins down a key part of the equivariant Balmer spectrum.

Trust

Claimed, not checked

Field

Topology

1 paper

The explanation

Take p = 2 and the group G = C₄ × C₂, with 8 elements. Here Cₖ is the cyclic group of k elements, the turns of a k-sided wheel. G has two different subgroups of order 2. One is H₁, generated by twice the generator of C₄. The other is H₂, the C₂ factor. Divide out by each one. G/H₁ is C₂ × C₂, which needs two cyclic steps to build. G/H₂ is C₄, which needs one step. The answer in this paper depends on exactly this count, 2 for H₁ and 1 for H₂.

Why count steps? Smith theory says that when a group of prime-power order acts on a finite space, mod-p acyclicity passes to the fixed pointsSource 1. Acyclic here means having no holes as seen by mod-p homology. The chromatic version uses Morava K-theories K(n), one for each height n. These see finer and finer layers of stable homotopy. The loss is how many heights higher you must check the fixed points of H to force acyclicity of the fixed points of G. A chain of cyclic steps gives an upper bound, one height per stepSource 2.

For abelian groups the answer was known. The loss equals the fewest generators of G/HSource 3. For other groups a simple lower bound can be too small. In the eight-element symmetry group of a square, a subgroup of order 2 that is not central has loss two. The simple lower bound there is oneSource 4. The obstacle was proving that the chain bound is sharp for every embedded subgroupSource 5. The usual character theory sees only abelian images, so it cannot reach this directlySource 6.

The paper claims that the optimal loss equals the shortest chain of cyclic steps from H to GSource 7. This holds at every prime and every height. So the loss does not depend on the heightSource 8. It includes the rational case at height zeroSource 9. A cyclic step of order p² still counts onceSource 10. This confirms the equality Kuhn and Lloyd proposedSource 11.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof combines delicate character theory with arithmetic constraints on profinite groups. No specialist review of the paper has been published.

What it doesn't show. This does not directly treat groups whose order is not a prime power; those need earlier reductions by Balmer and Sanders.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the proof holds. It completes a known equality for all finite p-groups. There is no Lean check and no published specialist review.

Words used

p-group: A finite group whose number of elements is a power of the prime p.
Fixed points: The points of a space that a group of symmetries leaves in place.
Smith theory: Classical results that carry homology properties of a space over to its fixed points.
Morava K-theory K(n): A homology theory that sees one chromatic height n of stable homotopy.
Acyclic: Having no holes as seen by a given homology theory.
Subnormal chain: A chain of subgroups where each is normal in the next one.
Loss: How many heights must be added at the subgroup to force acyclicity at the whole group.

The original paper

Cyclic length and chromatic fixed-point loss (PDF, opens in a new tab)
September 24, 2026

All quoted lines (11)

The classical starting point is Smith theory: for a finite complex with an action of a finite $p$ -group, mod- $p$ acyclicity passes to fixed points.
Cyclic length and chromatic fixed-point loss Back to text
The cyclic chromatic Smith theorem, applied along a chain, gives $r_n(G,H)\le\ell(G,H)$ .
Cyclic length and chromatic fixed-point loss Back to text
Barthel, Hausmann, Naumann, Nikolaus, Noel, and Stapleton determined the finite abelian $p$ -group case: the loss for $H\le G$ is the minimum number of generators of $G/H$ .
Cyclic length and chromatic fixed-point loss Back to text
In particular, a noncentral subgroup of order two in $D_8$ has loss two, although the elementary-quotient lower bound is one.
Cyclic length and chromatic fixed-point loss Back to text
The remaining obstacle is sharpness for an arbitrary embedded subgroup.
Cyclic length and chromatic fixed-point loss Back to text
Thus they cannot directly supply characters with full nonabelian image.
Cyclic length and chromatic fixed-point loss Back to text
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.
Cyclic length and chromatic fixed-point loss Back to text
Thus the optimal loss is independent of the height.
Cyclic length and chromatic fixed-point loss Back to text
The theorem includes rational fixed-point detection at $n=0$ .
Cyclic length and chromatic fixed-point loss Back to text
A cyclic quotient of order $p^2$ still counts only once.
Cyclic length and chromatic fixed-point loss Back to text
The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.
Cyclic length and chromatic fixed-point loss Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

Browse Topology Graph theory Open since the 1950s Claimed, not yet checked