The fusion-stable tom Dieck homomorphism

Sam K. Miller

REVIEW 2 major objections 5 minor 33 references

The fusion-stable tom Dieck homomorphism

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper proves that the fusion-stable tom Dieck homomorphism and sign homomorphism are surjective for every saturated fusion system, settling when the Lefschetz map for endotrivial complexes is surjective over all finite groups.

desk verdict The fusion-stable sign and tom Dieck homomorphisms are surjective; this short, clean note closes an open question and should be refereed. read the letter →

arxiv 2608.12499 v1 pith:SXSKFDCE submitted 2026-08-12 math.GR math.ATmath.RT

Sam K. Miller This is my paper · ORCID

classification math.GRmath.ATmath.RT MSC 19A22

keywords tomDieckhomomorphismfusionsystemBurnsideringcharacteristicidempotentBorel-SmithfunctionsLefschetzendotrivialcomplexesp-permutationmodules

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the classical surjectivity of the tom Dieck homomorphism—which sends a virtual real representation of a $p$-group to a unit of its Burnside ring—survives when one restricts to data compatible with a saturated fusion system. For any saturated fusion system $F$ on a $p$-group $S$, both the $F$-restricted sign map from $F$-stable Borel–Smith functions to $F$-stable Burnside-ring units and the $F$-restricted tom Dieck map from $F$-stable real representations are well defined and surjective. The motivation is representation theory: a corollary answers the open question posed in [MM26] by showing that the Lefschetz homomorphism from endotrivial complexes to the trivial source ring is surjective for every finite group exactly when the field is $\mathbb{F}_2$.

What carries the argument

The engine is the characteristic idempotent $\omega_F$ of the saturated fusion system $F$: a bifree idempotent in the $p$-localized double Burnside ring of $S$ that encodes the fusion system, together with the theory of $p$-biset functors (functors on finite $p$-groups induced by biset compositions). A proposition imported from [RY18] says that acting by $\omega_F$ on the $p$-localization of any $p$-biset functor has image exactly its $F$-stable subfunctor and fixes those elements. Applied to $\mathrm{RO}(-)$, $\mathrm{CF}^b(-)$, and $A(-)^\times$, this converts an arbitrary preimage into an $F$-stable one; because the tom Dieck map is a morphism of $p$-biset functors, the idempotent commutes with it. The sign map is identified with reduction modulo $2$ followed by the identification of $\mathrm{CF}(-,\mathbb{F}_2)$ with the unit group of $\mathrm{CF}(-)$, which shows it too is a morphism of $p$-biset functors.

What would settle it

A computational counterexample would settle the matter: for the $2$-fusion system of a small group such as $A_4$ or $\mathrm{SL}(2,3)$, enumerate all $F$-stable units of the Burnside ring and all $F$-stable real representation classes and check whether $\Theta_F$ hits every unit. A single missing unit would disprove Theorem 3.1; equivalently, one can test whether every preimage of an $F$-stable unit under the unrestricted tom Dieck map, after applying $\omega_F$ and clearing odd denominators, lands in $\mathrm{RO}(F)$.

Watch

Extended reading notes

Core claim

Theorem A: Let $F$ be a saturated fusion system over a $p$-group $S$. The $F$-restricted sign homomorphism $\mathrm{sgn}_F : \mathrm{CF}^b(F) \to A(F)^\times$ and the $F$-restricted tom Dieck homomorphism $\Theta_F : \mathrm{RO}(F) \to A(F)^\times$ are well-defined and surjective. The proof starts from the known surjectivity of the unrestricted tom Dieck map for $p$-groups, applies the characteristic idempotent $\omega_F$ to an arbitrary preimage to make it $F$-stable after $p$-localization, and then multiplies by an odd integer to clear denominators without changing the sign; the resulting element is an integral $F$-stable real representation mapping to the given unit. As a corollary, for a finite group $G$ the Lefschetz map $\mathrm{Pic}(K(G;\mathbb{F}_2)) \to O(T(\mathbb{F}_2 G))$ is surjective, and for a field $k$ of positive characteristic this Lefschetz map is surjective for all finite groups if and only if $k = \mathbb{F}_2$.

Load-bearing premise

The paper's argument rests on the statement, imported from [RY18], that the characteristic idempotent of a saturated fusion system acts on every $p$-localized $p$-biset functor with image exactly the $F$-stable subfunctor and fixes those elements; if that statement fails for real representations or Burnside units, the surjectivity construction collapses.

Editorial extensions

If this is right

Every $F$-stable unit of the Burnside ring is the tom Dieck image of an $F$-stable virtual real representation, so fusion-stable units are realized geometrically.
The Lefschetz homomorphism $\mathrm{Pic}(K(G;\mathbb{F}_2)) \to O(T(\mathbb{F}_2 G))$ is surjective for every finite group $G$.
For every positive-characteristic field $k \neq \mathbb{F}_2$, there is a finite group for which the Lefschetz map is not surjective; combined with the $\mathbb{F}_2$ case this gives the if-and-only-if classification.
The $F$-restricted sign homomorphism $\mathrm{CF}^b(F) \to A(F)^\times$ is surjective, answering the question in [MM26, Remark 4.2].

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same idempotent-and-odd-scaling argument would likely transfer surjectivity to any $p$-biset functor whose unrestricted map is surjective and whose target is insensitive to odd scaling, so the mechanism may be a general 'fusion-stable surjectivity transfers' principle.
Since the sign map is base change to $\mathbb{F}_2$, the theorem suggests the failure of Lefschetz surjectivity over other fields is a mod-$2$ phenomenon: non-surjectivity appears exactly when the target ring contains units invisible through $\mathbb{F}_2$-valued marks.
The explicit construction—apply $\omega_F$ to a preimage and clear odd denominators—could help build $F$-stable real representations with prescribed Euler characteristics, and might offer a route to an explicit inverse of the known isomorphism between $\mathrm{Pic}(K(S;k))$ and $\mathrm{CF}^b(S)$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this paper proves the fusion-stable sign and tom Dieck maps are surjective for saturated fusion systems, and uses that to settle a specific open question about Lefschetz maps. The result is new, the proof is a neat application of characteristic idempotents, and as far as I can tell the argument holds together. It deserves a serious referee.

What's actually new: Reeh–Yalcin had shown the dimension map is not surjective on the fusion-stable level; this paper shows the two maps that factor through it are. That's a clean, somewhat surprising completion, and the corollary closes the F2 case of the Lefschetz surjectivity question that Mazza–Miller left open. The main theorem is a short p-localization argument: take the known surjectivity for p-groups, hit a preimage with the characteristic idempotent to get an F-stable p-local element, then clear odd denominators. That strategy is well chosen and the cited machinery is the right machinery.

Credit: the paper doesn't pretend to prove more than it does. It black-boxes Ragnarsson–Stancu and Reeh–Yalcin cleanly, the functoriality checks are the right ones, and the applications section is honest about what depends on prior work. The exposition is a bit terse in places, but not obscurely so.

Soft spots are minor. The proof of Proposition 1.13 says 'routine verification' for a claim that is doing work; the base-change wording is imprecise (saying 2Z ⊂ Z is flat is, at best, a shortcut), and the step where a times a p-local F-stable element becomes globally F-stable deserves one more sentence. None of these break the argument. The whole proof leans on [RY18, Prop. 2.5], a published result that appears to be the right statement; if it failed for the unit functor A(−)^×, whose biset action is tensor induction rather than the additive one, the theorem would collapse, but I see no evidence it does. That dependency is normal for this area.

This is for people working on Burnside rings, fusion systems, and endotrivial complexes. A short contribution, not a new theory, but it closes a question with a transparent proof. I'd cite it and I'd send it to a referee without hesitation.

Referee Report

2 major / 5 minor

Summary. The paper proves Theorem A (Theorem 3.1): for a saturated fusion system F on a p-group S, the F-restricted sign homomorphism sgn_F : CF^b(F) → A(F)^× and the F-restricted tom Dieck homomorphism Θ_F : RO(F) → A(F)^× are well-defined and surjective. The proof uses Tornehave–Yalcin surjectivity for p-groups, applies the characteristic idempotent ω_F of F to obtain a p-local F-stable preimage via Reeh–Yalcin's Proposition 2.8, and then clears odd denominators, relying on the fact that the unit group of the Burnside ring of a p-group has exponent two. As a corollary, the paper shows that the Lefschetz homomorphism from the Picard group of the bounded homotopy category of p-permutation modules over F_2 to the trivial source ring is surjective for every finite group, and hence that for a field k of positive characteristic this surjectivity holds for all finite groups if and only if k = F_2.

Significance. The result is a significant and clean extension of classical p-group theorems to saturated fusion systems, and it answers an open question of Mazza–Miller. The proof is transparent and uses the right external tools (characteristic idempotents, p-local biset functors, Reeh–Yalcin's stabilization theorem), with the dependence on [RY18, Prop. 2.5] explicitly identified. The corollary gives a complete classification over fields of positive characteristic, which is a strong and falsifiable statement. The paper is concise, the main argument is internally consistent, and I found no ad hoc assumptions or invented entities. The main proof is appropriately conditional on published results rather than on its own conclusion.

major comments (2)

[Section 3, Corollary 3.3] The 'only if' direction for characteristic 2 relies on [Mil24, Example 7.7] being applicable to every field k with char(k)=2 and k≠F_2, but the Introduction states the earlier counterexample for A_4 only under the hypothesis that k contains a primitive third root of unity. Please state precisely whether Example 7.7 applies to all such k; if it does, give the exact statement (or a brief explanation), and if it does not, the 'if and only if' claim is incomplete.
[Section 3, proof of Theorem 3.1] After obtaining Θ_(p)(X'') = x in A(S)^×_(p), the proof concludes the global equality Θ_S(X'') = x. This conclusion requires the canonical map A(S)^× → A(S)^×_(p) to be injective; for p=2 this holds because A(S)^× is an elementary abelian 2-group, and for p odd the case is already trivial. Please add a sentence making this explicit, since it is the final step of the main theorem.

minor comments (5)

[Introduction] The mark homomorphism is written m_S : B(S) → CF(S), but the symbol B(S) is otherwise unused; it should be A(S).
[Introduction] The phrase 'results of Rees–Yalcin [RY18]' should read 'Reeh–Yalcin'.
[Section 1 and Section 1.1] There are small typos: 'for this apper' should be 'for this paper', and 'Borel-Smith are intimately related' should be 'Borel-Smith functions are intimately related'.
[Definition 1.8(d)] The Borel–Smith conditions say 'subquotient Q/P of G' but the functions are on subgroups of the p-group S; please write 'subquotient Q/P of S' for consistency.
[Proposition 1.13] The 'routine verification' of the first statement could be expanded by one sentence for self-containedness, although the base-change argument is convincing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is derived from external published results, and the self-citations in Corollary 3.3 are prior counterexamples or reductions, not the target claim.

full rationale

Walking the derivation chain of Theorem 3.1: after choosing an F-stable unit x, the proof invokes Tornehave–Yalçın surjectivity of the unrestricted Θ_S to get some X, then applies the characteristic idempotent ω_F via the p-biset functor action. The equality (Θ_(p)∘T_ωF)(X)=T_ωF(Θ_(p)(X))=T_ωF(x)=x is justified by biset functoriality plus [RY18, Prop. 2.5] applied to A(−)^×_(p); nothing here assumes the surjectivity of Θ_F being proved. The element X' = T_ωF(X) is F-stable by the same proposition applied to RO(−)_(p), and the denominator-clearing step uses only that a is odd, so the sign is unchanged. The cited [RY18, Prop. 2.5] and [RS13, Theorem C] are external published results, not self-citations, and they do not contain the target statement. The self-citations in Corollary 3.3—[MM26, Thm 4.1] for the reduction and [MM26, Thm 5.10] / [Mil24, Ex. 7.7] for counterexamples—are prior published results used as independent support or as falsifiable counterexamples, not as restatements of the conclusion. I found no equation in which a predicted quantity is defined as its own input, and no fitted parameter is renamed as a prediction. Hence no significant circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters or fitted constants appear; the paper is a purely algebraic proof. The central claim rests on established theorems about biset functors and fusion systems, most importantly the projection property of characteristic idempotents. The corollary also relies on same-author prior work [MM26, Mil24] for the negative cases, which is disclosed and not circular.

assumptions (8)

domain assumption Tornehave-Yalcin theorem: the tom Dieck homomorphism Theta_S : RO(S) to A(S)^x is surjective for every p-group S.
Used at the start of the proof of Theorem 3.1 to lift an arbitrary F-stable unit x to a virtual representation X.
domain assumption Existence and uniqueness of characteristic idempotents for saturated fusion systems (Ragnarsson-Stancu, Theorem 2.4).
Provides the idempotent omega_F that implements F-stabilization in the proof.
domain assumption Reeh-Yalcin Proposition 2.8: for a saturated fusion system F and any p-biset functor F, F(S)^F_(p) equals T_{omega_F}(F(S)_(p)), and T_{omega_F} fixes F-stable elements.
The load-bearing projection step; applied to RO(-)_(p) and A(-)^x_(p).
domain assumption The tom Dieck homomorphism Theta and the unit functor A(-)^x are morphisms of p-biset functors (Yoshida, Bouc, Yalcin).
Allows Theta_(p) to commute with the action of omega_F, which is essential in the main computation.
domain assumption For odd p, the unit group of the Burnside ring of a p-group is {+-[S/S]}.
Used to reduce Theorem 3.1 to the case p=2.
standard math RO(S) is torsion-free, so the localization map RO(S) to RO(S)_(p) is injective.
Implicitly used when concluding that the odd multiple of a p-local F-stable element is F-stable in RO(S).
domain assumption The equivalence in [MM26, Theorem 4.1] between surjectivity of the Lefschetz map Lambda and surjectivity of the F_S(G)-stable sign homomorphism, together with Boltje-Carman's description of O(T(kG)).
Bridges Theorem 3.1 to the positive direction of Corollary 3.3.
domain assumption The counterexamples for fields other than F_2: [MM26, Theorem 5.10] for odd characteristic and [Mil24, Example 7.7] for characteristic 2 fields strictly larger than F_2.
Provides the negative direction of the if and only if in Corollary 3.3.

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Cite this review

Pith. "Pith review of The fusion-stable tom Dieck homomorphism." pith.science (2026). https://pith.science/paper/SXSKFDCE

@misc{pith2026260812499,
  author       = {Pith},
  title        = {Pith review of: The fusion-stable tom Dieck homomorphism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXSKFDCE}},
  note         = {Machine review of arXiv:2608.12499}
}

abstract

Tornehave and Yal\c{c}in proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any $p$-group $S$. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on $S$. As a corollary, we close the main question posed by Mazza--Miller in arXiv:2508.07404 by showing that given a field $k$ of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of $p$-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if $k = \mathbb{F}_2$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Bouc , publisher =

S. Bouc , publisher =. Biset Functors for Finite Groups , year =

work page

[2] [2]

and Perepelitsky, P

Boltje, R. and Perepelitsky, P. , TITLE =. J. Algebra , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.jalgebra.2024.09.038 , URL =

work page doi:10.1016/j.jalgebra.2024.09.038 2025

[3] [3]

Boltje and R

R. Boltje and R. Carman , title =. J. Algebra , year =

work page

[4] [4]

Matsuda , title =

T. Matsuda , title =. J. Math. Soc. Japan , year = 1982, volume = 42, number = 1, pages =

work page 1982

[5] [5]

tom Dieck , title =

T. tom Dieck , title =

work page

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S. K. Miller , title =. J. Algebra , year = 2024, volume = 650, pages =

work page 2024

[7] [7]

S. K. Miller , title =

work page

[8] [8]

Puig , title=

L. Puig , title=. J. Algebra , year = 1988, volume = 116, pages =

work page 1988