{"id":"e1ad8854-6954-4f20-9472-cf8f66ccb902","arxiv_id":"2608.12499","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every saturated fusion system, the fusion-stable sign and tom Dieck homomorphisms are surjective, which completes the classification of when the Lefschetz map on endotrivial complexes is surjective for all finite groups.","lead":"The paper proves that the sign and tom Dieck homomorphisms remain surjective after restricting to fusion-stable subgroups of a p-group, extending a classical theorem. This settles a recent question and shows that a certain Lefschetz map on permutation-module complexes is onto for all finite groups exactly over the two-element field.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is sound conditional on the cited [RY18, Prop. 2.5], which is a published result.","rationale":"The reader's weakest assumption correctly identifies Proposition 2.8 as the load-bearing step. I re-read the proof of Theorem 3.1 and the surrounding biset-functor setup. Given Proposition 2.8 and the cited identifications in Remark 2.6, the argument is coherent: the p-local tom Dieck map is a biset-functor morphism, X' is F-stable in the p-local sense, clearing denominators by an odd integer preserves the sign because it acts trivially modulo 2, and the final equality lifts from Z_(p)-localization to integers because units of the Burnside ring have all marks ±1 and hence satisfy u^2=1, so localizing at 2 does not change A(S)^×. The only possible failure point is external: whether [RY18, Prop. 2.5] really applies to the unit functor and to RO(−), and whether [BC20] justifies the identification of F-stable units. These are published results, not flaws in the present derivation. The flatness remark in Proposition 1.13 is imprecise but not load-bearing, since the base-change map is a biset-functor morphism for elementary reasons. I therefore see no reason to alter the reader's ACCEPT verdict.","tokens_in":10368,"tokens_out":32928,"duration_ms":309672,"concrete_test":"Check the original statement of [RY18, Proposition 2.5] to confirm it applies verbatim to Z_(2)-linear biset functors A(−)^×(2) and RO(−)_(2), and confirm the [BC20] identification in Remark 2.6; as a computational cross-check, for the fusion system of A4 on V4 compute T_ωF on A(V4)^×(2) and RO(V4)_(2) and verify the image equals the F-stable part.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The construction is most vulnerable at the invocation of Proposition 2.8 ([RY18, Prop. 2.5]), which supplies both the equality F(S)^F_(p)=T_ωF(F(S)_(p)) and the fixed-point property. This is used twice, for RO(−)_(p) and for the unit functor A(−)^×_(p), whose biset action is generalized tensor induction rather than the additive Burnside action. The paper imports the result without reproof, and if the proposition's hypotheses were not met by either functor, or if Remark 2.6's identification of F-stable units with A(F)^× failed, Theorem 3.1 would collapse. However, this is an external dependence on a published proposition, and I found no internal inconsistency in the p-localization/clearing-denominators step: for p=2 the odd denominator a leaves the sign unchanged, and equality of an integral unit with x in Z_(2)-localization is global because A(S)^× has exponent 2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10519,"tokens_out":26050,"duration_ms":222878,"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":[{"comment":"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":"Section 3, Corollary 3.3"},{"comment":"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.","section":"Section 3, proof of Theorem 3.1"}],"minor_comments":[{"comment":"The mark homomorphism is written m_S : B(S) → CF(S), but the symbol B(S) is otherwise unused; it should be A(S).","section":"Introduction"},{"comment":"The phrase 'results of Rees–Yalcin [RY18]' should read 'Reeh–Yalcin'.","section":"Introduction"},{"comment":"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'.","section":"Section 1 and Section 1.1"},{"comment":"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.","section":"Definition 1.8(d)"},{"comment":"The 'routine verification' of the first statement could be expanded by one sentence for self-containedness, although the base-change argument is convincing.","section":"Proposition 1.13"}],"recommendation":"minor_revision","confidential_remarks":"The only point that gives me pause is the scope of the cited counterexample in [Mil24, Example 7.7] for fields of characteristic 2; the rest of the paper is sound. I would be comfortable with acceptance after the requested clarifications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe 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.\n\nWhat'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.\n\nCredit: 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.\n\nSoft 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.\n\nThis 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.","headline":"The fusion-stable sign and tom Dieck homomorphisms are surjective; this short, clean note closes an open question and should be refereed.","tokens_in":11079,"tokens_out":3883,"would_cite":true,"duration_ms":32774,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["tom Dieck homomorphism","fusion system","Burnside ring","characteristic idempotent","Borel-Smith functions","Lefschetz homomorphism","endotrivial complexes","p-permutation modules"],"falsifier":"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 $\\mathrm{RO}(F)$.","tokens_in":10121,"feed_emoji":"🧩","tokens_out":9854,"duration_ms":78100,"temperature":0.7,"pith_summary":"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 $\\mathbb{F}_2$.","feed_headline":"Fusion-stable Burnside units come from real representations","feed_subtitle":"The result pins down when the Lefschetz map for endotrivial complexes is surjective: exactly over F2.","key_machinery":"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.","core_discovery":"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 $k = \\mathbb{F}_2$.","pith_inferences":["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 $\\mathrm{CF}^b(S)$."],"forward_implications":["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]."],"supporting_citations":[{"why":"Supplies Proposition 2.5, the load-bearing statement that the characteristic idempotent stabilizes $p$-localized $p$-biset functors with image exactly the $F$-stable subfunctor.","marker":"[RY18]"},{"why":"Establishes the bijection between saturated fusion systems and characteristic idempotents, guaranteeing that $\\omega_F$ exists and is unique.","marker":"[RS13]"},{"why":"First proved that every saturated fusion system has a unique characteristic idempotent, the object used to $F$-stabilize preimages.","marker":"[Rag06]"},{"why":"Gives the original surjectivity of the unrestricted tom Dieck homomorphism for $p$-groups, the starting point of the proof.","marker":"[Tor84]"},{"why":"Provides the algebraic proof of surjectivity of the unrestricted tom Dieck map, making the $p$-local biset-functor argument available.","marker":"[Yal05]"},{"why":"Provides the biset-functor framework and the unit functor $A(-)^\\times$ structure, plus the inclusion of $A(-)^\\times$ into $\\mathrm{CF}(-,\\mathbb{F}_2)$ used to identify the sign map.","marker":"[Bou10]"},{"why":"Poses the question answered here and shows Lefschetz surjectivity is equivalent to surjectivity of the $F$-restricted sign map.","marker":"[MM26]"},{"why":"Gives the description of the trivial source ring used in the corollary connecting the sign map to the Lefschetz homomorphism.","marker":"[BC23]"}],"fun_headline_variants":["Fusion-stable real reps cover all Burnside units","Real reps hit every fusion-stable Burnside unit","Surjectivity of tom Dieck survives fusion","Fusion-stable tom Dieck is onto"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fusion-stable real reps cover all Burnside units","Real reps hit every fusion-stable Burnside unit","Surjectivity of tom Dieck survives fusion","Fusion-stable tom Dieck is onto"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000762,"raw_usage":{"total_tokens":3381,"prompt_tokens":942,"completion_tokens":2439,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":2377}},"tokens_in":558,"tokens_out":2439,"duration_ms":18655,"temperature":1.0,"reasoning_tokens":2377,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:09:08.007049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\mathrm{RO}(F)$.","supporting_citations":[],"review_version":1}