{"id":"46d2b495-71aa-463b-9e5d-7ce103053da6","arxiv_id":"1908.00779","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct a stable model structure on orthogonal G-spectra for every Lie group G, with equivalences tested on compact subgroups, whose homotopy category is compactly generated and represents equivariant K-theory and stable cohomotopy.","lead":"This monograph builds a full 'genuine' stable homotopy theory for symmetries of spaces under any Lie group, including infinite groups, where it previously existed only for compact groups. It shows that equivariant K-theory and stable cohomotopy fit into this framework, and links the theory to geometric group theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Com-cofibrant G-spaces are assumed G-homotopy equivalent to proper G-CW-complexes without proof; this is load-bearing for Theorem 1.2.9.","rationale":"The reader's weakest assumption points to the imported Com-model structure (Illman, Fausk, [60]). We agree this is the load-bearing region, but we found a more specific and less protected statement inside the paper: the unproven replacement assertion used in Theorem 1.2.9. The paper's own proof of Theorem 1.2.9 cannot go through without it, and it is not merely an imported lemma but an assertion in the text with no reference. The rest of the monograph (compact generation, t-structure, Mackey functor heart) is built on Theorem 1.2.22, so the gap is central. I do not claim the theorem is false; the cited sources likely contain the needed result, but the paper should state and prove or explicitly attribute the replacement assertion. Hence a conditional acceptance is appropriate: accept if the cited sources indeed cover the statement at this generality.","tokens_in":65415,"tokens_out":32082,"duration_ms":313806,"concrete_test":"Check [60, Prop. B.7] and [21, Prop. 2.11] for the statement: 'every Com-cofibrant G-space is G-homotopy equivalent to a proper G-CW-complex' for arbitrary Lie groups G. If neither source contains it, prove it: build a functorial proper G-CW-replacement and show the map to the original is a Com-equivalence. If a counterexample exists (e.g., a G-space whose Com-cofibrant replacement is not homotopy equivalent to a proper G-CW-complex), Theorem 1.2.9 and hence Theorem 1.2.22 are not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.2.22 (stable model structure) depends on Proposition 1.2.21, which requires the generating maps J_G^st to be pi*-isomorphisms. That in turn uses Corollary 1.2.10 (induction along a closed subgroup preserves pi*-isomorphisms), proved via Theorem 1.2.9. Theorem 1.2.9 is proved only for finite-dimensional proper (K x Gamma)-CW-complexes, then extended to 'a general Com-cofibrant (K x Gamma)-space' by the assertion that every Com-cofibrant G-space is G-homotopy equivalent to a proper G-CW-complex. That assertion appears in the text after Proposition 1.1.3 without proof or explicit citation. It is not a formal consequence of the model structure axioms; it is a concrete property of the Com-model structure. If it fails, the induction functor G ⋉_Gamma(-) need not preserve pi*-isomorphisms, so the maps in K_G may fail to be acyclic, and the small-object argument of Theorem 1.2.22 does not produce the claimed stable model structure. The reader flagged the imported [60, Prop. B.7]/[21, Prop. 2.11] as the weak spot; the current concern is the part of that import that is asserted internally rather than cited.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This monograph develops a genuine proper equivariant stable homotopy theory for arbitrary Lie groups. The authors construct a stable model structure on the category of orthogonal G-spectra whose weak equivalences are the pi_*-isomorphisms detected on all compact subgroups, prove that the homotopy category is compactly generated by the suspension spectra of G/H for compact H, and establish a rich change-of-groups formalism including total left derived restriction functors and homotopy invariance of the theory under multiplicative weak equivalences of Lie groups. For discrete groups, the heart of the preferred t-structure is identified with G-Mackey functors, the rational theory is shown to be algebraic, and the previously defined theories of equivariant stable cohomotopy and equivariant K-theory are proved to be represented by explicit orthogonal G-spectra.","tokens_in":65587,"tokens_out":7738,"duration_ms":71820,"significance":"If correct, this is a substantial foundational contribution: it extends genuine equivariant stable homotopy theory to all Lie groups, with transfers, Wirthmuller isomorphisms, and an RO(G)-grading analog, and it connects the theory to finiteness properties of discrete groups. The paper is commendably detailed: the model structure axioms are verified explicitly in Theorem 1.2.22, the reduction to maximal compact subgroups in Theorem 1.4.4 is proved using Abels' theorem, and the representability results for equivariant stable cohomotopy and K-theory are genuine comparisons of independently defined theories. The authors are also careful to identify which results are imported, such as the Com-model structure from [60, Prop. B.7] and [21, Prop. 2.11].","major_comments":[],"minor_comments":[{"comment":"The assertion that every Com-cofibrant G-space is G-equivariantly homotopy equivalent to a proper G-CW-complex is stated without proof or citation. This statement is load-bearing because it is used in the proof of Theorem 1.2.9, in the passage 'A general Com-cofibrant (K x Gamma)-space is (K x Gamma)-homotopy equivalent to a proper (K x Gamma)-CW-complex.' The assertion is true and follows from the explicit generating cofibrations in Proposition 1.1.3(ii) together with standard retract and cellular approximation arguments, so I do not regard this as a mathematical gap; nevertheless, the authors should add a proof sketch or a precise reference to make the dependency explicit.","section":"Section 1.1, paragraph after Proposition 1.1.3"},{"comment":"The proof refers forward to Theorem 1.4.1(ii) to show that the restriction along the diagonal embedding preserves cofibrancy. The forward reference is harmless because Theorem 1.4.1 is independent of the pushout-product statement, but a brief parenthetical remark would help the reader avoid the impression of circularity.","section":"Proposition 1.2.28, proof of (i)"},{"comment":"The notation map^H_*(S^V, X(V+W)) is used in the definition of stable fibration before the based mapping space with H-action is formally introduced in the surrounding text. A one-sentence explanation of the diagonal H-action on this mapping space would improve readability.","section":"Definition 1.2.15"}],"recommendation":"minor_revision","confidential_remarks":"This is a substantial and carefully written monograph; the central theorems appear correct and the level of detail is high. The only point worth emphasizing to the editor is the unproved claim about Com-cofibrant spaces being G-homotopy equivalent to proper G-CW-complexes, which is used in the proof of Theorem 1.2.9. Since the claim is standard and easily justified, this should not delay publication, but it should be fixed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious, mostly self-contained monograph that delivers what the abstract promises. The central new result, Theorem 1.2.22, is a genuine advance: a stable model structure on orthogonal G-spectra for every Lie group G, with pi*-isomorphisms as weak equivalences and explicit control over stable fibrations (they are exactly the G-Omega-spectra). Fausk had a model structure with the same weak equivalences, but the present one has a different cofibration/fibration pair and an explicit characterization of stable fibrations, which matters for applications. Theorem 1.4.31, homotopy invariance of the theory under weak equivalences of the ambient group, is a new phenomenon that has no content in the compact case, and the t-structure heart for discrete groups is cleanly identified with G-Mackey functors (Theorem 2.2.9). The identifications of equivariant stable cohomotopy and equivariant K-theory as represented theories are concrete and useful.\n\nThe reader's take is fair, and I share the moderate confidence. The one genuine soft spot is exactly where the stress-test points. The proof of Theorem 1.2.9 uses the assertion that every Com-cofibrant (K x Gamma)-space is (K x Gamma)-homotopy equivalent to a proper (K x Gamma)-CW-complex. This appears as a one-liner after Proposition 1.1.3 with no proof and no explicit citation. It is load-bearing: it feeds into Corollary 1.2.10, then Proposition 1.2.21, then Theorem 1.2.22. I believe the claim is true and almost certainly follows from the cited sources ([60, Prop. B.7] or [21, Prop. 2.11]), but the paper should say so explicitly. As written, a rigorous reader has to do the bookkeeping. That is a patchable omission, not a fatal flaw. The deferred proof of Proposition 1.3.7 (coherence of integer shifts) is minor. The paper cites Schwede's book and Luck's work as background tools, and the central theorems are proven from stated hypotheses; the self-citations are not carrying the load.\n\nNet: this is a reference-quality monograph for equivariant homotopy theorists and geometric group theorists. It deserves a serious referee. If I were the editor, I would send it out and ask the authors to close the Com-cofibrant gap with a proof or a precise citation.\n","headline":"Proper equivariant stable homotopy theory for all Lie groups: the central model structure is real and new, but one load-bearing assertion about Com-cofibrant spaces needs an explicit proof or citation.","tokens_in":66261,"tokens_out":3647,"would_cite":true,"duration_ms":38224,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P91"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper builds a genuine proper equivariant stable homotopy theory for every Lie group, with equivalences tested on compact subgroups.","keywords":["Lie group","equivariant homotopy theory","proper action","orthogonal spectra","stable model structure","Mackey functors","equivariant K-theory","RO(G)-grading"],"falsifier":"A reader could settle the foundational claim by checking the Com-model structure in the simplest noncompact case, the infinite cyclic group G=\\mathbb{Z}, whose only compact subgroup is trivial; there the alleged generating cofibrations reduce to \\mathbb{Z}\\times\\partial D^k\\to \\mathbb{Z}\\times D^k. If that set fails to generate the maps with the left lifting property against maps that are weak equivalences and Serre fibrations on underlying spaces, then Proposition 1.1.3 is false and the stable model structure of Theorem 1.2.22 collapses.","tokens_in":65121,"feed_emoji":"","tokens_out":14262,"duration_ms":138725,"temperature":0.7,"pith_summary":"This monograph builds a version of stable homotopy theory for every Lie group G, not just compact ones, in which homotopical information is tested on all compact subgroups of G. Its central claim is that the category of orthogonal G-spectra admits a symmetric monoidal stable model structure whose weak equivalences are the \\pi_*-isomorphisms, and whose triangulated homotopy category is compactly generated by the suspension spectra of the orbits G/H for compact subgroups H. If correct, this gives every Lie group the full apparatus of genuine equivariant stable homotopy theory\\u2014transfers, Wirthm\\u00fcller isomorphisms, and an analogue of an RO(G)-grading\\u2014and makes every orthogonal G-spectrum represent an equivariant cohomology theory on proper G-spaces. For infinite discrete groups, the theory connects equivariant stability to finiteness properties of the group and recovers equivariant stable cohomotopy and equivariant K-theory as represented cohomology theories.","feed_headline":"A genuine stable homotopy theory now exists for every Lie group","feed_subtitle":"Equivalences are tested on compact subgroups, and the homotopy category is generated by the orbits G/H.","key_machinery":"The central object is the category of orthogonal G-spectra: continuous functors from the category \\mathcal{O} of finite-dimensional inner product spaces, with morphism spaces the Thom spaces O(V,W), to based G-spaces, with G acting levelwise and diagonally on smash products. The load-bearing mechanism is the stable model structure of Theorem 1.2.22, obtained by localizing a level model structure. Its generating cofibrations are the maps G\\ltimes_H F_V\\wedge \\partial D^k_+ \\to G\\ltimes_H F_V\\wedge D^k_+, with H compact and V an H-representation; its stable fibrations are level fibrations satisfying homotopy cartesian squares that compare fixed-point evaluation X(W)^H with H-equivariant mapping spaces into X(V\\oplus W); and its fibrant objects are the G-\\$\\Omega$-spectra. The set of compact generators is \\{\\Sigma^\\infty_+ G/H \\mid H\\le G\\ \\text{compact}\\}. Beneath all of this sits the Com-model structure on G-spaces, in which equivalences and fibrations are tested on H-fixed points for compact subgroups and the cells are G/H\\times D^k with compact isotropy.","core_discovery":"On its own terms, the paper proves that for every Lie group G, the category \\mathrm{Sp}^G of orthogonal G-spectra carries a cofibrantly generated, proper, stable, topological, symmetric monoidal model structure (Theorem 1.2.22). The weak equivalences are the \\pi_*-isomorphisms: maps inducing isomorphisms on H-equivariant stable homotopy groups for every compact subgroup H of G. The fibrant objects are the G-\\$\\Omega$-spectra, and the triangulated homotopy category \\mathrm{Ho}(\\mathrm{Sp}^G) is compactly generated by the small generators \\Sigma^\\infty_+ G/H for compact H (Corollary 1.3.11). The whole structure is functorial in the group: restriction along continuous homomorphisms has a total left derived functor, inner automorphisms act trivially, homotopic homomorphisms induce isomorphic functors, and a weak equivalence of Lie groups induces an equivalence of homotopy categories. For discrete groups the heart of the preferred t-structure is the abelian category of G-Mackey functors, the rational theory is algebraic, and the represented cohomology theories on finite proper G-CW-complexes admit an explicit description in terms of fiberwise equivariant homotopy theory stabilized by G-vector bundles, which identifies equivariant stable cohomotopy and equivariant K-theory as represented theories.","pith_inferences":["Editorial inference: because weak equivalences of Lie groups induce equivalences of homotopy categories, stable equivariant computations for a Lie group G should be invariant under replacing G by a weakly equivalent model such as G\\times\\mathbb{R}, a reduction not developed as a computational device in the paper.","Editorial inference: the grading analogue by the Grothendieck group KO_G(EG) of G-vector bundles over EG invites a theory of equivariant orientations and characteristic classes indexed by such bundles, going beyond the paper's construction of Thom-space invertibility.","Editorial inference: the short exact sequence for countable locally finite groups reduces morphism groups in \\mathrm{Ho}(\\mathrm{Sp}^G) to towers of finite-subgroup equivariant spectra, giving a concrete route to computations for groups such as \\mathbb{Z} and the infinite dihedral group."],"forward_implications":["Every orthogonal G-spectrum represents an equivariant cohomology theory on proper G-spaces, so transfers, Wirthm\\u00fcller isomorphisms, and the other structures of genuine equivariant stable homotopy theory become available for every Lie group.","For almost connected Lie groups, restriction to a maximal compact subgroup is a Quillen equivalence, so the theory reduces to the classical compact case in that setting.","For discrete groups, the heart of the preferred t-structure is the category of G-Mackey functors; every such Mackey functor has an Eilenberg\\u2013Mac Lane spectrum, and rational G-spectra are classified by rational G-Mackey functors.","The G-sphere spectrum is a compact object when the universal proper G-space EG has a finite G-CW-model, and compactness can fail otherwise, so finiteness properties of the group are reflected in the equivariant stable category.","Equivariant stable cohomotopy and equivariant K-theory for discrete groups are shown to be represented by explicit orthogonal G-spectra, giving them all the structure of the ambient model category."],"supporting_citations":[{"why":"It supplies the Com-model structure on G-spaces, including the generating cofibrations G/H\\times\\partial D^k\\to G/H\\times D^k that all later model structures are built from.","marker":"[60, Prop. B.7]"},{"why":"It provides the level model structure on enriched functor categories that is localized to obtain the stable model structure.","marker":"[60, Prop. C.23]"},{"why":"It proves that the morphisms \\lambda_{H,V,W} are \\pi_*-isomorphisms, a fact used to build the generating acyclic cofibrations.","marker":"[48, III Lemma 4.5]"},{"why":"It identifies stable fibrations by a right lifting property with respect to the set K, a key step in proving Theorem 1.2.22.","marker":"[48, III Prop. 4.8]"},{"why":"It combines with Prop. 4.8 to show that a \\pi_*-isomorphism which is a stable fibration is a level equivalence, completing the model category axioms.","marker":"[48, III Cor. 4.11]"},{"why":"It supplies the equivariant triangulation theorem used to show that homogeneous spaces G/H are Com-cofibrant.","marker":"[27, Thm. 7.1]"},{"why":"It provides maximal compact subgroups with a linear slice in almost connected Lie groups, used in the reduction to compact equivariant theory.","marker":"[1, Thm. A.5]"},{"why":"It supplies the definition of G-Mackey functors for discrete groups that form the heart of the preferred t-structure.","marker":"[49, Sec. 3]"},{"why":"It defines equivariant stable cohomotopy, which the paper identifies with the theory represented by the G-sphere spectrum.","marker":"[38]"},{"why":"It defines equivariant K-theory via G-vector bundles, which the paper shows is represented by an orthogonal G-spectrum.","marker":"[42]"}],"fun_headline_variants":["Genuine proper stable homotopy for every Lie group","Proper equivariant stable homotopy for all Lie groups","Equivariant stable homotopy, tested on compact subgroups","Lie groups get genuine proper stable homotopy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported theorem that G-spaces form a cofibrantly generated model structure when equivalences and fibrations are tested on fixed points of compact subgroups, with the maps G/H\\times\\partial D^k\\to G/H\\times D^k as generators; if that theorem fails for noncompact Lie groups, the stable model structure on orthogonal G-spectra does not exist as claimed.","fun_headline_variants_meta":{"raw":{"variants":["Genuine proper stable homotopy for every Lie group","Proper equivariant stable homotopy for all Lie groups","Equivariant stable homotopy, tested on compact subgroups","Lie groups get genuine proper stable homotopy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001652,"raw_usage":{"total_tokens":6696,"prompt_tokens":1215,"completion_tokens":5481,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":831,"completion_tokens_details":{"reasoning_tokens":5414}},"tokens_in":831,"tokens_out":5481,"duration_ms":34081,"temperature":1.0,"reasoning_tokens":5414,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:33:17.136022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A reader could settle the foundational claim by checking the Com-model structure in the simplest noncompact case, the infinite cyclic group G=\\mathbb{Z}, whose only compact subgroup is trivial; there the alleged generating cofibrations reduce to \\mathbb{Z}\\times\\partial D^k\\to \\mathbb{Z}\\times D^k. If that set fails to generate the maps with the left lifting property against maps that are weak equivalences and Serre fibrations on underlying spaces, then Proposition 1.1.3 is false and the stable model structure of Theorem 1.2.22 collapses.","supporting_citations":[{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"It defines equivariant stable cohomotopy, which the paper identifies with the theory represented by the G-sphere spectrum."},{"cited_title":"Topology 40 (2001), 585–616","cited_arxiv_id":null,"evidence_quote":"It defines equivariant K-theory via G-vector bundles, which the paper shows is represented by an orthogonal G-spectrum."}],"review_version":1}