{"id":"a9bbd859-7efb-43d4-bc7c-46e52014c746","arxiv_id":"2506.14301","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors define a relative coarse index, prove it computes submanifold indices via a Thom class, and use it to build scalar curvature obstructions away from submanifolds.","lead":"This paper builds a coarse-geometric version of the relative index, extending Gromov and Lawson's classical invariant to noncompact manifolds. It uses this to construct wrong-way maps that detect obstructions to uniform positive scalar curvature from the index of a submanifold.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.2 invokes cap-product results that assume global bounded geometry, while the theorem only assumes bounded geometry near N; the proof needs a localized version or an added hypothesis.","rationale":"The paper is a careful, technically rich development of a relative coarse index and its application to submanifold obstructions. The central claim, Theorem 7.2, is conditional on the Thom class lying in the image of the coarse co-assembly map away from N; the reader identified this as the weakest assumption, and the paper itself poses it as Question 1.7 with verification only for multi-partitioned manifolds. That is a genuine scope limitation but not an internal inconsistency. My stress-test found a more immediate proof-level concern: the proof of Theorem 7.2 applies Theorem 6.12 and Lemma 6.11, both of which are stated under a global bounded-geometry assumption on M. Theorem 7.2 only assumes bounded geometry in an R-neighborhood of N, and no argument is given that the cap product and the localized E-theory formula remain valid under this weaker hypothesis. The step-function approximation used to prove Lemma 6.11(i) explicitly relies on bounded geometry; whether it can be localized is nontrivial. If it cannot, the theorem's hypotheses are insufficient and a global bounded-geometry assumption is required. This is a concrete, load-bearing gap in the proof, distinct from the acknowledged open lifting problem. I therefore partially agree with the reader: the lifting condition is a real limitation, but the bounded-geometry mismatch is the most immediate obstacle to the central claim as stated. The verdict remains CONDITIONAL, pending either a localized proof of the cap-product machinery or an added global bounded-geometry hypothesis in Theorem 7.2.","tokens_in":38672,"tokens_out":27170,"duration_ms":265296,"concrete_test":"Re-derive the proof of Lemma 6.11(i) under the hypotheses of Theorem 7.2 (bounded geometry only on an R-neighborhood of N). Concretely, for a finite-propagation T and f∈\\bar{c}_N(M), check whether the step-function approximation of f and the estimate [T,f]∈C*(N⊂M;A⊗B) can be established without controlling the geometry of M outside that neighborhood. If the approximation fails—for example on a manifold that is Euclidean near N but has a sequence of curvature spikes going to infinity—then Theorem 7.2 needs a global bounded geometry assumption; if it succeeds, record the localized proof and keep the theorem as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 7.2 uses Theorem 6.12 (equation (6.2)) and Lemma 6.11 to define the cap product x∩τ-tilde and to identify J/D_M,N;Cℓ_{1,r}K∘ι_*(τ-tilde) with ind(/D_M,N)∩τ-tilde. Both results are stated for a complete Riemannian manifold M of global bounded geometry. Theorem 7.2, however, assumes only that M has bounded geometry in some R-neighbourhood of N, together with uniform embeddedness of N and bounded geometry of the associated spinc line bundle. No argument is supplied that the step-function approximation in the proof of Lemma 6.11(i) — used to show [T,f]∈C*(N⊂M;A⊗B) for finite-propagation T and f∈\\bar{c}_N(M) — survives when the geometry of M is uncontrolled outside a neighbourhood of N. If the cap product genuinely requires global bounded geometry, Theorem 7.2 as stated is not proven and must either add 'M has bounded geometry' or provide a localized version of Lemma 6.11/Theorems 6.6 and 6.12. This concern is independent of the open Thom-class lifting condition (Question 1.7).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a coarse-geometric analogue of the Gromov–Lawson relative index, establishes its main formal properties, and uses it to construct wrong-way maps from the coarse index of a Dirac operator on a complete spin manifold to the coarse index of a Dirac operator on a K-oriented submanifold of arbitrary codimension. The main technical ingredients are: an L-local coarse index for operators invertible away from a closed subset (Section 3), a relative index theorem for operators that agree outside compact regions (Theorem 4.5), an E-theoretic formula for the relative index of twisted operators (Theorem 4.10), a Thom-class description of the relation between the index on M and the index on a submanifold N (Theorem 5.13), and a cap-product formula for the coarse index away from L (Theorem 6.12). These are combined in Theorem 7.2, which states that, under a lifting assumption on the Thom class through the coarse co-assembly map away from N, the wrong-way map sends ind(/D_M,N) to i_* ind(/D_N), giving a submanifold obstruction to uniform positive scalar curvature away from N. Theorem 7.5 verifies the lifting assumption for multi-partitioned manifolds, recovering results of Schick–Zadeh and Siegel.","tokens_in":38897,"tokens_out":9133,"duration_ms":101700,"significance":"If the main theorems are correct, the paper gives a substantial and useful framework: a relative coarse index with a clean E-theoretic calculus, a general Thom-class relation for submanifolds, and an abstract machinery for submanifold obstructions to uniform positive scalar curvature. The paper is careful and honest: it explicitly flags the unproved comparison with Roe's relative index and poses the lifting condition as Question 1.7, so the conditional nature of the central application is clear. The proofs are detailed and build systematically on the published framework of [Wul19] and on [Wul22b] for non-separable E-theory. The main weakness is a mismatch between the hypotheses of Section 6's cap-product machinery and those of Theorem 7.2, as detailed in the major comments.","major_comments":[{"comment":"The wrong-way map in Theorem 7.2 is x ↦ ∂(x ∩ τ̃), using the cap product from Lemma 6.11, and the proof identifies J/D_M,N;Cℓ_{1,r}K∘ι_*(τ̃) with ind(/D_M,N) ∩ τ̃ by citing Theorem 6.12, Eq. (6.2). Both Lemma 6.11 and Theorem 6.12 are stated and proved only for a complete Riemannian manifold M of global bounded geometry. Theorem 7.2, however, assumes only that M has bounded geometry in some R-neighbourhood of N. The proof of Lemma 6.11(i) relies on a step-function approximation of f that uses bounded geometry, and no argument is supplied that this approximation, or the equality (6.2), remains valid when the geometry of M is uncontrolled away from N. Consequently, the displayed calculation in the proof of Theorem 7.2 is not justified as stated. The authors should either add the hypothesis that M has global bounded geometry to Theorem 7.2 (and accordingly to Theorem 7.5), or prove a localized version of Lemma 6.11 and Theorem 6.12 in which bounded geometry near L suffices, and then verify that this localized version applies to the situation of Theorem 7.2.","section":"§7, Theorem 7.2; §6.2, Lemma 6.11 and Theorem 6.12"},{"comment":"The proof of Theorem 5.13 uses the assertion that 'an analogue of Theorem 2.17 for the localized E-theory classes, which can be shown by adapting the proof in [Wul19, Theorem 4.12]' implies invariance of J/D_M∥N;Cℓ_{0,r}K under the metric and connection modifications. This is a load-bearing step in passing from the restrictive Assumptions 5.1–5.3 to the general statement of Theorem 5.13, and neither the full statement nor a proof of this localized bordism invariance is supplied. Please provide the precise statement and proof, or a complete reference, for this localized version.","section":"§5.2, proof of Theorem 5.13"}],"minor_comments":[{"comment":"The authors state that they have not checked whether their relative coarse index agrees with Roe's relative index from [Roe16, Section 4]. Since both are called relative coarse indices, a short remark indicating the expected comparison, or at least the canonical maps between the two constructions, would be helpful to the reader; this does not affect the main results.","section":"§1, Introduction"},{"comment":"There are several typographical slips, including 'adressed' in Section 4.1 and 'preceed' in the proof of Theorem 3.8, and the spelling of 'coassembly' versus 'co-assembly' is inconsistent across Section 7. None of these affect the mathematics.","section":"§4.1 and §3"},{"comment":"In Definition 5.12, the Thom class is defined using a representative ψ_{R1} ∘ β|_{S_{R1}} and is said to be unique up to homotopy. The sentence explaining independence of the choice of R1 would benefit from explicitly noting that the inclusions A(U,∂U;Cℓ_{0,r}) for different tubular neighborhoods are compatible; the argument is likely routine, but a one-sentence clarification would improve readability.","section":"§5.2, Definition 5.12"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the global versus local bounded geometry mismatch in the proof of Theorem 7.2. This is a genuine gap in the stated result, but it is local in nature and could be fixed either by strengthening the hypothesis or by extending the Section 6 machinery. I do not see grounds for rejection: the framework is substantial, the authors are explicit about the open lifting condition and the unverified comparison with Roe's index, and the central ideas appear sound. A major revision that closes the bounded-geometry gap and supplies the missing localized bordism argument would make the paper acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, technically dense contribution to coarse index theory, but the main wrong-way map theorem (Thm 7.2) has a proof gap the authors should fix. The stress-test concern is legitimate: Thm 7.2 assumes only bounded geometry near N, but its proof invokes Thm 6.12 and Lemma 6.11, both stated for global bounded geometry. The step-function approximation in Lemma 6.11 that controls commutators into the ideal C*(L⊂M;A) uses the global bound, and the paper gives no localized argument. So as written, Thm 7.2 is not proven. The likely fix is to add 'M has bounded geometry' or to prove a local version of the cap product; this is independent of the open Thom-class lifting question (Q 1.7), which the authors flag honestly.\n\nWhat is genuinely new and good: the relative coarse index for noncompact subspaces, the relative index theorem, the E-theory formulas for twisted operators, and the Thom-class relation (5.7). The index away from L and the wrong-way map machinery are useful, and Thm 7.5 recovers Schick–Zadeh for multi-partitioned manifolds. The paper also treats Roe's earlier relative index and Bunke–Engel's construction fairly, noting that equality with Roe's version is unproved; that is a caveat, not a flaw.\n\nThe reader's conditional verdict is about right, with the added caveat from the stress test. I'd send this to a serious referee; the main theorem will need revision, but the framework deserves scrutiny. For a reader in coarse index theory or positive scalar curvature, the paper is worth engaging with despite the gap. Once the bounded geometry hypothesis is sorted, I'd cite it.","headline":"Useful framework with a real gap in Theorem 7.2's bounded geometry hypothesis; worth a serious referee, but needs fixing.","tokens_in":39444,"tokens_out":3681,"would_cite":true,"duration_ms":37472,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19K56","53C27","58J22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs a coarse relative index for Dirac operators and uses it to show that a nonzero submanifold Dirac index obstructs uniformly positive scalar curvature away from that submanifold.","keywords":["relative coarse index","coarse index theory","Roe algebra","wrong way maps","uniformly positive scalar curvature","Thom class","coarse co-assembly map","multi-partitioned manifolds"],"falsifier":"For a concrete multi-partitioned spin manifold, for example $M=N\\times\\mathbb{R}$ with $N$ a compact odd-dimensional spin manifold of nonzero index, compute $\\operatorname{ind}(/D_M,N)$ through the wrong way map and compare it with $i_*\\operatorname{ind}(/D_N)$ under the map of Theorem 7.5; the theorem predicts equality, so a nonzero discrepancy would falsify it. Equivalently, an $M$ of this type with $i_*\\operatorname{ind}(/D_N)\\neq 0$ that nevertheless admits a metric of uniformly positive scalar curvature away from $N$ in the same quasi-isometry class would disprove the obstruction claim.","tokens_in":38438,"feed_emoji":"📐","tokens_out":15052,"duration_ms":136807,"temperature":0.7,"pith_summary":"The paper builds a relative coarse index for Dirac operators on complete manifolds, extending the classical relative index from compactly supported differences to operators that differ only outside a closed subset. It proves the standard properties of this index, including a gluing theorem and a formula expressing the index of an operator twisted by the difference of two bundles as a pairing of K-homology with K-theory. The main application is a general wrong-way map in the K-theory of the Roe algebra, the C*-algebra of coarse operators on the manifold: for a submanifold N with K-oriented normal bundle, the coarse Dirac index of M away from N is mapped to the coarse Dirac index of N whenever the Thom class of N lifts through the coarse co-assembly map away from N. If that submanifold index is nonzero, the map gives an obstruction to metrics of uniformly positive scalar curvature away from N in the same quasi-isometry class. The lift is proved for multi-partitioned manifolds, recovering earlier partitioned-manifold index theorems.","feed_headline":"A nonzero submanifold index blocks uniform positive scalar curvature","feed_subtitle":"New relative coarse index turns Thom-class lifts into wrong-way maps that force scalar-curvature obstructions.","key_machinery":"The load-bearing construction is the relative coarse index $\\operatorname{ind}(D_1,D_2|\\Psi)$, obtained by gluing two Dirac operators along an isometric identification $\\Psi$ outside a closed set and taking the coarse index of the resulting operator. The calculational engine is the composition-product identity $\\operatorname{ind}(D\\|E,F)=J/D\\|L;B K\\circ(J E K-J F K)$ and the Thom-class identity $J/D_M\\|N;\\mathbb{C}\\ell_{0,r}K\\circ\\tau=i_*\\operatorname{ind}(/D_N)$. The final ingredient is the coarse co-assembly map away from $N$, $\\mu_N\\colon K(c_N(M;\\mathbb{C}\\ell_{1,r}))\\to K(A_0(N\\Subset M;\\mathbb{C}\\ell_{0,r}))$, the boundary map of the short exact sequence defining the stable Higson corona $c_N(M;B)$ as the quotient of functions of vanishing variation away from $N$ by functions supported near $N$. The cap product with this corona turns the lifted Thom class into the boundary map connecting the index away from $N$ to the submanifold index.","core_discovery":"The central claim is that the relative coarse index is a submanifold-index machine: for a complete spin manifold $M$ and a submanifold $N$ with K-oriented normal bundle, satisfying the paper's geometric hypotheses, the relation $J/D_M\\|N;\\mathbb{C}\\ell_{0,r}K\\circ\\tau = i_*\\operatorname{ind}(/D_N)$ holds (Theorem 5.13). Combined with the index away from $N$, this gives, whenever the Thom class $\\tau$ lies in the image of the coarse co-assembly map $\\mu_N$, a wrong way map $x\\mapsto\\partial(x\\cap\\tilde{\\tau})$ that sends $\\operatorname{ind}(/D_M,N)$ to $i_*\\operatorname{ind}(/D_N)$ (Theorem 7.2). The consequence is the paper's curvature obstruction: if $i_*\\operatorname{ind}(/D_N)\\neq 0$, then no metric of uniformly positive scalar curvature away from $N$ can lie in the same quasi-isometry class as the original metric. For multi-partitioned manifolds the paper proves the required lift of the Thom class and thereby constructs the wrong way map unconditionally (Theorem 7.5).","pith_inferences":["An implicit extension is that any positive answer to the paper's open question about the Thom-class lift would immediately turn every such submanifold with nonzero submanifold index into an obstruction to uniformly positive scalar curvature away from it; the most natural candidates are hypersurfaces with bounded-geometry K-oriented normal bundles.","The cap-product formulation suggests that the wrong-way map could be refined to secondary index invariants in the K-theory of the stable Higson corona, yielding vanishing results for higher relative index classes rather than only for the primary coarse index.","Because the paper works with arbitrary graded C*-algebra coefficients, the same machinery should transplant to twisted or equivariant settings, where a submanifold index obstruction could be tested in the equivariant or twisted Roe algebra."],"forward_implications":["If a metric on $M$ has uniformly positive scalar curvature away from $N$, then $\\operatorname{ind}(/D_M,N)=0$ (Corollary 6.3).","Whenever the Thom-class lift exists, $i_*\\operatorname{ind}(/D_N)\\neq 0$ rules out such metrics in the same quasi-isometry class as the original metric.","For multi-partitioned manifolds the condition is automatic, so the wrong way map exists unconditionally and sends $\\operatorname{ind}(/D_M,N)$ to $i_*\\operatorname{ind}(/D_N)$ (Theorem 7.5).","The relative coarse index is bordism invariant and, for twisted operators, depends only on the difference $J E K-J F K$ of the two twisting bundles, not on the bundles themselves.","The separated-hypersurface case reproduces the partitioned-manifold index theorem as a corollary, and the multi-partitioned case reproduces the earlier partitioned-manifold index theorems."],"supporting_citations":[{"why":"Provides the source of the relative index construction and its relative index theorem, which the paper coarsifies.","marker":"[GL83]"},{"why":"Supplies the A-linear Dirac operator framework, the E-theory composition-product formula, and the stable Higson corona methods used throughout.","marker":"[Wul19]"},{"why":"Introduces the coarse co-assembly map and Higson compactification that the paper generalizes to arbitrary closed subsets.","marker":"[EM06]"},{"why":"Provides an earlier relative coarse index and the invertibility-away-from-a-subset criterion used for local indices.","marker":"[Roe16]"},{"why":"Sets out the spin geometry, spin^c structures, and Thom class conventions underlying the spinor bundles and Thom class.","marker":"[LM89]"},{"why":"Contains the Bott periodicity and functional-calculus estimates used in proving the Thom-class identity.","marker":"[HKT98]"},{"why":"Proves the multi-partitioned manifold index theorem that the paper recovers through the co-assembly lift.","marker":"[SZ18]"},{"why":"Gives the alternative Mayer-Vietoris wrong way map construction for partitioned manifolds that the paper's construction generalizes.","marker":"[Sie12]"}],"fun_headline_variants":["Submanifold index obstructs uniform positive scalar curvature","Relative index forces submanifold scalar-curvature obstructions","Nonzero submanifold index blocks UPSC","Wrong-way maps from coarse index detect no UPSC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The obstruction machinery works only if the class built from the submanifold's normal bundle can be lifted through the map that connects the coarse geometry of M far from N to the local K-theory near N; the paper establishes this lift only for multi-partitioned manifolds and leaves the general case open.","fun_headline_variants_meta":{"raw":{"variants":["Submanifold index obstructs uniform positive scalar curvature","Relative index forces submanifold scalar-curvature obstructions","Nonzero submanifold index blocks UPSC","Wrong-way maps from coarse index detect no UPSC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001113,"raw_usage":{"total_tokens":4599,"prompt_tokens":871,"completion_tokens":3728,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":3661}},"tokens_in":487,"tokens_out":3728,"duration_ms":27455,"temperature":1.0,"reasoning_tokens":3661,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:17:21.566778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete multi-partitioned spin manifold, for example $M=N\\times\\mathbb{R}$ with $N$ a compact odd-dimensional spin manifold of nonzero index, compute $\\operatorname{ind}(/D_M,N)$ through the wrong way map and compare it with $i_*\\operatorname{ind}(/D_N)$ under the map of Theorem 7.5; the theorem predicts equality, so a nonzero discrepancy would falsify it. Equivalently, an $M$ of this type with $i_*\\operatorname{ind}(/D_N)\\neq 0$ that nevertheless admits a metric of uniformly positive scalar curvature away from $N$ in the same quasi-isometry class would disprove the obstruction claim.","supporting_citations":[],"review_version":1}