The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature

Alexander Engel; Christopher Wulff

REVIEW 2 major objections 3 minor 1 cited by

The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Useful framework with a real gap in Theorem 7.2's bounded geometry hypothesis; worth a serious referee, but needs fixing. read the letter →

arxiv 2506.14301 v1 pith:X4QN55OD submitted 2025-06-17 math.KT math.GTmath.OA

Alexander Engel , Christopher Wulff This is my paper · ORCID

classification math.KTmath.GTmath.OA MSC 19K5653C2758J22

keywords relativecoarseindextheoryRoealgebrawrongwaymapsuniformlypositivescalarcurvatureThomclassco-assemblymapmulti-partitionedmanifolds

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What 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.

The 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[§7, Theorem 7.2; §6.2, Lemma 6.11 and Theorem 6.12] 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.
[§5.2, proof of Theorem 5.13] 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.

minor comments (3)

[§1, Introduction] 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.
[§4.1 and §3] 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.
[§5.2, Definition 5.12] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central wrong-way map theorem is a genuine derivation from independent E-theory and cap-product results.

full rationale

After walking the claimed derivation chain, I find no step in which a prediction is equivalent to its inputs by construction, nor any fitted parameter renamed as a prediction. The paper's central wrong-way map theorem (Theorem 7.2) is a genuine consequence of independent ingredients: the relative index formula (Theorem 4.10), the Thom class computation (Theorem 5.13), the cap-product index theorem (Theorem 6.12), and the boundary diagram (7.1). Each of these is derived rather than posited as the conclusion. The Thom-class lifting assumption in Theorem 7.2 is explicitly stated and left as Question 1.7, then verified for multi-partitioned manifolds in Theorem 7.5; it is not a hidden input. The paper leans heavily on [Wul19] and, for non-separable E-theory exactness, [Wul22b], but those are prior results with proofs that do not contain the present theorem; under this ledger's rules they constitute independent support rather than circularity. Two non-circular concerns are noted but do not affect the verdict: the proof of Theorem 7.2 invokes Theorem 6.12 and Lemma 6.11, which are stated for manifolds of global bounded geometry, while Theorem 7.2 assumes only bounded geometry near N, and the paper explicitly leaves unchecked the equality of its relative index with Roe's. Neither is a definitional reduction or a fitted-input prediction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on the established coarse index and E-theory framework of [Wul19], on the non-separable E-theory of [Wul22b], and on geometric hypotheses such as bounded geometry, uniform embedding, and bounded geometry of S'_V. The one assumption that is both load-bearing and open in general is the liftability of the Thom class through the coarse co-assembly map away from N, which is only proved for multi-partitioned manifolds. No free parameters or invented physical entities are present.

assumptions (5)

standard math Framework of A-linear Dirac operators, E-theory classes, and composition formulas from Wulff [Wul19, Theorems 2.13, 2.17, 4.14]
The paper explicitly adopts [Wul19] as its foundation (Section 2) and uses its index pairing, bordism invariance, and E-theory classes as black boxes.
standard math E-theory for non-separable C*-algebras as developed in [Wul22b]
Used in Section 6 to obtain exact boundary E-theory classes and commutativity of diagrams; the paper notes the non-separable setting requires this model.
domain assumption Geometric hypotheses: N uniformly embedded, M has bounded geometry in an R-neighborhood of N, and the spinor Cℓ_{0,r}-bundle S'_V has bounded geometry
These are hypotheses of Theorems 5.13 and 7.2, introduced in Definition 5.7 and Lemma 5.11; they are geometric conditions on the ambient manifold and submanifold.
ad hoc to paper The Thom class tau lies in the image of the coarse co-assembly map away from N, mu_N
Stated as an assumption in Theorem 7.2 and posed as open Question 1.7; Theorem 7.5 verifies it only for multi-partitioned manifolds. The main obstruction machinery is conditional on it.
domain assumption K-orientability of the normal bundle V and existence of spinc structures
Required from Section 5 onward to define submanifold spinor bundles S_V and S'_V and the Thom class.

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Pith. "Pith review of The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature." pith.science (2026). https://pith.science/paper/X4QN55OD

@misc{pith2026250614301,
  author       = {Pith},
  title        = {Pith review of: The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4QN55OD}},
  note         = {Machine review of arXiv:2506.14301}
}

abstract

We provide a coarse version of the relative index of Gromov and Lawson and thoroughly establish all of its basic properties. As an application, we discuss a general procedure to construct wrong way maps on the $K$-theory of the Roe algebra mapping the coarse index class of the Dirac operator of a manifold to the one of a suitably embedded submanifold of arbitrary codimension, thereby establishing an abstract machinery to find obstructions to uniform positive scalar curvature coming from these submanifolds.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A partitioned manifold index theorem for noncompact hypersurfaces
math.DG 2025-07 accept novelty 6.0 of 10

For noncompact hypersurfaces satisfying mild geometric conditions, the partitioned index of a Dirac operator equals the index of the induced Dirac operator on the hypersurface.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [2]

Dualizing the coarse assembly map

Atlantis Stud. Dyn. Syst. Amsterdam: Atlantis Press, 2013.doi: 10.2991/978-94-6239-003-4. [EM06] H. Emerson and R. Meyer. “Dualizing the coarse assembly map”. In:J. Inst. Math. Jussieu5.2 (2006), pp. 161–186.doi: 10.1017/S147474800500023X. [Eng19] A. Engel. “UniformK-theory, and Poincaré duality for uniformK-homology”. English. In:J. Funct. Anal.276.7 (20...

work page doi:10.2991/978-94-6239-003-4 2006

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[Wul22b] C

doi: 10.3842/SIGMA.2022.057. [Wul22b] C. Wulff.Generalized asymptotic algebras andE-theory for non-separableC∗- algebras

work page doi:10.3842/sigma.2022.057 2022

[3] [135]

Cambridge University Press, 1988, pp

London Mathematical Society Lecture Note Series. Cambridge University Press, 1988, pp. 187–228. [Roe93] J. Roe.Coarse Cohomology and Index Theory on Complete Riemannian Mani- folds. Vol

work page 1988

[4] [1831]

EquivariantE-theory forC ∗-algebras

Lecture Notes in Math. Berlin: Springer, 2004, pp. 137–251. doi: 10.1007/978-3-540-39702-1_3. [GHT00] E. Guentner, N. Higson, and J. Trout. “EquivariantE-theory forC ∗-algebras”. In: Mem. Amer. Math. Soc.148.703 (2000), pp. viii+86.doi: 10.1090/memo/0703. [GL83] M. Gromov and H. B. Lawson. “Positive scalar curvature and the Dirac operator on complete Riem...

work page arXiv 2000

[5] [1970]

The coarse index class with support

[BE25] U. Bunke and A. Engel. “The coarse index class with support”. In:Tunisian J. Math. 7.2 (2025), pp. 339–378. [BL] U. Bunke and M. Ludewig. “Coronas and Callias type operators in coarse geome- try”. arXiv:2411.01646. [Ebe18] J. Ebert. “Elliptic regularity for Dirac operators on families of noncompact manifolds”

arXiv 2025

[6] [1989]

Positive curvature, partial vanishing theorems and coarse indices

[Roe16] J. Roe. “Positive curvature, partial vanishing theorems and coarse indices”. English. In: Proc. Edinb. Math. Soc., II. Ser.59.1 (2016), pp. 223–233. doi: 10.1017/S0013091514000236. [Roe88a] J. Roe. “An Index Theorem on Open Manifolds, I”. In:J. Differential Geom.27 (1988), pp. 87–113. [Roe88b] J. Roe. “An Index Theorem on Open Manifolds, II”. In:J...

work page doi:10.1017/s0013091514000236 2016

[7] [1996]

The Mayer-Vietoris Sequence for the Analytic Structure Group

doi: 10.1090/cbms/090. [Shu92] M. A. Shubin. “Spectral Theory of Elliptic Operators on Non-Compact Manifolds”. In: Astérisque 207 (1992), pp. 35–108. [Sie12] P. Siegel. “The Mayer–Vietoris sequence for the analytic structure group”. arXiv:1212.0241v1

work page Pith review arXiv 1992

[8] [2012]

Large scale index of multi-partitioned manifolds

[SZ18] T. Schick and M. E. Zadeh. “Large scale index of multi-partitioned manifolds”. In: J. Noncommut. Geom.12 (2018), pp. 439–456. [Tro00] J. Trout. “On gradedK-theory, elliptic operators and the functional calculus”. In: Illinois J. Math.44.2 (2000), pp. 294–309. [Wul12] C. Wulff. “Bordism invariance of the coarse index”. In:Proc. Amer. Math. Soc. 140....

work page doi:10.1090/s0002-9939-2012-11546-1 2018