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Rigidity of Higson coronas

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Pith's one-line read Under $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$, homeomorphic Higson coronas force coarse equivalence.

desk verdict Significant result with a repairable but load-bearing gap: the OCA application in Section 4 uses a non-Polish space of maps. read the letter →

arxiv 2502.10073 v1 pith:5EZYBRFB submitted 2025-02-14 math.LO math.MGmath.OA

Alessandro Vignati This is my paper · ORCID
classification math.LOmath.MGmath.OA MSC 03E3503E5003E6546L8554D40
keywords HigsoncoronacoarseequivalencerigidityOpenColouringAxiomMartin'suniformlylocallyfinitemetricspacesslowlyoscillatingfunctionsC*-algebras
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 Higson corona of a discrete metric space is its boundary at infinity: the leftover part of a canonical compactification after the space itself is removed. This paper asks how much of the original geometry survives in that boundary, and answers that under the set-theoretic axioms $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$ all of it does: two uniformly locally finite metric spaces — bounded-geometry spaces in which balls of any fixed radius are uniformly finite — with homeomorphic Higson coronas are coarsely equivalent, meaning their large-scale geometries agree. That statement is not provable in $\mathsf{ZFC}$ alone: under the Continuum Hypothesis there are many non-coarsely-equivalent spaces with isomorphic Higson coronas, so the axioms are doing genuine work. The proof shows that every isomorphism between the function algebras of Higson coronas is trivial in the paper's sense, namely induced by a coarse map of the underlying metric spaces.

What carries the argument

The central device is the coarse weak Extension Principle (cwEP), an analogue for Higson coronas of a weak Extension Principle for remainders of compactifications. A map satisfies cwEP when, off a nowhere-dense set, it is trivial, meaning it is induced by a coarse proper map of the underlying metric spaces. The proof of local triviality uses two specially chosen sparse sequences of annuli, the reduced products of finite-diameter metric spaces that they generate, and two imported lifting theorems: coordinate-fixing functions between such reduced products are of product form, and positive order-zero contractions from $\ell^\infty/c_0$ to itself lift on a nonmeager dense ideal. Once a homomorphism has product form on the two sparse sequences, a $\mathsf{ZFC}$ argument assembles the local coordinate data, using marker functions, into a single coarse proper map that induces the homomorphism.

What would settle it

Build, in a model of $\mathsf{OCA}$ plus $\mathsf{MA}_{\aleph_1}$, two uniformly locally finite metric spaces with homeomorphic Higson coronas that are not coarsely equivalent; any such pair would refute Theorem 1.4 directly. Alternatively, exhibit a coordinate-fixing function between the reduced products $N_f$ of slowly oscillating functions defined in Section 4 that is not of product form; that would break the specific application of the metric lifting theorem on which local triviality depends.

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Extended reading notes

Core claim

Formally, the main result (Theorem 1.4) is that, assuming $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$, if $X$ and $Y$ are uniformly locally finite metric spaces and their Higson coronas $\nu X$ and $\nu Y$ are homeomorphic, then $X$ and $Y$ are coarsely equivalent. The Higson corona $\nu X$ is the spectrum of the quotient $C^*$-algebra $C_\nu(X)=\mathrm{Ch}(X)/C_0(X)$, where $\mathrm{Ch}(X)$ is the algebra of bounded slowly oscillating functions. Along the way the paper proves a coarse weak Extension Principle for surjective maps, showing that any unital surjective $*$-homomorphism between such algebras is trivial off a nowhere-dense set, where a trivial homomorphism is one induced by a coarse proper map. It also proves an embedding version: a continuous injection between coronas restricts to a coarse embedding on a clopen subspace. Since a trivial isomorphism is induced by a coarse equivalence, Theorem 1.4 follows.

Load-bearing premise

The load-bearing premise is that two imported lifting results — one stated from the author's own unreviewed preprint — really apply to the reduced products of slowly oscillating functions used in Section 4; if either theorem fails, or covers fewer maps than claimed, the local triviality step that powers the rigidity theorem collapses.

Editorial extensions

If this is right

  • Under $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$, homeomorphic Higson coronas imply coarse equivalence for uniformly locally finite metric spaces, resolving the open rigidity question in that setting.
  • The rigidity statement is independent of $\mathsf{ZFC}$: it holds under $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$, and known constructions under the Continuum Hypothesis produce non-coarsely-equivalent spaces with isomorphic Higson coronas.
  • Every surjective $*$-homomorphism between Higson-corona algebras is trivial in the paper's sense, so its dual map comes from a coarse proper map off a nowhere-dense set.
  • A continuous injection between Higson coronas restricts to a coarse embedding from a clopen subspace, with the complement mapped to a nowhere-dense set.
  • The coarse weak Extension Principle for surjective maps follows from the same axioms, giving a structural description of all such homomorphisms.

Reading between the lines

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

  • A natural next step would be to test whether the same rigidity survives for non-uniformly locally finite or non-discrete bounded-geometry spaces, where the reduced-product machinery does not directly apply.
  • If the metric lifting theorem used here is later extended to weaker continuity assumptions, the hypothesis could perhaps be weakened from $\mathsf{OCA}$ plus $\mathsf{MA}_{\aleph_1}$ to $\mathsf{OCA}$ alone; the paper's comments suggest this is open.
  • The cwEP formulation may transfer to other coarse-geometry quotients, such as stable Higson coronas, where analogous rigidity questions remain open.
  • One could look for a $\mathsf{ZFC}$ example of non-coarsely-equivalent spaces with homeomorphic Higson coronas; finding one would show the axioms are necessary rather than merely sufficient.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the rigidity of Higson coronas for uniformly locally finite (u.l.f.) metric spaces. It introduces a notion of trivial *-homomorphisms between algebras of slowly oscillating functions modulo compact functions, and the coarse weak Extension Principle (cwEP). The main theorem (Theorem 1.4) asserts that under OCA and MA_ℵ1, if two u.l.f. metric spaces have homeomorphic Higson coronas, then they are coarsely equivalent. The proof architecture is: (i) in ZFC, show that a surjective *-homomorphism which is of product form on two sparse sequences is trivial (Section 3); (ii) under OCA+MA, show that every *-homomorphism between such algebras is of product form on the relevant sparse sequences (Theorem 4.1); (iii) combine to get the rigidity theorem and a companion theorem on injective maps (Theorem 1.5). The paper relies on two imported lifting results: Theorem 4.3 (De Bondt–Vignati, to appear) and Theorem 4.4, derived by a two-sentence sketch from [24] and [39].

Significance. If the main result is correct, it provides a positive answer to Question 1.3 under mild set-theoretic hypotheses, complementing the CH-based counterexamples of Protasov and Brian–Farah. The paper also introduces a useful framework (cwEP) and proves a substantial ZFC result (Theorem 3.12) showing that local triviality implies triviality for surjective maps. The architecture is clear, and the connection to known lifting theorems for reduced products is natural. However, the proof of the crucial local-existence step (Theorem 4.1) contains a serious gap in the application of OCA, and there is an off-by-one error in Claim 4.6. As written, the main theorem is not established.

major comments (3)
  1. [Section 4, after Claim 4.9] The application of OCA is invalid. The paper states that Maps(DX_i_n, DY_i_n), the set of all maps from DX_i_n to DY_i_n, 'is a Polish space with the sup-distance'. For nonempty finite X_i_n, DX_i_n = D^{X_i_n} is an uncountable compact metric space (a finite-dimensional cube), and the set of all functions from this uncountable space to any space containing two distinct points is not separable in the sup metric (e.g., the characteristic functions of singletons form an uncountable discrete family). Hence Maps(DX_i_n, DY_i_n) is not Polish, and the product ∏_n Maps(DX_i_n, DY_i_n) × N^N is not separably metrizable. Consequently, OCA cannot be applied to the colouring K^ε_0 as stated. This is load-bearing: the OCA application produces the uniform family of maps α_n used to define the product-form lift Λ in Claim 4.12. A repair would require either applying OCA directly to [N^N]^2 with a proof that K^ε_0 is open in the usual topology, or proving that the lifts α_{f,n} can be chosen from a separable family (e.g., continuous maps).
  2. [Claim 4.6] There is an off-by-one error in the proof. The set T is defined as {k_n + 1 | n ∈ N}, but the geometry of the annuli in Definition 3.10 shows that X^e_{k_n} is contained in the union of X^o_{k_n} and X^o_{k_n - 1} (since X^e_n = [2^{6n+1}, 2^{6n+5}] and X^o_{n-1} = [2^{6n-2}, 2^{6n+2}]), not in X^o_{k_n} ∪ X^o_{k_n+1}. The argument as written therefore does not establish the claimed containment. The proof should use T = {k_n - 1} (after passing to a subsequence with k_n > 0). This affects the proof that χ_{\tilde{U}} q_{N,i} =_{C0(Y)} q_{N,i}, which is used to show that U is clopen.
  3. [Theorem 4.4] The paper's proof of Theorem 4.4 is only a two-sentence sketch, and Theorem 4.3 is cited as 'to appear' without a proof or preprint reference. Since these lifting results are the entire engine behind the local-existence step (Theorem 4.1), the paper is not self-contained. In particular, it is not verified that the hypotheses of Theorem 4.3 are satisfied by the spaces N_f and ∏ D^{Y_n} / ⊕ D^{Y_n}, especially regarding the quotient equivalence relation used in the definition of N_f. The author should either include complete proofs of Theorems 4.3 and 4.4, or state and prove the precise special cases needed here, rather than referring to unpublished work.
minor comments (4)
  1. [Proposition 2.4, proof of (2)] In the proof of part (2), the sets A and B are both defined as {y'_n}. This is clearly a typo; one should be {y_n} and the other {y'_n}. As written, the displayed equality of norms is nonsensical.
  2. [Claim 4.11] In the definition of the relation R_n, the text reads 'if f(k) ≥ m then g(k) ≤ f(k)', but m is not defined; it should be 'if f(k) ≥ n'.
  3. [Definition 2.7] There is a duplicated word: 'we say the the coarse weak Extension Principle holds'.
  4. [Section 4, proof of Theorem 4.4] The construction of \tilde{ρ} from the values on characteristic functions is not fully rigorous: it is stated that \tilde{ρ} = ∑ \tilde{ρ}_n is positive and order zero, but the extension from the specified values to all of ℓ∞ is only implicit. Since simple functions are dense in ℓ∞, this can be filled in, but the details should be supplied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Higson-corona rigidity conclusion is not assumed in the imported lifting theorems, and the paper's local-to-global argument is a direct ZFC proof.

full rationale

The derivation is not circular. Theorem 1.4 is obtained from Theorem 4.2 (cwEP(s)) together with Proposition 2.4 and Corollary 2.5, where triviality is defined via the existence of a coarse map and proved by direct norm estimates (Lemma 2.3, Proposition 2.4). The set-theoretic hypotheses OCA and MA_aleph1 enter through two imported lifting theorems: Theorem 4.3 from [11] (De Bondt-Vignati) and Theorem 4.4 derived from [24] and [39]. These are general statements about coordinate-fixing maps between reduced products of separable metric spaces and about positive order-zero contractions on ell_infty/c0; neither mentions Higson coronas nor asserts the rigidity of C_nu(X), so neither is equivalent by construction to the target theorem. The self-citation is substantial, and one source ([11]) is an unreviewed to-appear preprint, which is an evidence-quality concern; under the stated rules, however, that does not amount to circularity. The local-triviality-to-triviality step in Section 3 is proved in ZFC with no imported rigidity input. There is a possible non-circular correctness gap in the OCA uniformization argument: the paper asserts that Maps(DX_i_n, DY_i_n) with the sup-distance is a Polish space, although the space of all maps between uncountable compact metric spaces is not separable; this is a proof-gap concern, not a definitional reduction of the conclusion to the assumptions. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the author's prior work is used to forbid alternatives.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters appear: the paper is pure mathematics, and the few numerical choices (the exponent 6 in Definition 3.10, the threshold 1/24 in Theorem 3.12, the function j in Claim 4.6) are structural constants, not fitted values. The proof assumes ZFC plus OCA and MA_ℵ1; the paper notes OCA implies b = ω_2 and contradicts CH, and both axioms are consistent relative to ZFC without large cardinals. Imported black boxes: Theorem 4.3 (metric lifting theorem of De Bondt-Vignati [11], an overlapping-author preprint), Theorem 4.4 (sketched from [24] and [39]), and the CH counterexamples of Protasov [28] and Brian-Farah [9] used to establish independence. Standard background includes Gelfand duality (which converts homeomorphic coronas into isomorphic C*-algebras), the lifting of projections in ℓ∞/c0, and Parovicenko's theorem for the dimension-zero remark.

assumptions (7)
  • standard math ZFC
    Background framework for all arguments. The independence claim is relative to ZFC models.
  • domain assumption OCA (Open Colouring Axiom)
    Assumed in Section 1.1; used directly in the colouring argument (Claim 4.10) and needed for Theorems 4.3 and 4.4. OCA implies b = ω_2 and contradicts CH.
  • domain assumption MA_ℵ1 (Martin's Axiom at ℵ1)
    Assumed in Section 1.1. Footnote 6 notes it is used only for the lifting theorems [24] and [11] and conjectures it may be unnecessary.
  • domain assumption Theorem 4.3, the De Bondt-Vignati metric lifting theorem [11]
    Used as a black box in Section 4 to turn coordinate-fixing maps into product-form lifts. The paper cites it as 'to appear in Comptes Rendus'; its hypotheses and correctness are not verified in the text.
  • domain assumption Theorem 4.4, lifting of order-zero contractions
    The proof is a two-sentence sketch citing [24, Theorem 4.6] and [24, Lemma 3.11]/[39, Proposition 2.18]. The stated positivity, order-zero, and strong-continuity properties are asserted, not derived.
  • domain assumption CH counterexamples of Protasov [28] and Brian-Farah [9]
    Used to show the rigidity statement fails under CH, giving independence from ZFC. [9] is cited as an arXiv preprint at the time of submission.
  • standard math Gelfand duality for commutative unital C*-algebras
    Bridges homeomorphic coronas to isomorphic function algebras Cν(X) = Ch(X)/C0(X), the setting of all the proofs. Standard.

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Pith. "Pith review of Rigidity of Higson coronas." pith.science (2026). https://pith.science/paper/5EZYBRFB

@misc{pith2026250210073,
  author       = {Pith},
  title        = {Pith review of: Rigidity of Higson coronas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5EZYBRFB}},
  note         = {Machine review of arXiv:2502.10073}
}
abstract

We show that under mild set theoretic hypotheses we have rigidity for algebras of continuous functions over Higson coronas, topological spaces arising in coarse geometry. In particular, we show that under $\mathsf{OCA}$ and $\mathsf {MA}_{\aleph_1}$, if two uniformly locally finite metric spaces $X$ and $Y$ have homeomorphic Higson coronas $\nu X$ and $\nu Y$, then $X$ and $Y$ are coarsely equivalent, a statement which provably does not follow from $\mathsf{ZFC}$ alone.

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