{"id":"c38a35c1-7e2f-480f-9d51-ce0ffa90b3f9","arxiv_id":"2411.13294","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Topological overlap profiles are introduced as higher-dimensional analogues of the separation profile, with new calculations and coarse geometric obstructions.","lead":"This paper defines a new family of coarse invariants, topological overlap profiles, that measure how close finite parts of a simplicial complex are to being high-dimensional topological expanders. It uses these profiles to prove new theorems, including that every bounded-degree 1-dimensional topological expander contains a classical graph expander, and to rule out coarse maps from horocyclic products of trees to products of hyperbolic spaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the monotonicity theorem depends on a cited connect-the-dots lemma, whose hypotheses appear matched, but a direct check of [DK18, Prop. 9.48] is the one verification worth running.","rationale":"The reader's weakest assumption correctly identifies the cited extension lemma as the main external dependency of Theorem 1.2. I independently traced how it is used: a regular 0-skeleton map is extended simplex-by-simplex using uniform k-connectivity, giving uniformly bounded simplex diameters; then bounded degree of X and Y and regularity of the map give the uniform simplex-intersection bounds needed for Proposition 3.11. This is the standard connect-the-dots argument, and the setup appears to satisfy it, so the concern is a verification task rather than a discovered flaw. I also checked the other potentially fragile points: the upper bound for TO^d_Y in Theorem 1.19 follows from Corollary 4.8 and the product theorem; the lower bound for horocyclic products follows from Theorem 1.24 and Theorem 1.23; and the Euclidean upper bound, while intricate and with some typographical roughness, is a fillable constructive argument. The unproved Proposition 1.20 and Theorem 1.21 are explicitly flagged as not the main focus and are not used in the main obstruction theorem. Hence I find no load-bearing concern that would change the conditional verdict.","tokens_in":33959,"tokens_out":29241,"duration_ms":337890,"concrete_test":"Verify the exact statement of [DK18, Proposition 9.48] and check that its hypotheses (e.g., bounded geometry, uniform k-connectivity in the diameter-bounded sense, and geodesicity of the metric (1)) are satisfied by the target products H^k x (H^2)^{d-1} x R^l; if the proposition requires a stronger contractibility condition, Theorem 1.2 and Theorem 1.19 need a replacement extension argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The paper's central claims (monotonicity Theorem 1.2 and the regular-map obstruction Theorem 1.19) rest on a coherent chain: uniform k-connectivity of the target permits a bounded-diameter extension of a regular 0-skeleton map via [DK18, Prop. 9.48], and the estimates in Corollary 6.6 and Corollary 4.8 give the required separation of lower and upper bounds. The only load-bearing step not proved in the paper is that cited extension lemma; it is standard, and the hypotheses used here (bounded degree, uniform connectivity for all k <= dim(X), local finiteness of the target) appear to match its standard form, so I do not regard this as a real flaw. The stated-but-unproved Proposition 1.20 and Theorem 1.21 are clearly marked as not the main focus and are not used in the main obstruction, so they do not weaken the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces two families of coarse invariants for bounded-degree simplicial complexes and, more generally, bounded-geometry metric spaces: the simplicial topological overlap profiles sTO^q_X(r) and the metric topological overlap profiles TO^q_X(r). It proves that these profiles are monotone under regular maps between complexes with suitable uniform connectivity, that they recover the cutwidth profile in dimension 1, and that dimension-1 topological expanders contain graphical expanders. In higher dimensions it computes TO^q(R^n) up to equivalence, gives product and fibration upper bounds, and obtains lower bounds for horocyclic products of trees and for symmetric spaces. The main advertised application is an obstruction to regular maps from horocyclic products of trees into products of hyperbolic graphs and doubling spaces, and into symmetric spaces of small corank.","tokens_in":34095,"tokens_out":23698,"duration_ms":245708,"significance":"If the main results stand, the paper introduces a useful quantitative coarse analogue of higher-dimensional topological expansion, with several natural and nontrivial calculations. The monotonicity theorem under regular maps is a genuine higher-dimensional extension of the separation-profile philosophy, and the Euclidean calculations and the horocyclic-product lower bounds are substantive. The paper also connects topological overlap to coarse constructions and thick embeddings, extending earlier work of Barrett and Hume. The proofs of the central monotonicity, Euclidean upper bound, and coarse-construction theorems are given in detail, and the main applications do not depend on fitting parameters or on circular assumptions. However, the statement and proof of Theorem 1.1(i) have a gap, and the abstract and Theorem 1.19 use inconsistent indexing for the corank obstruction; these issues need to be resolved before the paper is fully reliable.","major_comments":[{"comment":"The proof does not establish that the subcomplexes Z_r produced from the liminf condition have dimension at most q, yet Definition 3.4 defines a q-dimensional topological expander as a family of finite q-dimensional complexes. The proof only shows that |Z_r| is unbounded and that sTO^q(Z_r) is linear in |Z_r|; if dim(Z_r) > q, these complexes are not q-dimensional topological expanders in the stated sense. The paper itself leaves open the analogous skeleton-reduction question for q ≥ 3 after Proposition 3.16 and in Question 1.26. Therefore Theorem 1.1(i) is not proved as stated unless one assumes dim(X) ≤ q or changes the definition of q-dimensional expander to allow higher-dimensional complexes.","section":"§3.1, proof of Theorem 1.1(i)"},{"comment":"The corank indexing in the abstract is inconsistent with Theorem 1.19 and with its proof. Theorem 1.19 states, for the horocyclic product H^{d+1} of d+1 trees, that there is no regular map to a symmetric space of corank strictly less than d, and the proof uses the lower bound on TO^d together with the Corollary 1.15 upper bound that requires corank + 1 ≤ d. The abstract instead says that for every k ≥ 2 there is no regular map from the k-fold horocyclic product to any symmetric space of corank strictly less than k. If k = d+1, the abstract requires corank < d+1, which is weaker than the theorem; if k denotes the number of trees, then for k=2 the abstract would assert that the 2-fold horocyclic product (which is quasi-isometric to the real hyperbolic plane) admits no regular map to H^2, a symmetric space of corank 1, which is false. The indexing in the abstract and introduction should be corrected to match Theorem 1.19.","section":"Abstract and §1.7/§6.3, Theorem 1.19"}],"minor_comments":[{"comment":"The arrow directions and composition order for the rescaled projection π_R are written incorrectly: the text says π_R : R D^k → R S^k, but the subsequent preimage argument requires π_R : R S^k → R D^k and the definition in the proof of Corollary 5.2 should be F = G ∘ π_R rather than F = π_R ∘ G. This is local and correctable but should be fixed for readability.","section":"§5.1.1, Corollary 5.2 and preceding paragraph"},{"comment":"The display 'TO^d_{H^{d+1}}(r) ≳ r/log(1+r)^q' uses an undefined q and should presumably be r/log(1+r)^d, matching Theorem 1.18.","section":"§6.3, proof of Theorem 1.19"},{"comment":"In the proof, the functions f and g are chosen with Ov(f) ≤ sTO^q_X(r) and Ov(g) ≤ sTO^{q'}_X(r); these should be TO^q_X(r) and TO^{q'}_Y(r), respectively.","section":"§5.2, proof of Theorem 1.8"},{"comment":"There is a typo in the sentence 'there is some Z_r ≤ I': it should be Z_r ≤ X. The same typo appears in the proof of Corollary 4.6, where 'Z_r ≤ I' should be 'Z_r ≤ X'.","section":"§3.1, proof of Theorem 1.1(i)"},{"comment":"These two results are stated without proof and are not used in the main obstruction theorem. If they are intended as contributions, proofs or a precise reference to a sequel are needed; otherwise they should be presented clearly as announced results rather than as proposition/theorem statements.","section":"§1.8, Proposition 1.20 and Theorem 1.21"},{"comment":"The uniformity argument bounding the number of source simplices mapping to a given target simplex is terse: after choosing a vertex (ω_0,...,ω_d) of σ, the sentence 'the number of simplices in S containing n_j is at most 2 deg S, and σ′ must be one of these' is ambiguous because σ′ is not defined in that paragraph. Please clarify that this bounds the number of possible source simplices A, independent of the chosen vertex and of the target simplex.","section":"§6.2, Step 3 of Theorem 6.4"}],"recommendation":"major_revision","confidential_remarks":"I believe the main coarse-geometric results, especially Theorem 1.2, Theorem 1.7, and the horocyclic-product obstruction in Theorem 1.19, are likely correct, and the detailed arguments in Sections 5 and 6 are a strength of the paper. The two major comments concern the introductory characterization in Theorem 1.1(i) and a mismatch between the abstract and the stated corank obstruction in Theorem 1.19; both are fixable within the scope of the paper, but they affect the reliability of the headline claims. The unproved Proposition 1.20 and Theorem 1.21 should also be explicitly marked as deferred if they remain without proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"David, have a look at this one. Hume introduces topological overlap profiles sTO^q and TO^q, a coarse family that quantifies how close finite subcomplexes are to q-dimensional topological expanders, and proves they are monotone under regular maps between uniformly connected complexes. The framework is a natural higher-dimensional analogue of the separation/cutwidth profiles, and unlike some profile constructions it actually does work: Theorem 1.7 gives the full Euclidean calculation TO^q(R^n) ≃ r^{1-q/n}, and Corollary 1.5 (1D topological expanders contain graphical expanders) is a clean, new-looking discrete result. The dimension-1 recovery of the HHKL cutwidth profile is a sanity check that lands. The horocyclic-product lower bounds and the resulting obstruction Theorem 1.19 are substantial applications, not afterthoughts.\n\nThe central proofs look coherent. I read the monotonicity argument and the coarse-construction chain in Section 6; no circularity. The one load-bearing external input is the connect-the-dots extension lemma cited from DK18 Prop 9.48 used for Theorem 1.2. The hypotheses appear to match (bounded degree, uniform k-connectivity), and this is standard, but it is not proved in the paper. If you referee, that is the single verification worth doing.\n\nThe soft spots are the stated-but-unproved Proposition 1.20 and Theorem 1.21 on converting thick embeddings to coarse constructions. They are clearly marked as not the main focus and are not used in the main obstruction, so this is a minor presentation issue, not a logical gap in the core claims. The Euclidean upper bound in Theorem 1.7 is long and intricate; I could not fully machine-check it, but the strategy is standard Guth-style and the lower bound via Gromov waist is solid.\n\nCitation pattern is honest: prior work of HHKL, Hum17, BH24, HMT20/22 is used as benchmark and tool, not as conclusion. No fitting of parameters anywhere.\n\nWho is this for? Geometric group theorists and anyone working on high-dimensional expanders or coarse invariants. It deserves a serious referee. I would accept for review and send to someone who can check the DK18 lemma and the Euclidean construction in detail.","headline":"New coarse invariants with real content: the overlap profiles are worth taking seriously, and the Euclidean calculation plus the graph-expander containment are the payoffs.","tokens_in":34632,"tokens_out":1440,"would_cite":true,"duration_ms":15264,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","53C23","05E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines topological overlap profiles for bounded-degree complexes, proves they are monotone under regular maps, and uses them to obstruct regular maps from horocyclic products of trees into hyperbolic products and lower-corank…","keywords":["topological expanders","topological overlap profiles","regular maps","coarse geometry","horocyclic products of trees","separation profile","cutwidth profile","symmetric spaces"],"falsifier":"Construct a bounded-degree, uniformly $k$-connected simplicial complex $Y$ and a regular map from the 0-skeleton of some bounded-degree complex $X$ into $Y^0$ such that no continuous extension maps each simplex of $X$ into a uniformly bounded subset of $Y$; this would break Theorem 1.2. Alternatively, exhibit a regular map from the 0-skeleton of the $k$-fold horocyclic product of 3-regular trees into a product $(\\mathbb{H}^2)^{k-2}\\times H\\times D$ with $H$ hyperbolic bounded degree and $D$ doubling, which would falsify Theorem 1.19.","tokens_in":33711,"feed_emoji":"📐","tokens_out":7865,"duration_ms":69774,"temperature":0.7,"pith_summary":"This paper introduces a quantitative measure of topological expansion for bounded-degree simplicial complexes: the topological overlap profile, a family of sublinear functions that record, for each scale r, how many simplices must meet the preimage of some point under any continuous map to Euclidean space. The main structural result is monotonicity: under regular maps between complexes with uniform higher connectivity, the profile of the source is bounded by a constant multiple of the profile of the target, so the profile acts as a coarse-geometric obstruction. The paper computes the profile of Euclidean spaces exactly up to constants, recovers the cutwidth profile in dimension one, and proves that every one-dimensional bounded-degree topological expander contains a graphical expander. It then uses coarse constructions into horocyclic products of trees to show that these tree products are topologically thick, and as a consequence proves that for every k at least 2 the k-fold horocyclic product of 3-regular trees admits no regular map into certain hyperbolic products or into symmetric spaces of corank less than k.","feed_headline":"New profile turns expansion into a map-blocking invariant","feed_subtitle":"Higher-dimensional separation-like profiles show the k-fold tree product cannot live in lower-rank symmetric spaces.","key_machinery":"The central object is the topological overlap profile: $sTO^q_X(r)$ is the maximal, over subcomplexes $Z$ of $X$ with at most $r$ vertices, of the minimal possible value of the largest number of simplices of $Z$ whose images under a map $Z \\to \\mathbb{R}^q$ meet a common point; the metric version $TO^q_X(r)$ counts covering balls of radius 1 instead of simplices. The main proof mechanism is a connect-the-dots lemma, cited from a standard geometric group theory text, which turns a regular map between 0-skeleta into a continuous map between the complexes with uniformly bounded simplex images whenever the target is uniformly $k$-connected for all $k$ up to the source dimension. Lower bounds are produced by coarse constructions, continuous maps that send skeleta to skeleta with uniformly bounded preimages of simplices, whose volume into the horocyclic product of trees is controlled by $r\\ln(1+r)^q$.","core_discovery":"The central claim is that topological expansion of bounded-degree complexes can be quantified by the profiles $sTO^q_X(r)$ and $TO^q_X(r)$, and that these profiles behave like the separation profile: they are monotone under regular maps between complexes with uniform connectivity up to the source dimension, and they are explicitly computable in key cases. Specifically, the paper claims $TO^q_{\\mathbb{R}^n}(r) \\simeq r^{1-q/n}$ for $1 \\le q < n$ and bounded for $q \\ge n$, that $sTO^1$ coincides up to constants with the cutwidth profile, and that a 1-dimensional bounded-degree topological expander necessarily contains a graphical expander. For the horocyclic product $H^{d+1}$ of $d+1$ copies of the 3-regular tree, it claims $sTO^q_{H^{d+1}}(r) \\gtrsim r/\\ln(1+r)^q$ for $q \\le d$, and consequently that there is no regular map from its 0-skeleton to any product $H \\times (\\mathbb{H}^2)^{d-1} \\times D$ with $H$ a bounded-degree hyperbolic graph and $D$ doubling, nor to any symmetric space whose non-compact factor has corank strictly less than $d$.","pith_inferences":["A natural extension of the monotonicity theorem would be to quasi-isometries under the same connectivity hypotheses, a step the paper does not take.","The Euclidean calculation suggests a testable recipe for other nilpotent groups: project a ball onto a low-dimensional factor along cosets and compare covering volumes, as the paper does for Heisenberg groups.","The coarse-construction lower bound for tree products may adapt to other buildings and to groups acting cocompactly on them, yielding rigid-rank obstructions beyond symmetric spaces.","The paper leaves open whether topological overlap profiles are determined by the $q$-skeleton for $q \\ge 3$; a positive or negative answer would clarify how much of the phenomenon is purely combinatorial."],"forward_implications":["Regular-map monotonicity makes $sTO^q$ and $TO^q$ genuine coarse invariants: any lower bound on the source profile rules out the existence of a regular map into a target with a smaller profile.","The Euclidean computation $TO^q_{\\mathbb{R}^n}(r) \\simeq r^{1-q/n}$ gives a sharp quantitative form of waist- and width-volume-type inequalities for maps to $\\mathbb{R}^q$.","Dimension-one equivalence with cutwidth means all known cutwidth and separation profile bounds for graphs transfer directly to topological overlap.","Every bounded-degree one-dimensional topological expander contains a graphical expander, so the two notions of expansion are not independent in dimension one.","The non-existence of regular maps from horocyclic tree products into hyperbolic products and lower-corank symmetric spaces provides a new family of coarse non-embeddability results."],"supporting_citations":[{"why":"Supplies the connect-the-dots extension lemma that turns a regular map between 0-skeleta into a continuous map with uniformly bounded simplex images, the key step in Theorem 1.2.","marker":"[DK18, Proposition 9.48]"},{"why":"Gromov's waist inequality provides the lower bound in the Euclidean calculation of Theorem 1.7.","marker":"[Gro83]"},{"why":"Guth's width-volume inequality furnishes the technical upper bound for the Euclidean profile in Theorem 1.7.","marker":"[Gut06]"},{"why":"Defines the cutwidth profile and supplies the equivalence result and many upper bounds used throughout, especially in Section 4.","marker":"[HHKL]"},{"why":"Provides the separation-profile characterization of expanders and the subcomplex Cheeger bound used to extract a graphical expander from a topological expander.","marker":"[Hum17]"},{"why":"Introduces thick embeddings and coarse wirings for graphs, which this paper generalizes to simplicial complexes and coarse constructions.","marker":"[BH24]"},{"why":"Provides biLipschitz embeddings of products of trees into rank-$d$ symmetric spaces, used to deduce topological thickness of those symmetric spaces.","marker":"[BN24]"},{"why":"Constructs bounded-degree topological expanders in every dimension, used to convert coarse-construction volume bounds into lower bounds on $sTO^q$ via Theorem 1.23.","marker":"[EK15]"},{"why":"Bonk-Schramm embedding of hyperbolic graphs into Euclidean space is used in Theorem 1.19 to pass from a hypothetical regular map to a contradiction.","marker":"[BS00]"},{"why":"Assouad's embedding theorem for doubling metric spaces is used in Theorem 1.19 to embed the doubling factor $D$ into some $\\mathbb{R}^l$.","marker":"[Ass83]"}],"fun_headline_variants":["Topological expansion profiles block coarse embeddings","New invariants obstruct embeddings of tree products","Exact profiles for Euclidean topological expansion","Horocyclic tree products fail to embed in symmetric spaces","Expansion profiles make coarse embeddings impossible"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The monotonicity theorem depends on a cited extension lemma asserting that a regular map between 0-skeleta extends to a continuous map between complexes with each simplex mapped into a uniformly bounded ball, provided the target is uniformly $k$-connected up to the source dimension; the paper cites this lemma rather than proving it in the bounded-degree, not necessarily locally finite setting.","fun_headline_variants_meta":{"raw":{"variants":["Topological expansion profiles block coarse embeddings","New invariants obstruct embeddings of tree products","Exact profiles for Euclidean topological expansion","Horocyclic tree products fail to embed in symmetric spaces","Expansion profiles make coarse embeddings impossible"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000278,"raw_usage":{"total_tokens":1761,"prompt_tokens":1159,"completion_tokens":602,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":775,"completion_tokens_details":{"reasoning_tokens":536}},"tokens_in":775,"tokens_out":602,"duration_ms":6549,"temperature":1.0,"reasoning_tokens":536,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:35:37.589796+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a bounded-degree, uniformly $k$-connected simplicial complex $Y$ and a regular map from the 0-skeleton of some bounded-degree complex $X$ into $Y^0$ such that no continuous extension maps each simplex of $X$ into a uniformly bounded subset of $Y$; this would break Theorem 1.2. Alternatively, exhibit a regular map from the 0-skeleton of the $k$-fold horocyclic product of 3-regular trees into a product $(\\mathbb{H}^2)^{k-2}\\times H\\times D$ with $H$ hyperbolic bounded degree and $D$ doubling, which would falsify Theorem 1.19.","supporting_citations":[],"review_version":1}