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arxiv: 2502.00122 · v2 · submitted 2025-01-31 · 🧮 math.AT · math.MG

Homotopy connectivity of v{C}ech complexes of spheres

Henry Adams , Ekansh Jauhari , Sucharita Mallick This is my paper

Pith reviewed 2026-05-23 04:00 UTC · model grok-4.3

classification 🧮 math.AT math.MG
keywords Čech complexeshomotopy connectivityBorsuk graphsspheresconicitycoveringshomological dimensionnerve
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The pith

The homotopy type of the Čech complex of an n-sphere changes infinitely many times as the scale r varies over (0, π) for every n at least 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes upper and lower bounds on the homotopy connectivity of intrinsic Čech complexes of the n-sphere at each scale r between 0 and π. The upper bound is expressed using the chromatic number of the Borsuk graph of the sphere while the lower bound follows from the conicity of Čech complexes built on sufficiently dense finite subsets. These matching bounds imply that the homotopy type must switch at infinitely many values of r, and the authors conjecture that only countably many distinct types appear overall. A reader would care because the result shows that the topological information captured by these complexes on even the simplest curved spaces is highly sensitive to the precise value of the scale parameter.

Core claim

The paper shows that for n greater than or equal to 1 the homotopy type of the Čech complex of S^n at scale r changes infinitely many times as r varies over the interval (0, π). The upper bound on connectivity comes from the chromatic numbers of Borsuk graphs of spheres and is sharp for n equals 1. The lower bound is obtained from the conicity of Čech complexes of sufficiently dense finite subsets of S^n. The same techniques also give a lower bound on the homological dimension of Čech complexes of finite subsets in terms of their packings.

What carries the argument

Conicity in the sense of Barmak of the Čech complexes of sufficiently dense finite subsets of S^n, which supplies the lower bound on homotopy connectivity in terms of coverings.

If this is right

  • The homotopy connectivity is bounded above by the chromatic number of the Borsuk graph of the sphere at the given scale.
  • For n equals 1 the upper bound coming from the Borsuk graph is achieved.
  • The homological dimension of Čech complexes of finite point sets on the sphere is bounded below by a quantity derived from their packings.
  • There exist infinitely many distinct homotopy types as r varies, and the total number of distinct types is conjectured to be countable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The jumps in homotopy type are likely to occur at isolated scales because chromatic numbers and covering numbers are discrete quantities.
  • The same covering-based control of connectivity could be applied to Čech complexes on other Riemannian manifolds to detect scale-dependent topology.
  • Persistent homology computations on spherical data sets would encounter repeated changes in the underlying complex topology and therefore require dense sampling of the scale parameter.

Load-bearing premise

The lower bound on connectivity at most scales requires that Čech complexes of sufficiently dense finite subsets of the sphere are conical.

What would settle it

An explicit computation of the homotopy connectivity of the Čech complex of S^1 or S^2 at a dense sequence of scales r in (0, π) that finds only finitely many distinct values would show that the bounds do not force infinitely many changes.

Figures

Figures reproduced from arXiv: 2502.00122 by Ekansh Jauhari, Henry Adams, Sucharita Mallick.

Figure 1
Figure 1. Figure 1: Intervals where C( ˇ S 2 ; r) may have connectivity k−1, given by Theorem 1.1. The blue endpoints are plotted using only approximate values of covS2 (2k + 2) or of 2 covS2 (k + 1); see [41]. For a metric space (X, d), the r-covering number, denoted numCoverX(r), is the smallest integer n such that X can be covered by n closed balls of radius r > 0. Rearranging the inequalities in Theorem 1.1 gives the foll… view at source ↗
Figure 2
Figure 2. Figure 2: The homotopy types of C( ˇ S 1 ; r) as r varies [1] are indicated by black bars. Theorem 1.1 gives intervals where C( ˇ S 1 ; r) may have connectivity k − 1, which are indicated by colored bars. Left endpoints of orange bars (when k −1 is even) are tight. We also study the homological dimension of Cech complexes of (finite subsets) of spheres. The ˇ Z-homological dimension of a topological space Y , denote… view at source ↗
Figure 3
Figure 3. Figure 3: We have χ(Bor(S & 2 ; δ)) = 4 for all s2 < δ < π, as illustrated by the covering of S 2 by four sets each of diameter s2. The problem of bounding χ(Bor(S n ; δ)) for 0 < δ < π close to π is classical. One version of the Borsuk–Ulam theorem (the Lyusternik–Schnirel’man–Borsuk covering theorem) gives, and is in fact equivalent to, the statement that χ(Bor[S n ; δ]) ≥ n + 2 for all δ < π; see, for example, [1… view at source ↗
read the original abstract

Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $\pi$. The intrinsic \v{C}ech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of \v{C}ech complexes of spheres at each scale between $0$ and $\pi$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of \v{C}ech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the \v{C}ech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,\pi)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of \v{C}ech complexes of finite subsets of $S^n$ in terms of their packings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to control the homotopy connectivity of the intrinsic Čech complex Čech(S^n,r) of the n-sphere (geodesic metric, diameter π) via coverings. Upper bounds derive from chromatic numbers of Borsuk graphs (sharp for n=1). Lower bounds are obtained from conicity (Barmak) of Čech complexes on sufficiently dense finite subsets P⊂S^n. These imply that the homotopy type of Čech(S^n,r) changes infinitely many times as r varies over (0,π); the authors conjecture only countably many changes. An additional lower bound on homological dimension of Čech complexes of finite subsets is given in terms of packings.

Significance. If the bounds are established, the work supplies explicit scale-dependent control on the homotopy type of these complexes, yielding the new result of infinitely many changes. The combination of Borsuk-graph chromatic numbers for upper bounds and conicity for lower bounds is a coherent strategy, and the infinite-variation statement would be of interest in persistent homology and topological data analysis on spheres.

major comments (2)
  1. [Lower bound via conicity (abstract; relevant sections on finite subsets)] The lower-bound argument (abstract and the sections deriving connectivity from conicity) obtains high connectivity of Čech(S^n,r) from conicity of Čech(P,r) for dense finite P⊂S^n. Čech(P,r) is a subcomplex of Čech(S^n,r), but contractibility of a subcomplex supplies no automatic lower bound on the connectivity of the supercomplex. Any claim that Čech(S^n,r) is homotopy equivalent to Čech(P,r) (or that the colimit over denser P preserves the connectivity lower bound) therefore requires an explicit approximation, nerve theorem, or deformation-retract argument; this step is load-bearing for the infinite-changes result.
  2. [Upper bound via Borsuk graphs] § on upper bounds: the sharpness claim for n=1 is stated but the precise identification between the Borsuk-graph chromatic number and the connectivity drop must be verified against the definition of the Čech complex; any gap here would affect the claimed sharpness.
minor comments (2)
  1. [Introduction] Notation for the intrinsic Čech complex could be introduced with an explicit definition or reference to the nerve construction in the first section.
  2. [Abstract and conclusion] The conjecture that there are only countably many changes is stated in the abstract but receives no further discussion or supporting heuristic; moving it to a remark or adding a brief paragraph would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We respond point-by-point to the major comments below, acknowledging where additional justification is needed and outlining the planned revisions.

read point-by-point responses

  1. Referee: [Lower bound via conicity (abstract; relevant sections on finite subsets)] The lower-bound argument (abstract and the sections deriving connectivity from conicity) obtains high connectivity of Čech(S^n,r) from conicity of Čech(P,r) for dense finite P⊂S^n. Čech(P,r) is a subcomplex of Čech(S^n,r), but contractibility of a subcomplex supplies no automatic lower bound on the connectivity of the supercomplex. Any claim that Čech(S^n,r) is homotopy equivalent to Čech(P,r) (or that the colimit over denser P preserves the connectivity lower bound) therefore requires an explicit approximation, nerve theorem, or deformation-retract argument; this step is load-bearing for the infinite-changes result.

    Authors: The referee correctly identifies that the passage from conicity of Čech(P,r) for dense finite P to a connectivity lower bound for Čech(S^n,r) requires an explicit justification, as subcomplex connectivity does not automatically transfer. In the current manuscript the argument relies on taking limits of denser and denser finite subsets, but this step is only sketched. We will revise the relevant sections (including the abstract if needed for precision) to include a rigorous approximation argument: specifically, we will show that for any fixed r the Čech complex Čech(S^n,r) is the direct limit (in the sense of increasing unions under the Hausdorff metric on point sets) of the Čech(P,r) as the mesh of P tends to zero, and that homotopy connectivity is preserved under this limit by a standard stability result for nerves of ball coverings (invoking the nerve theorem for the open cover by r-balls and a controlled deformation retract between nearby complexes). This will make the infinite-changes result fully rigorous. revision: yes

  2. Referee: [Upper bound via Borsuk graphs] § on upper bounds: the sharpness claim for n=1 is stated but the precise identification between the Borsuk-graph chromatic number and the connectivity drop must be verified against the definition of the Čech complex; any gap here would affect the claimed sharpness.

    Authors: We agree that the sharpness statement for n=1 benefits from an explicit verification tying the Borsuk-graph chromatic number directly to the definition of the intrinsic Čech complex. For the circle, the Borsuk graph B(S^1,θ) with θ=2r has edges between antipodal pairs at geodesic distance >θ; its chromatic number jumps between 2 and 3 at the critical scales where the Čech complex on S^1 ceases to be path-connected or simply connected. We will add a short dedicated paragraph (or subsection) in the upper-bounds section that computes both quantities side-by-side for the circle, confirming that the connectivity drop occurs exactly when the chromatic number increases, thereby establishing sharpness without gap. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from external results

full rationale

The paper's upper bound on homotopy connectivity derives from chromatic numbers of Borsuk graphs (external combinatorial result). The lower bound invokes Barmak's prior conicity theorem applied to Čech complexes of finite dense subsets of S^n (external reference, not self-citation). The claim of infinitely many homotopy type changes follows directly from these independent bounds without reduction to fitted parameters or self-referential definitions. No load-bearing self-citation chains or ansatzes smuggled via citation are present. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions in algebraic topology (Borsuk graphs, chromatic numbers, Barmak conicity) without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Chromatic numbers of Borsuk graphs of spheres control upper bounds on connectivity
    Invoked for the upper bound on homotopy connectivity.
  • domain assumption Conicity of Čech complexes for sufficiently dense finite subsets (Barmak)
    Invoked for the lower bound on connectivity.

pith-pipeline@v0.9.0 · 5757 in / 1219 out tokens · 78943 ms · 2026-05-23T04:00:26.033536+00:00 · methodology

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Reference graph

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