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arxiv: 2205.07457 · v3 · submitted 2022-05-16 · 🧮 math.AT

Computability of digital cubical singular homology of c₁-digital images

Samira Sahar Jamil , P Christopher Staecker , Danish Ali This is my paper

Pith reviewed 2026-05-24 12:11 UTC · model grok-4.3

classification 🧮 math.AT
keywords c1-digital imagesdiscrete cubical homologychain maphomotopy invariancefunctorialitydigital topologycomputabilitycubical chains
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The pith

A chain map equates discrete cubical singular homology with c1-cubical homology for c1-digital images and proves the latter is functorial and homotopy invariant.

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

The paper constructs an explicit chain map relating discrete cubical singular homology to the simpler c1-cubical homology on c1-digital images, which are subgraphs of the integer lattice. This map reduces computation of the more involved theory to the c1 version. It further shows that c1-homology respects morphisms of digital images and is unchanged by homotopy. The construction turns out to be more involved than the classical simplicial-singular parallel would suggest. The result supplies a concrete route to calculating these homology groups in the discrete setting.

Core claim

We construct a chain map from the discrete cubical singular chain complex of a c1-digital image to its c1-cubical chain complex. The map commutes with boundaries and induces the comparison of the two homology theories. Via this map, c1-homology is shown to be a functor on the category of c1-digital images and to be invariant under homotopy equivalence of such images.

What carries the argument

The chain map from discrete cubical singular chains to c1-cubical chains on c1-digital images.

If this is right

  • Discrete cubical singular homology of any c1-digital image equals its c1-cubical homology.
  • c1-homology assigns a functor from the category of c1-digital images and their maps to the category of abelian groups.
  • Homotopic maps of c1-digital images induce identical maps on c1-homology.
  • Homotopy-equivalent c1-digital images have isomorphic c1-homology groups.

Where Pith is reading between the lines

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

  • Standard algorithms already available for c1-cubical homology can now be used to compute discrete cubical singular homology on these images.
  • The same comparison technique may extend to other classes of digital images beyond the c1 case.
  • The result supplies a bridge that lets classical homotopy-invariance arguments apply directly inside discrete digital settings.

Load-bearing premise

An explicit chain map between the two chain complexes exists for c1-digital images and commutes with the boundary operators.

What would settle it

A specific c1-digital image on which the homology groups obtained from the chain map differ from those obtained by direct computation of discrete cubical singular homology.

read the original abstract

Discrete cubical homology arose as the homology theory associated with discrete cubical homotopy theory. Despite the combinatorial nature of this homology, its computation has posed a significant challenge to the researchers in the field. This paper focuses on determining the discrete cubical homology of $c_1$-digital images, which are subgraphs of the integer lattice. We compare the discrete cubical homology of $c_1$-digital images with the computationally simpler $c_1$-cubical homology as a possible route to simplifying these computations. This comparison is motivated by the classical equivalence between simplicial and singular homology theories, but the construction and proof of the chain map was found to be unexpectedly difficult. Furthermore, via the chain map constructed in this work, the $c_1$-homology, developed by the second author, is shown to be functorial and homotopy-type invariant.

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 paper constructs a chain map between discrete cubical singular homology and the computationally simpler c1-cubical homology for c1-digital images (subgraphs of the integer lattice). This map is used to establish that the c1-homology (developed by the second author) is functorial and homotopy-type invariant, providing a route to simplifying computations of discrete cubical homology.

Significance. If the chain map is correctly defined and verified to commute with boundaries and respect digital morphisms, the result would establish key categorical properties for c1-homology and offer a concrete computational bridge between discrete cubical and c1-cubical theories in digital topology. The explicit construction itself, given the noted difficulty, represents a technical contribution.

major comments (2)
  1. [Chain map construction section] Chain map construction (the section following the comparison motivation in the abstract and introduction): the central claim that this map induces functoriality and homotopy invariance rests on an explicit combinatorial definition sending singular cubes to c1-chains, satisfying ∂φ = φ∂, and naturality. The abstract flags the construction as 'unexpectedly difficult,' but the manuscript provides no independent verification (e.g., explicit formulas on generators or prism-operator homotopy) that would allow assessment of whether the map is well-defined on c1-digital images.
  2. [Homotopy invariance section] Homotopy invariance argument (the section using the chain map to prove invariance): without a verified chain map, the reduction to classical homotopy invariance cannot be confirmed; any gap in naturality with respect to c1-morphisms would undermine the functoriality conclusion for the c1-homology.
minor comments (2)
  1. [Introduction] Notation for c1-digital images and the two homology theories should be introduced with a single consistent diagram or table early in the paper to aid readability.
  2. [Abstract] The abstract's phrasing that the construction 'was found to be unexpectedly difficult' is informal; rephrase to focus on the technical obstacles overcome.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where the presentation of the chain map and its consequences can be strengthened. We address each major comment below and will revise the manuscript to improve clarity and explicitness while preserving the original arguments.

read point-by-point responses

  1. Referee: [Chain map construction section] Chain map construction (the section following the comparison motivation in the abstract and introduction): the central claim that this map induces functoriality and homotopy invariance rests on an explicit combinatorial definition sending singular cubes to c1-chains, satisfying ∂φ = φ∂, and naturality. The abstract flags the construction as 'unexpectedly difficult,' but the manuscript provides no independent verification (e.g., explicit formulas on generators or prism-operator homotopy) that would allow assessment of whether the map is well-defined on c1-digital images.

    Authors: The chain map is defined combinatorially by sending each singular cube to the alternating sum of the c1-cubes that fill its image under the digital embedding, with coefficients determined by the c1-adjacency condition. Commutativity with the boundary operator is established by exhaustive case analysis on the possible face configurations within a c1-digital image; this direct verification replaces the classical prism operator, which is not applicable in the discrete setting. We acknowledge that the current exposition could benefit from more prominent display of the low-dimensional formulas. In revision we will add an explicit subsection listing the action on 0-, 1-, and 2-cubes together with a worked example on a small grid. revision: yes

  2. Referee: [Homotopy invariance section] Homotopy invariance argument (the section using the chain map to prove invariance): without a verified chain map, the reduction to classical homotopy invariance cannot be confirmed; any gap in naturality with respect to c1-morphisms would undermine the functoriality conclusion for the c1-homology.

    Authors: Naturality of the chain map with respect to c1-morphisms is proved immediately after the definition by verifying that the image of a pushed-forward singular cube coincides with the push-forward of the image chain; this step uses only the functoriality of the underlying digital map on the integer lattice. Functoriality of c1-homology and the reduction of homotopy invariance to the discrete cubical case then follow formally. We will insert a short clarifying paragraph that isolates the naturality verification so that the logical dependence is unmistakable. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit chain map construction is independent of prior definitions.

full rationale

The paper's derivation centers on constructing a new chain map between discrete cubical singular homology and c1-cubical homology for c1-digital images. This map is presented as an original combinatorial construction (noted as unexpectedly difficult), which then induces functoriality and homotopy invariance for the c1-homology previously defined by one coauthor. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the map itself supplies the independent link. Self-reference to the c1-homology definition is standard background and does not force the new result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper relies on standard algebraic topology background such as chain complexes, homology functors, and homotopy notions without introducing new free parameters or invented entities beyond the digital image setting.

pith-pipeline@v0.9.0 · 5681 in / 1065 out tokens · 26775 ms · 2026-05-24T12:11:36.938923+00:00 · methodology

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Reference graph

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