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Primitive path homology

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Pith's one-line read The paper introduces a primitive path homology theory for digraphs and proves that it coincides with the established path homology exactly on asymmetric digraphs, with a two-vertex example showing the two theories differ in general.

desk verdict Primitive path homology is a genuinely new digraph invariant that provably matches GLMY path homology on asymmetric digraphs; worth refereeing despite several deferred proofs. read the letter →

arxiv 2411.18955 v1 pith:J4ZV76IW submitted 2024-11-28 math.AT

Jingyan Li , Yuri Muranov , Jie Wu , Shing-Tung Yau This is my paper · ORCID
classification math.AT MSC 55N3505C2005C3805C2555U15
keywords primitivepathhomologydigraphasymmetricchaincomplexclusterfixedtailvertexhead
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

This paper introduces a primitive path homology theory for finite simple directed graphs, defined by a leaner boundary condition than the usual path homology: a chain is kept only if its alternating face sum remains inside the allowed paths of the graph. Its main theorem states that for asymmetric digraphs, those with no pair of opposite arrows, the primitive chain complex is naturally isomorphic to the ordinary path chain complex, so the two homology theories agree. The agreement is sharp: in the complete two-vertex digraph, where both arrows are present, primitive homology is $\mathbb{Z}$ in degree 1 while path homology is 0. The construction also yields homology theories of paths with a fixed tail vertex, a fixed head vertex, or both, and these theories are functorial only on asymmetric digraphs; an example shows the naturality diagram can fail when the target graph has two-way arrows.

What carries the argument

The primitive chain complex $\Pi_*(G) = \{w \in A_n(G) : \partial w \in A_{n-1}(G)\}$ uses the ordinary alternating face differential $\partial$ on the free module of allowed elementary paths of length $n$. The load-bearing fact is that asymmetry of the digraph rules out the repeated-consecutive-vertex fragment $iji$ inside allowed paths; this makes the projection from the free path module to its quotient by irregular paths injective on allowed paths, so the condition defining $\Pi$ coincides with the condition defining the standard path complex $\Omega$. The proof of Theorem 3.7 is carried by a commutative diagram comparing the two complexes through this projection, and Proposition 3.9 shows that membership in $\Pi_n$ is equivalent to each individual face $\partial_m(w)$ staying allowed, which is what makes the tail/head/cluster decompositions possible.

What would settle it

Compute the first primitive path homology group of the complete digraph on two vertices with arrows in both directions: the paper predicts $\mathbb{Z}$, while the standard path homology group is 0. For the main theorem, search for any asymmetric digraph $G$ and any $n \geq 0$ for which the natural map $\Pi_n(G) \to \Omega_n(G)$ is not an isomorphism, since Theorem 3.7 asserts no such example exists.

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Extended reading notes

Core claim

The central claim is Theorem 3.7: for every asymmetric digraph $G$, the primitive chain complex $\Pi_*(G)$ is naturally isomorphic to the path chain complex $\Omega_*(G)$, and therefore the primitive path homology $H_*(G)$ equals the path homology $\mathrm{H}_*(G)$. The isomorphism is the identity on allowed elementary paths in each dimension; it is natural because, with no opposite arrows present, no allowed path contains a fragment $iji$, so the difference between the two boundary conditions disappears. For general digraphs the theories diverge, as shown in Example 3.2 where the complete two-vertex digraph has primitive $H_1 = \mathbb{Z}$ but path $\mathrm{H}_1 = 0$. The paper also proves functoriality of the primitive theory on the subcategory of asymmetric digraphs, obtains a direct-sum decomposition of each $\Pi_n(G)$ into tail-fixed, head-fixed, and $(a,b)$-cluster summands, and relates primitive $(a,b)$-cluster homology of a directed suspension to primitive homology shifted by two dimensions. One structural limitation is noted in Remark 4.8: the $(a,b)$-cluster submodules do not themselves form a chain complex, so cluster homology must be built through a separate complex.

Load-bearing premise

The entire comparison rests on the digraph having no pair of opposite arrows; if even one two-way arrow appears, the primitive and path complexes may disagree and the induced maps on them need not commute.

Editorial extensions

If this is right

  • On every asymmetric digraph, primitive path homology equals the established path homology, so calculations on such graphs can use the simpler primitive boundary condition.
  • Primitive $(a,b)$-cluster homology of the directed suspension $\mathrm{Sd}\,G$ is isomorphic to the primitive homology of $G$ shifted down by two dimensions.
  • The tail-fixed, head-fixed, and cluster decompositions give a direct-sum splitting of $\Pi_n(G)$, allowing primitive homology computations to be organized by tail, head, or both.
  • The tail-fixed and head-fixed primitive homology theories are functorial on asymmetric digraphs and satisfy the cone and suspension shift relations of Theorem 6.10.
  • Since a digraph homotopy equivalent to a point via path homology can still have nonzero primitive $H_1$, primitive path homology is not homotopy invariant in the same sense as path homology.

Reading between the lines

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

  • If asymmetry is exactly where the two theories agree, the difference between primitive and path homology on general digraphs could be interpreted as a measure of two-cycle content, a reading the paper does not pursue.
  • The fixed-tail and fixed-head complexes are natural candidates for relative homology groups of a vertex pair; a long exact sequence relating $\Pi_*$, $\Theta^{[a,\cdot]}_*$, and $\Theta^{[\cdot,b]}_*$ would be a plausible next step not taken here.
  • Because the boundary condition is purely combinatorial, primitive path homology may be noticeably easier to compute algorithmically than path homology; benchmarking on random asymmetric digraphs would test this expectation.
  • In characteristic 2 the alternating signs that produce the two-vertex counterexample vanish, so testing whether the isomorphism of Theorem 3.7 survives there is a concrete check of how essential the signs are.
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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 / 5 minor

Summary. The paper introduces a primitive path homology theory for simple digraphs, defined by the submodule Π_n(G) = {w ∈ A_n(G) | ∂w ∈ A_{n-1}(G)} inside the free path module Λ_*(V_G), in contrast with the GLMY path homology Ω_*(G) defined in the quotient module R_*(V_G). The central theorem (Theorem 3.7) states that for every asymmetric digraph G, the chain complexes Π_*(G) and Ω_*(G) are naturally isomorphic, so the two homology theories coincide on the subcategory AD; Example 3.2 shows they differ in general, e.g., for the complete two-vertex digraph the primitive H_1 is Z while the path H_1 is 0. The paper then develops cluster digraphs, primitive (a,b)-cluster homology, primitive tail/head-fixed homology, establishes suspension and inverse-digraph results (Theorems 3.11, 5.12, 6.10), and states functoriality on AD (Theorem 7.1).

Significance. If the results are correct, the paper provides a genuinely new variant of path homology that agrees with the established GLMY theory on asymmetric digraphs but is distinct in the presence of two-way arrows. The main comparison theorem is a clean and useful statement, and the fixed-end and cluster constructions offer potential tools for studying paths with prescribed endpoints. The paper contains many worked examples that illustrate the behavior of the new invariants. However, the novelty is incremental relative to the existing GLMY framework, and the manuscript's reliance on 'similar' proofs and several incomplete arguments reduces its reliability in its current form.

major comments (3)
  1. [Theorem 3.7, around Eq. (3.20)] The proof of the natural isomorphism Π_*(G) ≅ Ω_*(G) is missing a key justification. The text claims that 'Σ r_β e_β ∈ A_{n-1}(G) if and only if Σ r_β[e_β] ∈ η_{n-1}(A_{n-1}(G))', but this equivalence is not immediate. From p_{n-1}(∂w) = η_{n-1}(v) one obtains ∂w − v ∈ I_{n-1}, which does not by itself imply ∂w ∈ A_{n-1}. The missing step is the observation that in an asymmetric digraph every face of an allowed elementary path is regular, so the classes [e_β] form a basis of their span in R_{n-1}; this makes p_{n-1} injective on the span of the regular paths appearing in ∂w. Please add this argument to make the proof complete.
  2. [Proposition 3.9, Eq. (3.31)] The claim that S_{n-1}^m(G)∩S_{n-1}^l(G) = 0 for m ≠ l is false in general. An elementary path can have two non-arrow 'skip' pairs at different positions and can arise both as ∂_m(p) and ∂_l(q) for different allowed paths p and q. For example, in the asymmetric graph with vertices 0,1,2,3,4,5 and arrows 0→1, 1→2, 2→3, 3→4, 0→5, 5→1, 2→5, 5→3, the path e_{0,1,2,3,4} equals ∂_1(e_{0,5,1,2,3,4}) and also ∂_3(e_{0,1,2,5,3,4}), and it satisfies both (0,2) ∉ E and (2,4) ∉ E. The proof of Proposition 3.9 therefore needs a different argument.
  3. [Theorem 7.1] The functoriality theorem for the cluster, tail, and head primitive homologies is stated with a one-line proof that says 'Similar to the proof of Proposition 3.5'. This is a load-bearing result for Sections 5 and 6, and the verification that f_♯ preserves the d-condition defining Θ^{[a,b]}_* (and the analogous conditions for Θ^{[a,·]}_* and Θ^{[·,b]}_*) requires checking the compatibility of the differential d with f_♯ and the boundary cases, especially n=1 and n=2. Please provide the details or a precise reduction to the already-proved cases.
minor comments (5)
  1. [Example 5.13] The sentence 'It is an easy exercise for readers to check similarly above that H[a,b]_2(H) = 0 for i≥5' contains a typo and a delegated computation: it should likely read 'H[a,b]_i(H) = 0 for i≥5', and the vanishing claim should be either proved or explicitly marked as an exercise with enough details to be reproducible.
  2. [Notation throughout] The two homology theories are both denoted by H_* (or H_n) in the text, which makes sentences like 'the groups H_n(G) and H_n(G) are not isomorphic' in Example 3.2 unintelligible. Please use distinct notation, e.g., a different font or a superscript for the primitive theory, and apply it consistently.
  3. [Section 3, Eq. (3.2)] There is a typo in the reference 'By (3.1) and (2.3, for any digraph...'; the closing parenthesis should appear after (2.3).
  4. [Proposition 3.9 proof] In the final line of the proof, '∂_m^n (w)' should be '∂_m(w)'. The superscript n appears to be a typographical error.
  5. [Abstract and Introduction] There are grammatical errors and missing words, e.g., 'Let a, b two different vertices of a digraph' should be 'Let a and b be two different vertices of a digraph'. A careful language edit would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.7 is proved from the definitions with the asymmetry hypothesis used exactly where needed; self-citations are background, not load-bearing.

full rationale

The central derivation is self-contained. The primitive complex Π_∗(G) (eq. 3.1) is defined by the condition ∂w∈A_{n−1}(G) inside the unreduced module Λ_∗(V_G), while Ω_∗(G) (eq. 2.3) imposes the same condition after passing to the regular quotient R_∗=Λ_∗/I_∗. Theorem 3.7 does not assume its conclusion: for asymmetric G every allowed path has no subpath of the form iji, so every face in (3.17) is regular, the projection p_n is injective on Π_n, and the equivalence in (3.20) proves Π_n≅Ω_n exactly. The proof is carried out in the paper; it does not import Theorem 3.7 from a citation. Lemma 3.3 (naturality of f♯ against ∂) is proved by a complete case analysis, and Proposition 3.5's proof is self-contained even though it begins 'Follows from Lemma 3.3 and [7, Lemma 2.3]'. The cited [6, Lemma 2.2] is used only to prove the cluster decomposition of the known Ω-complex (Theorem 2.3), not to establish the new comparison. Theorem 3.11, 5.10, 5.12, 6.10 and 7.1 are delegated as 'similar' proofs or proved in full; omitted details are brevity, not circularity. Example 3.2 shows the two theories genuinely differ outside AD, so the primitive theory is not a rename of GLMY path homology. No fitted parameter, no equation reduces the new object to an input by construction, and no load-bearing premise rests on a self-citation. Score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted constants or tuned parameters appear anywhere; the paper is a pure construction. The external inputs are the GLMY path complex and its cluster decomposition lemma, plus standard chain-complex facts. No new physical or ontological entities are postulated.

assumptions (3)
  • standard math The GLMY path homology theory, in particular the cluster decomposition lemma of [6] restated as Theorem 2.3, is correct and can be used as a black box.
    Invoked in Section 2 and used to prove Corollary 2.4 and the direct-sum decompositions in Sections 3 and 4.
  • standard math The free chain complex Lambda_* with the alternating face differential satisfies d^2=0, and the quotient by irregular paths R_* is a chain complex.
    Used in Section 3 to define Omega_n and in the proof of the isomorphism Pi_n to Omega_n.
  • domain assumption All digraphs are finite, simple, and have nonempty vertex sets; maps may collapse arrows to vertices.
    Stated at the beginning of Section 2; finiteness and simplicity are needed for the basis arguments and for A_n intersect I_n = 0.

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Pith. "Pith review of Primitive path homology." pith.science (2026). https://pith.science/paper/J4ZV76IW

@misc{pith2026241118955,
  author       = {Pith},
  title        = {Pith review of: Primitive path homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4ZV76IW}},
  note         = {Machine review of arXiv:2411.18955}
}
abstract

In this paper we introduce a primitive path homology theory on the category of simple digraphs. On the subcategory of asymmetric digraphs, this theory coincides with the path homology theory which was introduced by Grigor'yan, Lin, Muranov, and Yau, but these theories are different in general case. We study properties of the primitive path homology and describe relations between the primitive path homology and the path homology. Let $a,b$ two different vertices of a digraph. Our approach gives a possibility to construct primitive homology theories of paths which have a given tail vertex $a$ or (and) a given head vertex $b$. We study these theories and describe also relationships between them and the path homology theory.

Figures

Figures reproduced from arXiv: 2411.18955 by the authors.

Figure 1
Figure 1. Digraph cube I 3 . We have A [0,7] i = 0 for i 6= 3 and A [0,3] 3 = he0137, e0237, e0157, e0457, e0467, e0267i. We have the following differential d: A [0,7] 3 → Λ [0,7] 2 : d(e0137) = e037 − e017, d(e0237) = e037 − e027, d(e0157) = e057 − e017, d(e0457) = e057 − e047, d(e0467) = e067 − e047, d(e0267) = e067 − e027. (5.8) We can check directly that the kernel of d is generated by the element w = e0137 − e0237 − e015… view at source ↗
Figure 2
Figure 2. The planar digraph G from Example 5.7 (iii). We have A [0,8] n =    0 for n = 0, he08i ∼= Z for n = 1, he028, e038i ∼= Z 2 for n = 2, he0178, e0158, e0258, e0368, e0468, e0478i ∼= Z 6 for n = 3, 0 for n ≥ 4. (5.9) We have the following differential in dimensions 1 and 2: d(e08) = 0, d(e028) = d(e038) = e08. (5.10) In dimension 3, we have the following differential d: A [0,8] 3 → Λ [0,7] 2 : d(e0178) =… view at source ↗

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Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages · cited by 1 Pith paper

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