Quasimonophobic graphs and degree spectral sequences in discrete cubical homology

Samira Sahar Jamil , Mark Behrens

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Pith reviewed 2026-05-07 00:40 UTC · model grok-4.3

classification 🧮 math.AT math.CO

keywords homologycubicaldegreediscretespectralgraphgraphsinjective

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The pith

Quasimonophobicity on graphs forces the degree spectral sequence of discrete cubical homology to vanish in selected bidegrees and identifies injective homology with the homology of the filled CW complex, enabling explicit computation of H_2 on Greene sphere graphs.

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

Graphs can be studied with algebraic tools that count higher-dimensional cubes built from their edges. The authors filter these cubes by a numerical degree that records how injective their faces are. From this filtration they obtain a spectral sequence that begins with ordinary graph homology and converges to a stricter injective version. They define a combinatorial property called quasimonophobicity that makes many pages of the spectral sequence zero and simultaneously shows that the injective homology equals the ordinary homology of a space in which all subcubes have been filled in. This machinery is applied to a family of sphere-like graphs to obtain concrete homology groups in dimension two.

Core claim

Quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees and that H_n^{inj}(G) is isomorphic to the homology of the CW complex obtained by filling in subcubes of the graph; these results are applied to compute H_2(G_n^{sph}).

Load-bearing premise

That the combinatorial condition of quasimonophobicity, once imposed, is sufficient to guarantee both the spectral-sequence vanishing and the CW-complex isomorphism without additional hidden restrictions on the graphs or on the choice of singular cubes.

read the original abstract

We introduce the degree filtration on the discrete cubical chain complex of a graph, defined in terms of the maximal injective dimension of the facets of singular $n$-cubes, and study the degree spectral sequence which arises from this filtration. This spectral sequence interpolates between the discrete cubical homology of a graph $H_n(G)$ and the injective homology $H_n^{inj}(G)$, a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Babson et al. we introduce the combinatorial condition of quasimonophobicity on graphs, and show quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees, and implies $H_n^{inj}(G)$ is isomorphic to the homology of the CW complex obtained by ``filling in'' subcubes of the graph. These results are applied to compute $H_2(G_n^{sph})$ for the Greene sphere graphs $G^{sph}_n$.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; the paper necessarily relies on the standard axioms of algebraic topology (chain complexes, spectral sequences, singular cubes) and on the prior framework of Babson et al., but no explicit free parameters, ad-hoc axioms, or invented entities are visible.

pith-pipeline@v0.9.0 · 5445 in / 1178 out tokens · 20625 ms · 2026-05-07T00:40:31.009631+00:00 · methodology

Quasimonophobic graphs and degree spectral sequences in discrete cubical homology

Core claim

Load-bearing premise

discussion (0)