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PreprintCC BY 4.0arXiv

Quasimonophobic graphs and degree spectral sequences in discrete cubical homology

  • Samira Sahar Jamil
  • Mark Behrens
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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 Hn(G) and the injective homology Hninj(G), a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Greene and the first author, 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 Hninj(G) is isomorphic to the homology of the CW complex obtained by ``filling in'' subcubes of the graph. These results are applied to compute H2(Gnsph) for the Greene sphere graphs Gsphn.

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Cite

BibTeXRIS
BibTeX
@misc{jamil2026quasimonophobic,
  title = {{Quasimonophobic graphs and degree spectral sequences in discrete cubical homology}},
  author = {Samira Sahar Jamil and Mark Behrens},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.03894},
  eprint = {2605.03894},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.03894},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quasimonophobic graphs and degree spectral sequences in discrete cubical homology
AU  - Samira Sahar Jamil
AU  - Mark Behrens
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2605.03894
UR  - https://arxiv.org/abs/2605.03894
AB  - 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 Greene and the first author, 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.
ER  -