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Poincaré Duality and Quadratic Refinements over Laurent Rings

  • Błażej Ruba
  • Bowen Yang
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Abstract

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

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Cite

BibTeXRIS
BibTeX
@misc{ruba2026poincaré,
  title = {{Poincaré Duality and Quadratic Refinements over Laurent Rings}},
  author = {Błażej Ruba and Bowen Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25767},
  eprint = {2609.25767},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25767},
}
RIS
TY  - UNPB
TI  - Poincaré Duality and Quadratic Refinements over Laurent Rings
AU  - Błażej Ruba
AU  - Bowen Yang
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25767
UR  - https://arxiv.org/abs/2609.25767
AB  - We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.
ER  -