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Mathematics > Algebraic Topology

arXiv:2212.04182 (math)
[Submitted on 8 Dec 2022]

Title:Computing Cohomology Rings in Cubical Agda

Authors:Thomas Lamiaux, Axel Ljungström, Anders Mörtberg
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Abstract:In Homotopy Type Theory, cohomology theories are studied synthetically using higher inductive types and univalence. This paper extends previous developments by providing the first fully mechanized definition of cohomology rings. These rings may be defined as direct sums of cohomology groups together with a multiplication induced by the cup product, and can in many cases be characterized as quotients of multivariate polynomial rings. To this end, we introduce appropriate definitions of direct sums and graded rings, which we then use to define both cohomology rings and multivariate polynomial rings. Using this, we compute the cohomology rings of some classical spaces, such as the spheres and the Klein bottle. The formalization is constructive so that it can be used to do concrete computations, and it relies on the Cubical Agda system which natively supports higher inductive types and computational univalence.
Subjects: Algebraic Topology (math.AT); Logic in Computer Science (cs.LO)
Cite as: arXiv:2212.04182 [math.AT]
  (or arXiv:2212.04182v1 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.2212.04182
Related DOI: https://doi.org/10.1145/3573105.3575677

Submission history

From: Anders Mörtberg [view email]
[v1] Thu, 8 Dec 2022 10:43:52 UTC (110 KB)
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