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The geometric cobordism hypothesis

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arxiv 2111.01095 v3 pith:TNT3KELT submitted 2021-11-01 math.AT math-phmath.CTmath.MPmath.QA

classification math.ATmath-phmath.CTmath.MPmath.QA
keywords geometricstructuresfieldtheoriesarbitrarycobordismextendedfully
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We prove a generalization of the cobordism hypothesis of Baez--Dolan and Hopkins--Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures, principal bundles with connections, or geometric string structures. Our methods rely on the locality property for fully extended functorial field theories established in arXiv:2011.01208, reducing the problem to the special case of geometrically framed bordism categories. As an application, we upgrade the classification of invertible fully extended topological field theories by B\"okstedt--Madsen and Schommer-Pries to nontopological field theories, generalizing the work of Galatius--Madsen--Tillmann--Weiss to arbitrary geometric structures.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Free phases of Majorana fermions: Tenfold ways compared

    math-ph 2025-07 conditional novelty 6.0 of 10

    Neutral free fermion SPT phases protected by a real Z2-graded C*-algebra A are classified by the real K-theory group K_2(A^op), unifying charged and neutral tenfold-way classifications via Morita equivalence.

  2. Is Crane--Yetter fully extended?

    math-ph 2025-06 conditional novelty 6.0 of 10

    Fully extended invertible 4D TQFTs valued in braided fusion categories form a Z/6-extension of the Witt group, so Crane-Yetter has six inequivalent point-refinements for fixed modular data.

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