nLab
extended cobordism

**manifolds** and **cobordisms** ## Definitions * manifold * coordinate chart, atlas * orientation, spin structure, string structure, fivebrane structure * smooth manifold, Riemannian manifold, complex manifold * cobordism * B-bordism * extended cobordism * cobordism category * (∞,n)-category of cobordisms * functorial quantum field theory * Thom spectrum * cobordism ring * genus * infinite dimensional manifold * Banach manifold, Hilbert manifold, ILH manifold, Frechet manifold, convenient manifold ## Genera and invariants * signature genus, Kervaire invariant * A-hat genus, Witten genus ## Theorem * Thom's transversality theorem * Pontrjagin-Thom construction * Galatius-Tillmann-Madsen-Weiss theorem * geometrization conjecture, * Poincaré conjecture * elliptization conjecture * cobordism hypothesis-theorem

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*** **functorial quantum field theory** ## Contents * cobordism category * cobordism * extended cobordism * (∞,n)-category of cobordisms * Riemannian bordism category * cobordism hypothesis * generalized tangle hypothesis * classification of TQFTs * FQFT * extended TQFT * CFT * vertex operator algebra * TQFT * Reshetikhin–Turaev model / Chern-Simons theory * HQFT * TCFT * A-model, B-model, Gromov-Witten theory * homological mirror symmetry * FQFT and cohomology * (1,1)-dimensional Euclidean field theories and K-theory * (2,1)-dimensional Euclidean field theory * geometric models for tmf * holographic principle of higher category theory * holographic principle * AdS/CFT correspondence * quantization via the A-model

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Contents

Idea

In so far as a cobordism connects its boundary components with each other, the idea of an extended cobordism is that its (pieces of) boundary components are themselves extended cobordisms between their (pieces of) boundary components.

The idea arose in the context of extended quantum field theories – which were originally thought of as representations of ordinary cobordism categories – when it was realized that a decent description of these QFTs requires assigning data to pieces of arbitrary codimension to a manifold. In the context of QFT this is often thought of as an incarnation of the notion of locality in physics, which says that every data assigned to a chunk of space must already be fixed by what is assigned to all pieces of any of its decomopositions.

Indeed, the idea of extended cobordisms led to the generalized tangle hypothesis which takes this localization concept to its extreme by asserting, roughly, that the representation of fully extended cobordisms is already entirely determined by what is assigned to the point.

Formalizations

For a long time only partial progress was made with formalizing the idea of extended cobordisms, which are expected to form an infinity-category or omega-category of sorts. One approach using the notion of Trimble n-category as

  • Eugenia Cheng and Nick Gurski, Towards an nn-category of cobordisms (tac)

Another formalization is by Marco Grandis, using multi-cospans, see

This is in principle very general, indicating that a general extended cobordisms should be a multi-cospan in Top. But details are worked out only in dimension up to 2.

In this dimension the resulting structure is closely related to the bicategory of cobordisms described in

More recently

This generalizes an idea which for n=2n=2 was for instance also applied by Chris Schommer-Prier, see

Revised on June 26, 2010 13:13:24 by Urs Schreiber (134.100.32.208)