The generalized tangle hypothesis is a refinement of the cobordism hypothesis.

History

The original tangle hypothesis was formulated in

• John Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory 1995 (arXiv)

as follows:

In the limit $k\to \mathrm{\infty }$, this gives:

In extended toplogical quantum field theory, which is really the representation theory of these cobordism $n$-categories, we expect:

Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain:

Further discussion can be found here:

• Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory. PhD thesis, Sheffield (2008) (arXiv)

More recently Mike Hopkins and Jacob Lurie have claimed (see Hopkins-Lurie on Baez-Dolan) to have formalized and proven this hypothesis in the context of (infinity,n)-categories modeled on complete Segal spaces. See:

• Jacob Lurie, On the classification of topological field theories (pdf)

where an (infinity,n)-category of cobordisms is defined and shown to lead to a formalization and proof of the cobordism hypothesis. Lurie explains his work here:

• Jacob Lurie, TQFT and the cobordism hypothesis, videos of 4 lectures at the Geometry Research Group, Mathematics Department, University of Texas Austin.

Lecture notes for Lurie’s talks should eventually appear at the Geometry Research Group website.

Statement of the generalized tangle hypothesis

The $k$-tuply monoidal $n$-category of $G$-structured $n$-tangles in the $(n+k)$-cube is the fundamental $(n+k)$-category with duals of $(MG,Z)$.

• $MG$ is the Thom space of group $G$.
• $G$ can be any group equipped with a homomorphism to $O(k)$. (comment)

Further resources

• Jacob Lurie, TQFT and the Cobordism Hypothesis (video, notes)

Discussion

• Café