Cobordism conjecture

The cobordism conjecture is one of Baez and Dolan established guess from the mathematics with which the higher category theory to the classification of topological quantum field theories to be applied.

Building on a work by Jacob Lurie , David Ayala and John Francis published an article on ArXiv in 2017 in which the evidence of the cobordism conjecture is traced back to the correctness of a conjecture about factoring homology .

Formulation of the presumption

For every symmetric monoidal -category in which dual objects and adjoint morphisms can be formed, one has a bijection between the -valent symmetric monoidal functors of the cobordism category and the objects of . ${\displaystyle (\infty ,n)}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$

motivation

Symmetric monoidal functors from the cobordism category correspond to topological quantum field theories. The cobordism conjecture for topological quantum field theories is the analog of the Eilenberg-Steenrod axioms for homology theories . The Eilenberg-Steenrod axioms state that a homology theory is uniquely determined by its value for the point , and the cobordism conjecture says that a topological quantum field theory is uniquely determined by its value for the point: the bijection between -valent symmetricals formulated above monoidal functors and the objects of shall be defined by evaluation on the point. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$

literature

• Jacob Lurie: On the classification of topological field theories . Preprint 2009, arxiv : 0905.0465
• David Ayala, John Francis: The cobordism hypothesis , Preprint 2017, arxiv : 1705.02240