Cobordism hypothesis

Remarks

A symmetric monoidal functor $Z \colon \mathrm{Bord}_n^{\mathrm{fr}} \to \mathcal{C}$ is a fully extended framed topological quantum field theory. The theorem says that such a theory is determined by its value $Z(\mathrm{pt})$ on a point, and that every fully dualizable object occurs. The point is fully dualizable in the bordism category, because bending a bordism gives the duality data; this is why $Z(\mathrm{pt})$ must be fully dualizable. The hypothesis cannot be dropped: an object of $\mathcal{C}$ that is not fully dualizable is not the value on a point of any such theory. Baez and Dolan proposed the hypothesis in 1995 in the language of $n$-categories. Lurie published a detailed outline of a proof in 2009 in the language of $(\infty, n)$-categories. For $n = 1$ and $\mathcal{C}$ the category of vector spaces over a field, the fully dualizable objects are the finite-dimensional spaces $V$. The theory with $Z(\mathrm{pt}) = V$ assigns to a circle the number $\dim V$, the trace of the identity of $V$. There are versions for other tangential structures, such as orientations, in which framings are replaced by the structure and the object carries a homotopy fixed point structure.

Going deeper

For a student

Classical two-dimensional field theories are classified by commutative Frobenius algebras, which uses the decomposition of surfaces into pairs of pants. The cobordism hypothesis extends such classifications to all dimensions by requiring the theory to be defined down to points, using higher categories. It ties higher category theory to manifold topology: the framed bordism $(\infty, n)$-category is the free symmetric monoidal $(\infty, n)$-category with duals on one object. Its proof outline relies on Morse theory to decompose bordisms into elementary handles.