Proof of Cobordism Hypothesis

2 Outline

• • First we reformulate the cobordism hypothesis in an inductive way. Let $X,\xi$ be a space with a principle $O(n)$-bundle $\xi$ as in the construction of the bordism category. Let $X_0$ be the sphere bundle of associated to the bundle, with a map $f$ to $X$. $f^{\ast }\xi$ splits canonically as $\mathbb{R}\oplus {\xi }_0$, where ${\xi }_0$ is the bundle of vectors orthogonal to the vector on the sphere. Adding the trivial copy of $\mathbb{R}$ gives an inclusion ${\mathrm{Bord}}_{n-1}^{X_0,{\xi }_0}\to {\mathrm{Bord}}_n^{X,\xi }$. We will reformulate the universal property of ${\mathrm{Bord}}_n^{X,\xi }$ in terms of how to extend functors along this inclusion. The universal property will say that we need to specify what the field theory does on a disk of dimension $n$ with a nondegeneracy condition on the morphism corresponding to the disk.

• • The next step is only for simplicity (rather than necessity): we will reduce to proving the cobordism hypothesis for the unoriented bordism category ${\mathrm{Bord}}_n$. It isn’t too surprising that we can do this, since the unoriented bordism category is universal in the sense that its choice of $X,\xi$ is universal, however the actual implementation is a bit more subtle.

• • Since the inclusions ${\mathrm{Bord}}_n\to {\mathrm{Bord}}_{n+1}$ are highly connected (their $k$ morphisms agree for small $k$) and lots of duals exist, it is possible to capture the data of the inclusions in terms of $(\mathrm{\infty },1)$-categorical data, which is nice because $(\mathrm{\infty },1)$-categories are easier to work with.

• • The most important step in the proof will use a version of Morse theory due to Igusa to prove the cobordism hypothesis for another category ${\mathrm{Bord}}_n^{ff}$. In particular, we can understand bordisms in terms of handle attachments and cancellations, which will give generators and relations for ${\mathrm{Bord}}_n^{ff}$ in terms of ${\mathrm{Bord}}_{n-1}$. The difference between ${\mathrm{Bord}}_n^{ff}$ and ${\mathrm{Bord}}_n$ is nonexistent, and is can be thought of as the irrelevence of the choice of Morse function.

• • The last step of the proof is one that really shouldn’t have to be there, namely proving that ${\mathrm{Bord}}_n^{ff}$ is equivalent to ${\mathrm{Bord}}_n$. Now, there are independent proofs of thie fact [kupers2018applications; eliashberg2011space]. The proof given in [lurie2009classification] involves understanding an obstruction theory for $(\mathrm{\infty },n)$-categories, knowing from some Morse theory that they are equivalent in a range of dimensions, and doing cohomological computations similar to those in [galatius2009homotopy] to show that the two agree in general.