Proof of Cobordism Hypothesis

2 Outline

Here is an outline of the steps of the proof:

• 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 \(X\). \(f^*\xi \) splits canonically as \(\RR \oplus \xi _0\), where \(\xi _0\) is the bundle of vectors orthogonal to the vector on the sphere. Adding the trivial copy of \(\RR \) gives an inclusion \(\Bord _{n-1}^{X_0,\xi _0} \to \Bord _{n}^{X,\xi }\). We will reformulate the universal property of \(\Bord _{n}^{X,\xi }\) 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 \(\Bord _n\). 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 \(\Bord _n \to \Bord _{n+1}\) 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 \((\infty ,1)\)-categorical data, which is nice because \((\infty ,1)\)-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 \(\Bord _n^{ff}\). In particular, we can understand bordisms in terms of handle attachments and cancellations, which will give generators and relations for \(\Bord _n^{ff}\) in terms of \(\Bord _{n-1}\). The difference between \(\Bord _n^{ff}\) and \(\Bord _n\) 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 \(\Bord _n^{ff}\) is equivalent to \(\Bord _n\). Now, there are independent proofs of thie fact [kupers2018applications; eliashberg2011space]. The proof given in [lurie2009classification] involves understanding an obstruction theory for \((\infty ,n)\)-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.