Proof of Cobordism Hypothesis

6 The Index Filtration

\(\Omega ^{n-2}\Bord _{n-1}\) is the category \(\Cob _{t}^{un}(n-1)\) of closed unoriented manifolds of dimension \(n-2\) and bordisms between them. The map \(\Bord _{n-1} \to \Bord _n\) corresponds via the above proposition to the map \(\Cob _{t}^{un}(n-1)\) sending an \(n-2\) manifold \(M\) to the category \(B(M)\), where the objects are \(n-1\)-manifolds equipped with a diffeomorphism of their boundary with \(M\), and the maps are bordisms that are trivial on the boundary between such manifolds. To analyse this category, we will use Morse theory.

Let’s recall the basic idea of Morse theory: Suppose you have a compact manifold \(N\) and you choose a smooth Morse (i.e generic) function \(f\) from \(N\) to the real numbers \(\RR \). Morse means that its derivative vanishes at isolated points such that the Hessian is nondegenerate. We can consider the process of building up \(N\) from its descending manifolds \(N^d_{r}\) which is the preimage under \(f\) of \((-\infty ,r]\). The diffeomorphism type of \(N^d_{r}\) is empty for small \(r\) and \(N\) for large \(r\), and only changes when a critical point of \(f\) happens (because apart from that \(f\) is a proper submersion). By the nondegeneracy hypothesis, the diffeomorphism type changes at the critical points in a very predictable way, namely by handle attachments.

A \(d\)-handle \(H\) in dimension \(n\) is a copy of \(D^{n-d}\times D^d\) with an attaching map \(\phi : D^{n-d}\times S^{d-1} \to N\) onto a manifold with boundary \(N\). We can glue the handle \(H\cup _\phi N\) along \(\phi \) and smooth the corners to obtain another smooth manifold with boundary, a process we refer to as attaching a handle. Every time we pass a critical point, our manifold changes by a handle attachment, and moreover the attaching data is determined by local data of the critical point. For example \(d\) is the index of the Hessian as a symmstric bilinear form over the reals.

The perspective we want to take is that choosing a Morse function really is a way of presenting \(N\) as a bordism. For example, by choosing a Morse function with critical points who take on distinct values we obtain the result that every bordism is a composite of handle attachments, as we can break up \(N\) into \(f^{-1}([a_i,a_{i+1}])\), where only one critical point occurs within each interval. When we attach a \(d\)-handle, we do surgery on the boundary of the manifold, replacing a copy of \(D^{n-d}\times S^{d-1}\) with \(S^{n-d-1}\times D^{d}\).

A more refined version of Morse theory, called Cerf theory, explains how to move between different Morse functions \(f_0,f_1\) via a family \(f_t\) that fail to be Morse at finitely many values in predictable ways. One can describe all the possible moves entirely at the level of handles. the two possible moves are essentially changing the time when a handle is attached, isotoping the attaching map \(\phi \) of a handle, and cancelling or uncancelling a pair of \(d\) and \(d-1\) handles whose cores intersect geometrically once.

However what we really want is even more sophisticated than Cerf theory: we want an understanding of the whole space of ways to present a bordism, not just generic paths between bordisms. To do this we use Igusa’s theory of framed functions. A framed function is like a Morse function with slightly worse singularities allowed (those that occur when cancelling handles), and a identification of the negative index part of the Hessian at each critical point with a standard negative definite form. We can replace \(B(M)\) with \(B(M)^{ff}\) where the \(1\)-morphisms in \(B(M)^{ff}\) are equipped with framed functions. There is a natural map \(B(M)^{ff}\) to \(B(M)\) that forgets the structure. By the category unfolding equivalence, the assignment \(M \mapsto B(M)\) corresponds to a map \(\Bord _{n-1} \to \Bord _{n}^ff\), and moreover we get a map \(\Bord _{n}^ff \to \Bord _n\).

The point is to use Igusa’s theory to prove that the cobordism hypothesis holds for the map \(\Bord _{n-1} to \Bord _n^{ff}\), and then show that \(\Bord _n^{ff}\) and \(\Bord _n\) agree.

\(\Bord _n^{ff}\) is good because it is as if each morphism comes with a presentation in terms of handle attachments. To study it, we will filter between \(\Bord _n\) and \(\Bord _n^{ff}\) via a category \(F_k\), which will unfold to the assignment \(M \mapsto B_k(M)\) where \(B_k(M)\) is the category of bordisms for which all ciritcal points are index \(\leq k\). For \(k\geq n\) this is just \(B^{ff}(M)\) and for \(k < 0\), there are no critical points so all the bordisms are trivial. Thus the \(F_k\) interpolate between \(\Bord _{n-1}\) and \(\Bord _n^{ff}\), and are called the index filtration. It will turn out that only index \(0\) and index \(1\) are important with respect to mapping into fully dualizable categories.

Given this filtration, the main things proven about it are essentially a version of handle calculus:

• The functor \(B_0: \Omega ^{n-2}\Bord _{n-1} \to \Cat _{(\infty ,1)}\) is freely generated as a lax symmetric monoidal functor by the \(O(n)\)-equivariant morphism given by a disk in \(B_k(\phi )\). Note that there is no nondegeneracy condition as well as no dualizability condition on the target. This is quite reasonable to expect, since for index \(0\), we can only add in disks that are disjoint to the rest of the bordism, and this is a well defined operation up to a \(O(n)\) action. This implies that \(F_0\) is freely generated as a symmetric monoidal \((\infty ,n)\)-category from \(\Bord _n\) by a morphism given by a disk.
• For \(k>0\), the functor \(B_k: \Omega ^{n-2}\Bord _{n-1} \to \Cat _{(\infty ,1)}\) is generated from \(B_{k-1}\) by an \(O(n-k)\)-equivariant \(1\)-morphism (handle attachment of index \(k\)) subject to one relation i.e \(O(n-k)\)-equivariant \(2\)-morphism (handle cancellation between index \(k\) and \(k-1\)).
• The next claim is that if \(C\) is symmetric monoidal \(Fun^{\otimes }(F_1,C) \to Fun^{\otimes }(F_0,C)\) is fully faithful with essential image the functors such that the morphism corresponding to the disk is nondegenerate. This quite reasonably follows from the previous, since the only relation obtained when adding a \(1\)-handle is cancellation with \(0\)-handles, and we can copy the proof of the \((1,1)\)-categorical cobordism hypothesis in dimension \(1\) using \(0\) and \(1\)-handles to see that this just adds in the relation that the disk is nondegenerate.
• The final claim is that if \(C\) additionally has all duals, then for \(k\geq 1\), \(Fun^{\otimes }(F_{k+1},C) \to Fun^{\otimes }(F_k,C)\) is an equivalence for \(k \geq 1\). This follows from the same claim as before, but the reason is more subtle, and involves thinking more carefully about how the unfolding of categories works. The point is that the handle cancellations of \(k-1\) and \(k\)-handles impose another nondegeneracy condition, that is redundant when mapping to \(C\), since duals are essentially unique. Alternatively, the rest of the handles can be thought of as being there to make the category \(\Bord _n\) itself have duals.