Proof of Cobordism Hypothesis

7 Obstruction theory

We know at this point that \(\Bord _n^{ff}\) has the right universal property, but we need to know that our choice of framed function didn’t matter. It is true that given a bordism, the space of framed functions is contractible. However, this fact is actually equivalent to the fact that \(\Bord _{n}^{ff} \to \Bord _{n}\) is an equivalence. Igusa showed some connectivity bounds on the space of framed functions.

The reason we can show that \(\Bord _n^{ff} \to \Bord _n\) is an equivalence is because of the connectivity bound that Igusa showed as well as some cohomology computations generalizing work of GMTW. The goal is to set up an obstruction theory for \((\infty ,n)\) categories that generalizes the Postnikov tower, and use it to show the equivalence. To see why we would need both a connectivity bound as well as cohomology computations, we can make an analogy with the case \(n=0\): it is not true that a map of spaces that is an equivalence if it is just an equivalence on homology. One also needs to check that it is an equivalence on the fundamental groupoid, which is a connectivity statement.

Here, we will use a generalization of the Postnikov tower, namely the \((m,n)\)-truncation of an \((\infty ,n)\)-category (or the homotopy \(m\)-category). The analog of Eilenberg-Maclane spaces and cohomology will arise from the category \(\Loc (C)\) of local systems on \(C\), which for an \((\infty ,n)\) category has an inductive definition. For example \(n=0\) it is just a functor into abelian groups, and for \(n=1\) we require a functor from \(\Map (x,y)\) into abelian groups as well as a map local systems on \(\Map (x,y)\times \Map (y,z)\) from the product of the pullbacks of the local system on \(\Map (x,y)\) and \(\Map (y,z)\) to the pullback of the local system on \(\Map (x,z)\).

The definition of cohomology on a local system is also inductive, and can be thought of as the homotopy classes of sections of a fibration of \((\infty ,n)\)-categories whose fibres are Eilenberg-MacLane spaces. There are a few subtleties, for example that we really would like to work with a local system compatible with the symmetric monoidal structure (since we are essentially doing stable homotopy theory for \(n\)-categories).

The resulting obstruction theory says that a map \(C \to C'\) of symmetric monoidal \((\infty ,n)\)-categories is an equivalence iff it is an equivalence on homotopy \((n+1,n)\)-categories and induces an isomorphism on cohomology in any local system. In the case of interest, the first claim comes from a reult of Igusa on connectivity of the space of framed functions on a bordism. The cobordism hypothesis can be used to identify the relative cohomology of the pair \((\Bord _n,\Bord _{n-1})\) with a degree shifted cohomology of \(BO(n)\) for appropriate coefficient systems. This result is then true for \(\Bord _n^{ff}\), so it suffices to show it is also true for \(\Bord _n\).

This relative cohomology for constant local systems is also the relative cohomology of the pair \((\Sigma ^{n} MTO(n),\Sigma ^{n-1}MTO(n-1))\). But the cofibre of these spectra is indeed \(\Sigma ^{\infty +n}_+ BO(n)\), by cohomology calculations in the work of GMTW. The paper claims that their methods can be generalized to show that the relative cohomology agrees for arbitrary coefficient systems, which completes the proof.