Proof of Cobordism Hypothesis

1 Introduction

The cobordism hypothesis gives a classification of extended topological field theories on manifolds with certain structures.

Fix a space \(X\) with a rank \(n\) vector bundle \(\xi \). Let \(\Bord _n^{X,\xi }\) denote the symmetric monoidal \((\infty ,n)\) category, where \(i\)-morphisms for \(i\leq n\) are manifolds \(M^i\) with corners with an \((X,\xi )\) structure. This means we think of the manifold as a bordism between manifolds between \(i-1\)-dimensional manifolds, and the \((X,\xi )\) structure is a map \(M^i \to X\) and an identification of \(\RR ^{n-i}\) plus the tangent bundle of \(M\) with \(\xi \). For \(i>n\), the morphisms are diffeomorphisms, isotopies between diffeomorphisms, etc. We can think of \(X\) as the classifying space \(BG\) of a (topological) groupoid, so that lifting a morphism from \(BO(n)\) to \(BG=X\) can be thought of as giving a \(G\) structure on a rank \(n\) bundle.

Let \(\tilde {X}\) be the frame bundle on \(X\) coming from the map to \(BO(n)\). The cobordism hypothesis is:

Theorem 1.1 (Baez-Dolan, Hopkins-Lurie [lurie2009classification]). \(Fun^{\otimes }(\Bord _n^{X,\xi },C) = \Map _{O(n)}(\tilde {X},C^{\cong })\), where \(C\) is a symmetric monoidal \((\infty ,n)\)-category with duals, and \(C^{\cong }\) is the groupoid of invertible maps.

The isomorphism can be thought of as being given by evaluation at the space of connected \(0\)-dimensional manifolds equipped with a trivialization of the tangent space stabilized to dimension \(n\) (which is parameterized by \(\tilde {X}\)). The \(O(n)\) action on \(C^{\cong }\) comes from the case when \(X\) is a point.

A special case of this result is the result due to Galatius-Madsen-Tillmann-Weiss which describes invertible field theories. Just as before we have a map \(BG \to BO(n)\), and we consider the Thom spectrum \(MTG\) of the negative of the canonical bundle.

Theorem 1.2 (GMTW [galatius2009homotopy]). \(Fun^{\otimes }(\Bord _{n}^{BG},X) = \Map (\Sigma ^nMTG,X)\), where \(X\) is a spectrum.

Note that we can take \(n \to \infty \) to get \(\Bord _{\infty }^{X,\xi }\), an \((\infty ,\infty )\) category of bordisms. This is a category that is stable in two ways: We have already stabilized with respect to dimension, meaning that we only care about the stable tangent bundles of manifolds. We have also stabilized with respect to embedding dimension, we can think of the manifolds involved as embedded in \(\RR ^\infty \). This is reflected on the categorical side by having the category be symmetric monoidal.

Note that given a compatible family of maps to \(BO(n)\), we can let \(n\) go to \(\infty \), to describe the geometric realization of the \(\infty ,\infty \)-bordism category as a Thom spectrum where the group has been stabilized eg \(MTO\), \(MTU\). This looks similar to the result of Pontryagin-Thom, but is subtly different: the definition of bordism category for field theories involves manifolds with structures on their stable tangent bundles, but the definition of standard cobordism spectra like \(MO\) and \(MU\) involves manifolds with structures on their stable normal bundles. In many cases these two coincide: \(MO = MTO, MSO= MTSO, MU = MTU, MSpin = MTSpin\), but not always: \(MPin_+=MTPin_-\).

There is also a way to destabilize the cobordism hypthesis with respect to embedding dimension. This yields the tangle hypothesis, which is essentially the same as the cobordism hypothesis, except all manifolds involved are embedded, and the categories aren’t symmetric monoidal, but rather \(k\)-fold monoidal (ie have \(k\) deloopings).