Proof of Cobordism Hypothesis

4 Reduction to Unoriented Case

\(\Bord _n=\Bord _n^{BO(n)}\) will be the unoriented bordism category.

The right hand side of the cobordism hypothesis is \(\Map _{O(n)}(\tilde {X},C^{\cong })\), which is the same as \(\Map _{O(n)}(EO(n),(C^{\cong })^{\tilde {X}}\). So naively, we could try to reduce to the unoriented case by replacing \(C\) by a category \(C^{X,\xi }\) such that \((C^{X,\xi })^{\cong } = (C^{\cong })^{\tilde {X}}\), and such that \(Fun^{\otimes }(\Bord _n^{X.\xi },C) = Fun^{\otimes }(\Bord _n,C^{X,\xi })\), and the general case would follow immediately.

This strategy doesn’t work exactly, but a relative version of it does: There is a fibration of symmetric monoidal \((\infty ,n)\)-categories \(Fam_n(C) \to Fam_n = Fam_n(*)\) and a map \((X,\xi ): Fun^{\otimes }(\Bord _n,Fam_n)\) such that lifts of \((X,\xi )\) to \(Fam_n(C)\) are the same as elements of \(Fun^{\otimes }(\Bord _n^{(X,\xi )},C)\). This is summarized in the below diagram.

(-tikz- diagram)

The objects category \(Fam_n(C)\) are \(C\)-values local systems of functors from a space. \(n\)-morphisms are \(n\)-fold correspondences between such local systems (the correspondences make sure that \(Fam_n(C)\) has duals), and the symmetric monoidal structure is the product.

In particular \(Fam_n(*)\) is the category whose objects are topological spaces and \(n\)-morphisms are \(n\)-fold correspondences between topological spaces. We can produce the functor \((X,\xi )\) by sending a \(k\)-morphism \(M^k\) to the classifying space for \((X,\xi )\)-structures on \(M^k\).

From this correspondence, it should not be surprising that one can use the cobordism hypothesis for \(\Bord _n\) to deduce the inductive form of it for \(\Bord _n^{X,\xi }\).