Topological Gravity in Minkowski

The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A striking amount of work on pseudoisotopy theory [Hatcher, Waldhausen, Cohen-Carlsson-Goodwillie-Hsiang-Madsen . . . ] can be formulated as a TQFT in this framework. The resulting theory is far from trivial even in the case of Minkowski space, when the relevant three-manifold is the standard sphere. Topological gravity [18] extends Graeme Segal’s ideas about conformal field theory to higher dimensions. It seems to be very interesting, even in extremely restricted geometric contexts: §1 basic definitions 1.1 A cobordism W : V0 → V1 between d-manifolds is a D = d + 1-dimensional manifold W together with a distinguished diffeomorphism ∂W ∼= V op 0 ∐ V1 ; a diffeomorphism Φ : W → W ′ of cobordisms will be assumed consistent with this boundary data. Cob(V0, V1) is the category whose objects are such cobordisms, and whose morphisms are such diffeomorphisms. Gluing along the boundary defines a composition functor # : Cob(V , V )×Cob(V, V ) → Cob(V, V ) . The two-category with manifolds as objects and the categories Cob as morphisms is symmetric monoidal under disjoint union. The categories Cob are topological groupoids (all morphisms are invertible), with classifying spaces