Topological gravity in dimensions two and four

Recent work on gravity in two dimensions has a natural generalization to four dimensions. 1. Basic definitions 1.1 The (symmetric monoidal) two-category (Gravity)d+1 has objects: compact oriented d-manifolds, with • morphisms V0 → V1 : (d+ 1)-manifolds W with ∂W ∼= V op 0 t V1, and • diffeomorphisms W̃ →W as two-morphisms. The category Mor(V0, V1) with cobordisms from V0 to V1 as objects and diffeomorphisms (equal to the identity on the boundary) as morphisms, is a hom-object in this two-category. Disjoint union defines the monoidal structure, and the category has an orientation-reversing adjoint equivalence with its opposite. 1.2 The topological category (Gravity)d+1 has compact Riemannian d-manifolds as objects, and the spaces