Pretty Good Gravity

A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. 1. Gravity categories A cobordism category has manifolds as objects, and cobordisms as morphisms. Such categories were introduced by Milnor [14], but following Segal’s definition of conformal field theory [23] and Atiyah’s subsequent abstraction of the notion of topological quantum field theory [1] they have been studied very widely. Recently, Tillmann [25] has demonstrated the utility of certain closely related two-categories; the definition below is based on her ideas. Definition A gravity two-category has • (closed) manifolds as objects, • cobordisms as morphisms, and • isomorphisms of these cobordisms, equal to the identity on the boundary, as two-morphisms. There are many possible variations on this theme, and I will not try for maximal generality. If the objects of the category have dimension d (so the cobordisms are (d + 1)-dimensional) then I will say that the gravity (two-)category is (d+ 1)dimensional. I will assume that manifolds are smooth, compact and oriented, but not necessarily connected, and (following Segal) I understand the empty set to be a manifold of any dimension. 1.1 If V and V ′ are d-manifolds, a morphism W : V → V ′ is (the germ of) an orientation-preserving diffeomorphism (Vop ∪ V )× [0, 1) ∼= ν(∂W ) of the manifold on the left with a collar neighborhood of the boundary of the (d+1)manifoldW ; the subscript op signifies reversed orientation. The morphism category Mor(V, V ) has such cobordisms as its objects; it is a topological category, in which the space of morphisms between two cobordisms W and W̃ consists of orientationDate: 4 July 2000. 1991 Mathematics Subject Classification. 19Dxx, 57Rxx, 83Cxx. The author was supported in part by the NSF.