A rudimentary theory of topological 4D 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 some relatively accessible examples 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 [22], but following Segal’s definition of conformal field theory [29] and Atiyah’s subsequent abstraction of the notion of topological quantum field theory [1] they have been studied very widely. Recently, Tillmann [31] has shown the utility in this context of certain closely related two-categories (which generalize the classical notion of category, by admitting morphism-objects which are themselves categories). The following definition is based on her ideas. e-print archive: http://xxx.lanl.gov/math.DG/0007018