A RUDIMENTARY THEORY OF TOPOLOGICAL FOUR-DIMENSIONAL GRAVITY

ABSTRACT. Atheory of topological gravity is ahomotopy-theoretic representa-tion of the Segal-Tillmann topologification of atw0-category with cobordisms as morphisms. This note describes arelatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. 1. GRAVITY CATEGORIES Acobordism 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. $\mathrm{H}\epsilon \mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}$, Tillmann [25] has demonstrated the utility of certain closely related tw0-categories; the definition below is based on her ideas. Definition Agravity tw0-category has $\bullet $ (closed) manifolds as objects, $\bullet $ cobordisms as morphisms, and $\bullet $ isomorphisms of these cobordisms, equal to the identity on the boundary, as tw0-morphisms. There are many possible variations on this theme, and Iwill 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 $(\mathrm{t}\mathrm{w}\mathrm{o}-)\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{y} $ is $(d+1)-$ dimensional. Iwill assume that manifolds are smooth, compact and oriented, but not necessarily connected, and (following Segal) Iunderstand the empty set to be amanifold of any dimension. 1.1 If $V $ and $V$’are $d$-manifolds, amorphism $W $ : $Varrow V’$ is (the germ of) an orientation-preserving diffeomorphis