Spin Cobordism Categories in Low Dimensions Nitu Kitchloo and Jack Morava

The Madsen-Tillmann spectra defined by categories of threeand four-dimensional Spin manifolds have a very rich algebraic structure, whose surface is scratched here. For Michael Atiyah, in deep gratitude. 1. Cobordism categories 1.1 Many variations and generalizations are possible, but to begin, consider the topological two-category DCobord whose objects are oriented smooth closed d-manifolds (D = d+1), with the topological category DCobord(V, V ) of morphisms having as objects, D-dimensional cobordisms W : V → V ′ from V to V ; for our purposes this will mean an identification ∂W ∼= Vop ∐ V , extended to a small neighborhood of the boundary. The twomorphisms will be orientation-preserving diffeomorphisms of such cobordisms, which equal the identity near the boundary. The composition functor DCobord(V, V )×DCobord(V , V ) → DCobord(V, V ) is defined by glueing outgoing to incoming boundaries. A topological category C is a kind of simplicial space, and so has a geometric realization |C|; for example, |DCobord(V, V )| = ∐ [W :V→V ′] BDiff0 (W ) is the union, indexed by diffeomorphism classes of cobordisms W from V to V , of the classifying spaces of the groups of orientation-preserving diffeomorphisms of W which equal the identity near the boundary. We’ll write D|Cobord| for the topological category with closed d-manifolds as objects, and the classifying spaces above as morphism objects; it is symmetric monoidal (under disjoint union). Such categories have an impressive history [3, 16, 17, 21] in topology and physics. This paper applies the recent breakthroughs of [9] which (in great Date: 9 November 2006. Both authors were supported by the NSF. 1 2 NITU KITCHLOO AND JACK MORAVA generality) identify the classifying spectra of such categories. The formalism of Galatius, Madsen, Tillmann, and Weiss frames these categories somewhat differently: they work with a category CD of manifolds embedded in a high-dimensional Euclidean background, but the description used above is equivalent, and is convenient in physics. 1.2 A topological transformation group G × X → X has an associated homotopy-to-geometric quotient map X//G := EG×G X → pt×G X = X/G which defines a kind of resolution BDiff ∼ EDiff ×Diff Metrics → pt×Diff Metrics of the moduli space of Riemannian metrics [8] on a manifold. For a closed manifold the action of the diffeomorphism group on the space of metrics is proper; for surfaces of genus > 1, for example, its isotropy groups are not just compact but finite, making the map a rational homology equivalence. This resolution defines a monoidal functor D|Cobord| → GravityD to a topological category with moduli spaces of metrics as its morphism objects. The Einstein-Hilbert functional