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Topological gravity in Minkowski space

The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A rather striking amount of work on pseudoisotopy theory [Hatcher, Waldhausen, Cohen-Carlsson-Goodwillie-Hsiang-Madsen...] can be formulated as a (generalized) TQFT in this framework. The resulting theory is far from trivial even in the case of Minkowski space, when the relevant three-manifold is the standard sphere.

Topological gravity [18] extends Graeme Segal's ideas about conformal field theory to higher dimensions. It seems to be very interesting, even in extremely restricted geometric contexts: §1 basic definitions 1.1 A cobordism W : V 0 → V 1 between d-manifolds is a D = d + 1-dimensional manifold W together with a distinguished diffeomorphism a diffeomorphism Φ : W → W ′ of cobordisms will be assumed consistent with this boundary data.
Cob(V 0 , V 1 ) is the category whose objects are such cobordisms, and whose morphisms are such diffeomorphisms. Gluing along the boundary defines a composition functor # : The two-category with manifolds as objects and the categories Cob as morphisms is symmetric monoidal under disjoint union.
The categories Cob are topological groupoids (all morphisms are invertible), with classifying spaces BDiff(W rel ∂) .
The topological gravity category has these objects as hom-spaces: it is a (symmetric monoidal) topological category. 1.2 A theory of topological gravity is a representation of such a category in some simpler monoidal category, e.g. Hilbert spaces, or spectra.
The homotopy-to-geometric quotient map defines a functor from the topological gravity category to a category with the spaces Met/Diff as morphism objects; these are the spaces of states in general relativity (and is a kind of Morse function upon them).
In Segal's conformal field theory, the corresponding objects are moduli spaces of (complex structures on) Riemann surfaces. Indeed if W = Σ is a Riemann surface of genus > 1, its group of diffeomorphisms is homotopically discrete: the map is a homotopy equivalence. The mapping class group acts with finite isotropy on Teichmüller space, so when d = 1 the homotopy-to-geometric quotient is close to a rational homology equivalence. Three-manifolds under Spin cobordism have very interesting connections with the theory of even unimodular lattices [8,16], and the methods of [6] identify the classifying spectrum of this category with the desuspension of BSpin(4) by the vector bundle associated to the standard four-dimensional representation of the spin group. Because of well-known coincidences in low-dimensional geometry, Spin(4) ∼ = SU(2)×SU(2), so we can identify its classifying space with the product of two copies of infinite-dimensional quaternionic projective space, and the vector bundle defined by the standard representation with the tensor product (over H) of the resulting two canonical quaternionic line bundles L ± ; thus The generators of π 0 Ω ∞ M T Spin(4) ∼ = Z 2 can be identified with the signature and Euler characteristic, or alternately with the number of hyperbolic and E 8 factors in the middle-dimensional intersection form [2] of a spin cobordism.

2.2
There are other extremely interesting variant constructions in dimension four: contact three-manifolds under Spin c cobordism define a natural context for Seiberg-Witten theory, while Lorentz cobordism [20] incorporates an arrow of time; but this note is concerned with 3-manifolds up to h-cobordism: The trivial h-cobordism W = V × I, where I is an interval, is an interesting example. In dimensions ≥ 5, the s-cobordism theorem classifies h-cobordisms by elements of the Whitehead group and there are invariants for parametrized h-cobordisms taking values in higher homotopy groups of certain pseudoisotopy spaces, which have been studied by Hatcher, Waldhausen, Igusa, . . . This category has a monoidal structure, but it is relatively trivial, so that it is natural to assume that the manifolds V are connected.

2.3
Here I will be concerned mostly with the case V = S 3 : by Minkowski space I really mean the universal cover S 3 × R of Penrose's (and others') conformal compactification S 3 × ±1 S 1 of Minkowski space; this contains, in particular, a copy of Einstein's static universe [11]. Its time-like intervals define trivial h-cobordisms of S 3 .
Note that there are lots of wild S 3 × R's: remove a point from a fake R 4 . It would be very interesting to construct a semigroup of such things, under some kind of boundary gluing, as Segal did with topological annuli; current work of Gompf [10 §7, cf. also [3]] seems close to this. It is not clear at the moment if nontrivial smooth h-cobordisms of the three-sphere exist; the question is closely connected to the smooth four-dimensional Poincaré conjecture. §3 double categories 3.1 Boundary value problems involve the interplay between diffeomorphisms of a manifold and diffeomorphisms of its boundary. Tillmann [21] suggests that double categories provide a natural framework for such questions. In this context, the primary objects are certain rectangular diagrams W : with cobordisms displayed horizontally, and diffeomorphisms (which preserve some boundary framing) presented vertically; these can be patched together in either direction. More recently, Getzler [7] has used manifolds, together with suitable (eg separating) codimension one submanifolds, to define morphisms in such contexts; this seems particularly suited to the millefeuille examples of Gompf, which (if I understand correctly) can be regarded as smooth h-cobordisms between topological, but not necessarily smooth, three-spheres.

3.2
In any case, the double category D of trivial h-cobordisms between ordinary three-spheres is already extremely interesting. I don't know how to associate a topological category to a double category in general, but in this case pseudoisotopy theory defines an equivalence with the two-category having manifolds V as its objects, and Cerf's group C(V ) [13 §6.2] of pseudoisotopies (regarded as a category with one object) as its category of automorphisms: These pseudoisotopies are diffeomorphisms of the cylinders V × I, equal to the identity map on V × 0. There is a fibration of groups, and concordance The classifying space BC(V ) is thus a monoid, and the topological category associated to this rectification of D defines an ad hoc topologification (with one object for each V , and the topological monoid BC(V ) for its space of endomorphisms). The classifying B 2 C(V ) space of that topological category is the totalization of the bisimplicial space defined by the category of trivial h-cobordisms of V .
which looks like a presentation of this ad hoc classifying space for a double category as a fibration |Vertical| → |Horizontal| → |Double| built from classifying spaces for its component (vertical and horizontal) morphisms; but I don't know enough about double categories to guess if this might be an instance of something more general. §4 about A(S n ) 4.1 Through the efforts of many researchers, a great deal is known about the algebraic K-theory of spaces; in particular, if X is simply connected (and of finite type) its A-theory can be calculated (at least p-locally [4 §1.3]) from the topological cyclic homology [14 §7.3.14] of S 0 [ΩX].
Since this pretends to be a paper about physics, however, I will be content with some remarks about A * (X) ⊗ Q, which is accessible in more elementary terms. [I want to record here my thanks to Bruce Williams and Bjorn Dundas for walking me through a great deal of literature in this field, without suggesting that they bear any responsiblity for the excesses of this paper.] 4.2 In particular, old results [12] of Hsiang and Staffeldt imply that (when n > 1) the rationalization of A(S n ) splits as a copy of A(pt) ⊗ Q ∼ = K alg (Z) ⊗ Q and the suspension of what is essentially the (reduced) topological cyclic homology of S n , which can be computed effectively as the abelianization ofH * (ΩS n , Q) regarded as a graded Lie algebra; hence π * (ΩWh d (S n )) ⊗ Q ∼ = K alg * +1 (Z) ⊗ Q ⊕H * (ΩS n , Q) ab . The Whitehead product structure on a wedge of spheres is rationally free, so the graded Lie algebra structure has nontrivial commutators only when n is even. When n = 2m + 1 is odd, the rational homology is polynomial on a single generator x 2m ; it follows that is spanned as a rational vector space by elements x l 2 of degree 2l and elements ζ k of degree 4k corresponding to the odd zeta-values ζ(2k +1) which appear as regulators in Borel's calculations of K alg 4k+1 (Z) ⊗ Q.
This can be made more precise; when X is simply-connected then a reduced version Ω Wh(X) of loops on the Whitehead space is closely connected to a similarly reduced version Q( LX hT ) of (the infinite loopspace defined by) the suspension spectrum of the homotopy quotient (by its natural circle action) of the free loopspace of X.

4.3
The construction Q = Ω ∞ Σ ∞ sends a space to the infinite loopspace representing its suspension spectrum: this sends the rational homology of a space to its symmetric algebra. The cohomological invariants defined by the space of trivial h-cobordisms of the three-sphere thus resemble the 'big' phase spaces [9] studied in quantum cohomology: for example, the stable rational homology of the Riemann moduli space is essentially with the symmetric algebra on the homology of CP ∞ , and is thus a polynomial algebra with one generator of each even degree.
The rational cohomology of the infinite loopspace Ω ∞+1 A(S 3 ) seems similar in many ways: it is again a polynomial algebra, now with one set of generators indexed by even integers, the other by integers ≡ 0 modulo four. Physicists see these symmetric algebras as Fock representations associated to certain polarized symplectic vector spaces. In our context this seems to be related to an 'almost' splitting HC per ∼ HC ⊕ Hom Q[u] (HC, Q[u]) , of periodic cyclic homology [5] These representations have symmetries closely related to the Virasoro algebra, which lead [19] to interesting integrable systems.
This connection between 4D topological gravity and the equivariant free loopspace of the three-sphere resembles in many ways a purely mathematical instance of a phenomenon physicists [1] call 'holography', in which one physical model on the interior of a manifold is described by some other model on its boundary. Rather than proceed any further with speculations along these lines, I'd like to close by raising a mathematical question: A trivial h-cobordism between three-spheres is an example of a four-dimensional spin cobordism; this defines a monoidal functor, and hence a morphism