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I recently made the final edits to my paper “Positivity of the universal pairing in 3 dimensions”, written jointly with Mike Freedman and Kevin Walker, to appear in Jour. AMS. This paper is inspired by questions that arise in the theory of unitary TQFT’s. An $n+1$-dimensional TQFT (“topological quantum field theory”) is a functor $Z$ from the category of smooth oriented $n$-manifolds and smooth cobordisms between them, to the category of (usually complex) vector spaces and linear maps, that obeys the (so-called) monoidal axiom $Z(A \coprod B) = Z(A) \otimes Z(B)$. The monoidal axiom implies that $Z(\emptyset)=\mathbb{C}$. Roughly speaking, the functor associates to a “spacelike slice” — i.e. to each $n$-manifold $A$ — the vector space of “quantum states” on $A$ (whatever they are), denoted $Z(A)$. A cobordism stands in for the physical idea of the universe and its quantum state evolving in time. An $n+1$-manifold $W$ bounding $A$ can be thought of as a cobordism from the empty manifold to $A$, so $Z(W)$ is a linear map from $\mathbb{C}$ to $Z(A)$, or equivalently, a vector in $Z(A)$ (the image of $1 \in \mathbb{C}$).

Note that as defined above, a TQFT is sensitive not just to the underlying topology of a manifold, but to its smooth structure. One can define variants of TQFTs by requiring more or less structure on the underlying manifolds and cobordisms. One can also consider “decorated” cobordism categories, such as those whose objects are pairs $(A,K)$ where $A$ is a manifold and $K$ is a submanifold of some fixed codimension (usually $2$) and whose morphisms are pairs of cobordisms $(W,S)$ (e.g.  Wilson loops in a $2+1$-dimensional TQFT).

In realistic physical theories, the space of quantum states is a Hilbert space — i.e. it is equipped with a nondegenerate inner product. In particular, the result of pairing a vector with itself should be positive. One says that a TQFT with this property is unitary. In the TQFT, reversing the orientation of a manifold interchanges a vector space with its dual, and pairing is accomplished by gluing diffeomorphic manifolds with opposite orientations. It is interesting to note that many $3+1$-dimensional TQFTs of interest to mathematicians are not unitary; e.g. Donaldson theory, Heegaard Floer homology, etc. These theories depend on a grading, which prevents attempts to unitarize them. It turns out that there is a good reason why this is true, discussed below.

Definition: For any $n$-manifold $S$, let $\mathcal{M}(S)$ denote the complex vector space spanned by the set of $n+1$-manifolds bounding $S$, up to a diffeomorphism fixed on $S$. There is a pairing on this vector space — the universal pairing — taking values in the complex vector space $\mathcal{M}$ spanned by the set of closed $n+1$-manifolds up to diffeomorphism. If $\sum_i a_iA_i$ and $\sum_j b_jB_j$ are two vectors in $\mathcal{M}(A)$, the pairing of these two vectors is equal to the formal sum $\sum_{ij} a_i\overline{b}_j A_i\overline{B}_j$ where overline is complex conjugation on numbers, and orientation-reversal on manifolds, and $A_i\overline{B}_j$ denotes the closed manifold obtained by gluing ${}A_i$ to $\overline{B}_j$ along $S$.

The point of making this definition is the following. If $v \in \mathcal{M}(S)$ is a vector with the property that $\langle v,v\rangle_S = 0$ (i.e. the result of pairing $v$ with itself is zero), then $Z(v)=0$ for any unitary TQFT $Z$. One says that the universal pairing is positive in $n+1$ dimensions if every nonzero vector $v$ pairs nontrivially with itself.

Example: The Mazur manifold $M$ is a smooth $4$-manifold with boundary $S$. There is an involution $\theta$ of $S$ that does not extend over $M$, so $M,\theta(M)$ denote distinct elements of $\mathcal{M}(S)$. Let $v = M - \theta(M)$, their formal difference. Then the result of pairing $v$ with itself has four terms: $\langle v,v\rangle_S = M\overline{M} - \theta(M)\overline{M} - M\overline{\theta(M)} + \theta(M)\overline{\theta(M)}$. It turns out that all four terms are diffeomorphic to $S^4$, and therefore this formal sum is zero even though $v$ is not zero, and the universal pairing is not positive in dimension $4$.

More generally, it turns out that unitary TQFTs cannot distinguish $s$-cobordant $4$-manifolds, and therefore they are insensitive to essentially all “interesting” smooth $4$-manifold topology! This “explains” why interesting $3+1$-dimensional TQFTs, such as Donaldson theory and Heegaard Floer homology (mentioned above) are necessarily not unitary.

One sees that cancellation arises, and a pairing may fail to be positive, if there are some unusual “coincidences” in the set of terms $A_i\overline{B}_j$ arising in the pairing. One way to ensure that cancellation does not occur is to control the coefficients for the terms appearing in some fixed diffeomorphism type. Observe that the “diagonal” coefficients $a_i\overline{a}_i$ are all positive real numbers, and therefore cancellation can only occur if every manifold appearing as a diagonal term is diffeomorphic to some manifold appearing as an off-diagonal term. The way to ensure that this does not occur is to define some sort of ordering or complexity on terms in such a way that the term of greatest complexity can occur only on the diagonal. This property — diagonal dominance — can be expressed in the following way:

Definition: A pairing $\langle \cdot,\cdot \rangle_S$ as above satisfies the topological Cauchy-Schwarz inequality if there is a complexity function $\mathcal{C}$ defined on all closed $n+1$-manifolds, so that if ${}A,B$ are any two $n+1$-manifolds with boundary $S$, there is an inequality $\mathcal{C}(A\overline{B}) \le \max(\mathcal{C}(A\overline{A}),\mathcal{C}(B\overline{B}))$ with equality if and only if $A=B$.

The existence of such a complexity function ensures diagonal dominance, and therefore the positivity of the pairing $\langle\cdot,\cdot\rangle_S$.

Example: Define a complexity function $\mathcal{C}$ on closed $1$-manifolds, by defining $\mathcal{C}(M)$ to be equal to the number of components of $M$. This complexity function satisfies the topological Cauchy-Schwarz inequality, and proves positivity for the universal pairing in $1$ dimension.

Example: A suitable complexity function can also be found in $2$ dimensions. The first term in the complexity is number of components. The second is a lexicographic list of the Euler characteristics of the resulting pieces (i.e. the complexity favors more components of bigger Euler characteristic). The first term is maximized if and only if the pieces of $A$ and $B$ are all glued up in pairs with the same number of boundary components in $S$; the second term is then maximized if and only if each piece of $A$ is glued to a piece of $B$ with the same Euler characteristic and number of boundary components — i.e. if and only if $A=B$.

Positivity holds in dimensions below $3$, and fails in dimensions above $3$. The main theorem we prove in our paper is that positivity holds in dimension $3$, and we do this by constructing an explicit complexity function which satisfies the topological Cauchy-Schwarz inequality.

Unfortunately, the function itself is extremely complicated. At a first pass, it is a tuple $c=(c_0,c_1,c_2,c_3)$ where $c_0$ treats number of components, $c_1$ treats the kernel of $\pi_1(S) \to \pi_1(A)$ under inclusion, $c_2$ treats the essential $2$-spheres, and $c_3$ treats prime factors arising in the decomposition.

The term $c_1$ is itself very interesting: for each finite group $G$ Witten and Dijkgraaf constructed a real unitary TQFT $Z_G$ (i.e. one for which the resulting vector spaces are real), so that roughly speaking $Z_G(S)$ is the vector space spanned by representations of $\pi_1(S)$ into $G$ up to conjugacy, and $Z_G(A)$ is the vector that counts (in a suitable sense) the number of ways each such representation extends over $\pi_1(A)$. The value of $Z_G$ on a closed manifold is roughly just the number of representations of the fundamental group in $G$, up to conjugacy. The complexity $c_1$ is obtained by first enumerating all isomorphism classes of finite groups $G_1,G_2,G_3 \cdots$ and then listing the values of $Z_{G_i}$ in order. If the kernel of $\pi_1(S) \to \pi_1(A)$ is different from the kernel of $\pi_1(S) \to \pi_1(B)$, this difference can be detected by some finite group (this fact depends on the fact that $3$-manifold groups are residually finite, proved in this context by Hempel); so $c_1$ is diagonal dominant unless these two kernels are equal; equivalently, if the maximal compression bodies of $S$ in $A$ and $B$ are diffeomorphic rel. $S$. It is essential to control these compression bodies before counting essential $2$-spheres, so this term must come before $c_2$ in the complexity.

The term $c_3$ has a contribution $c_p$ from each prime summand. The complexity $c_p$ itself is a tuple $c_p = (c_S,c_h,c_a)$ where $c_S$ treats Seifert-fibered pieces, $c_h$ treats hyperbolic pieces, and $c_a$ treats the way in which these are assembled in the JSJ decomposition. The term $c_h$ is quite interesting; evaluated on a finite volume hyperbolic $3$-manifold $M$ it gives as output the tuple $c_h(M) = (-\text{vol}(M),\sigma(M))$ where $\text{vol}(M)$ denotes hyperbolic volume, and $\sigma(M)$ is the geodesic length spectrum, or at least those terms in the spectrum with zero imaginary part. The choice of the first term depends on the following theorem:

Theorem: Let $S$ be an orientable surface of finite type so that each component has negative Euler characteristic, and let ${}A,B$ be irreducible, atoroidal and acylindrical, with boundary $S$. Then $A\overline{A},A\overline{B},B\overline{B}$ admit unique complete hyperbolic structures, and either $2\text{vol}(A\overline{B}) > \text{vol}(A\overline{A})+\text{vol}(B\overline{B})$ or else $2\text{vol}(A\overline{B}) = \text{vol}(A\overline{A}) + \text{vol}(B\overline{B})$ and $S$ is totally geodesic in $A\overline{B}$.

This theorem is probably the most technically difficult part of the paper. Notice that even though in the end we are only interested in closed manifolds, we must prove this theorem for hyperbolic manifolds with cusps, since these are the pieces that arise in the JSJ decomposition. This theorem was proved for closed manifolds by Agol-Storm-Thurston, and our proof follows their argument in general terms, although there are more technical difficulties in the cusped case. One starts with the hyperbolic manifold $A\overline{B}$, and finds a least area representative of the surface $S$. Cut along this surface, and double (metrically) to get two singular metrics on the topological manifolds $A\overline{A}$ and $B\overline{B}$. The theorem will be proved if we can show the volume of this singular metric is bigger than the volume of the hyperbolic metric. Such comparison theorems for volume are widely studied in geometry; in many circumstances one defines a geometric invariant of a Riemannian metric, and then shows that it is minimized/maximized on a locally symmetric metric (which is usually unique in dimensions $>2$). For example, Besson-Courtois-Gallot famously proved that a negatively curved locally symmetric metric on a manifold uniquely minimizes the volume entropy over all metrics with fixed volume (roughly, the entropy of the geodesic flow, at least when the curvature is negative).

Hamilton proved that if one rescales Ricci flow to have constant volume, then scalar curvature $R$ satisfies $R' = \Delta R + 2|\text{Ric}_0|^2 + \frac 2 3 R(R-r)$ where $\text{Ric}_0$ denotes the traceless Ricci tensor, and $r$ denotes the spatial average of the scalar curvature $R$. If the spatial minimum of $R$ is negative, then at a point achieving the minimum, $\Delta R$ is non-negative, as are the other two terms; in other words, if one does Ricci flow rescaled to have constant volume, the minimum of scalar curvature increases (this fact remains true for noncompact manifolds, if one substitutes infimum for maximum). Conversely, if one rescales to keep the infimum of scalar curvature constant, volume decreases under flow. In $3$ dimensions, Perelman shows that Ricci flow with surgery converges to the hyperbolic metric. Surgery at finite times occurs when scalar curvature blows up to positive infinity, so surgery does not affect the infimum of scalar curvature, and only makes volume smaller (since things are being cut out). Consequently, Perelman’s work implies that of all metrics on a hyperbolic $3$-manifold with the infimum of scalar curvature equal to $-6$, the constant curvature metric is the unique metric minimizing volume.

Now, the metric on $A\overline{A}$ obtained by doubling along a minimal surface is not smooth, so one cannot even define the curvature tensor. However, if one interprets scalar curvature as an “average” of Ricci curvature, and observes that a minimal surface is flat “on average”, then one should expect that the distributional scalar curvature of the metric is equal to what it would be if one doubled along a totally geodesic surface, i.e. identically equal to $-6$. So Perelman’s inequality should apply, and prove the desired volume estimate.

To make this argument rigorous, one must show that the singular metric evolves under Ricci flow, and instantaneously becomes smooth, with $R \ge -6$. A theorem of Miles Simon says that this follows if one can find a smooth background metric with uniform bounds on the curvature and its first derivatives, and which is $1+\epsilon$-bilipschitz to the singular metric. The existence of such a background metric is essentially trivial in the closed case, but becomes much more delicate in the cusped case. Basically, one needs to establish the following comparison lemma, stated somewhat informally:

Lemma: Least area surfaces in cusps of hyperbolic $3$-manifolds become asymptotically flat faster than the thickness of the cusp goes to zero.

In other words, if one lifts a least area surface $S$ to a surface $\tilde{S}$ in the universal cover, there is a (unique) totally geodesic surface $\pi$ (the “osculating plane”) asymptotic to $\tilde{S}$ at the fixed point of the parabolic element corresponding to the cusp, and satisfying the following geometric estimate. If $B_t$ is the horoball centered at the parabolic fixed point at height $t$ (for some horofunction), then the Hausdorff distance between $\tilde{S} \cap B_t$ and $\pi \cap B_t$ is $o(e^{-t})$. One must further prove that if a surface $S$ has multiple ends in a single cusp, these ends osculate distinct geodesic planes. Given this, it is not too hard to construct a suitable background metric. Between ends of $S$, the geometry looks more and more like a slab wedged between two totally geodesic planes. The double of this is a nonsingular hyperbolic manifold, so it certainly enjoys uniform control on the curvature and its first derivatives; this gives the background metric in the thin part. In the thick part, one can convolve the singular metric with a bump function to find a bilipschitz background metric; compactness of the thick part implies trivially that any smooth metric enjoys uniform bounds on the curvature and its first derivatives. Hence one may apply Simon, and then Perelman, and the volume estimate is proved.

The Seifert fibered case is very fiddly, but ultimately does not require many new ideas. The assembly complexity turns out to be surprisingly involved. Essentially, one thinks of the JSJ decomposition as defining a decorated graph, whose vertices correspond to the pieces in the decomposition, and whose edges control the gluing along tori. One must prove an analogue of the topological Cauchy-Schwarz inequality in the context of (decorated) graphs. This ends up looking much more like the familiar TQFT picture of tensor networks, but a more detailed discussion will have to wait for another post.