Positivity of the universal pairing in $3$ dimensions

Associated to a closed, oriented surface $S$ is the complex vector space with basis the set of all compact, oriented $3$-manifolds which it bounds. Gluing along $S$ defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented $3$-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary $(2+1)$-dimensional TQFTs.

The proof involves the construction of a suitable complexity function $c$ on all closed $3$-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that $c(AB) \le \max(c(AA),c(BB))$ for all $A,B$ which bound $S$, with equality if and only if $A=B$.

The complexity function $c$ involves input from many aspects of $3$-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic $3$-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic $3$-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic $3$-manifolds due to Agol-Storm-Thurston.