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.