Associated to a closed, oriented surface
S
S
is the complex vector space with basis the set of all compact, oriented
3
3
-manifolds which it bounds. Gluing along
S
S
defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented
3
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
)
(2+1)
-dimensional TQFTs.
The proof involves the construction of a suitable complexity function
c
c
on all closed
3
3
-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that
c
(
A
B
)
≤
max
(
c
(
A
A
)
,
c
(
B
B
)
)
c(AB) \le \max (c(AA),c(BB))
for all
A
,
B
A,B
which bound
S
S
, with equality if and only if
A
=
B
A=B
.
The complexity function
c
c
involves input from many aspects of
3
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
3
-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic
3
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
3
-manifolds due to Agol-Storm-Thurston.