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) <= 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.Comment: 83 pages, 21 figures; version 3: incorporates referee's comments and
correction
