Positivity of the universal pairing in 3 dimensions

TL;DR

This paper proves the positivity of a Hermitian pairing associated with 3-manifolds bounded by a surface, revealing insights into topological information extractable from 2+1 dimensional TQFTs.

Contribution

It introduces a complexity function satisfying a topological Cauchy-Schwarz inequality, establishing the pairing's positivity and deriving new results in hyperbolic 3-manifold topology.

Findings

The pairing is positive for nonzero vectors.

Gluing hyperbolic 3-manifolds along certain surfaces affects volume minimally only under specific conditions.

A new complexity function c satisfies a topological Cauchy-Schwarz inequality.

Abstract

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.