Positivity of the universal pairing in 3 dimensions

Abstract

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.

The minimal volume orientable hyperbolic 2-cusped 3-manifolds

“…We then apply a volume estimate from [2] and volume estimates of Miyamoto [13] giving lower bounds on volumes of hyperbolic manifolds with totally geodesic boundary to get the volume lower bound on int(M ) (Theorem 3.5). A recent sharp estimate [3] implies that we may characterize completely the case of equality and identify the two smallest volume manifolds with two cusps by a simple combinatorial analysis (Theorem 3.6). The arguments of [2,3] depend strongly on Perelman's proof of the geometrization conjecture (see [19,18,12,14,6,15]).…”

“…A recent sharp estimate [3] implies that we may characterize completely the case of equality and identify the two smallest volume manifolds with two cusps by a simple combinatorial analysis (Theorem 3.6). The arguments of [2,3] depend strongly on Perelman's proof of the geometrization conjecture (see [19,18,12,14,6,15]). …”

On the origin and development of subfactors and quantum topology

“…The meaning of these theories remains somewhat obscure in general, though they can give classical invariants like enumerating representations of the fundamental group of the manifold. Taking the abstraction one level further, Calegari, Freedman, and Walker in [CFW08] have shown that TQFTs are powerful enough to separate 3-manifolds, though in dimension 4 there are counterexamples to such an approach.…”

Grothendieck’s problem for 3-manifold groups

“…This question has arisen, at least implicitly, in a totally different direction, in recent work of Calegari, Freedman and Walker, [9]. This paper addresses the properties of a "universal pairing" on a complex vector space that arises from looking at all compact oriented 3-manifolds that a fixed closed oriented surface bounds.…”

“…This paper addresses the properties of a "universal pairing" on a complex vector space that arises from looking at all compact oriented 3-manifolds that a fixed closed oriented surface bounds. To that end, in [9] a complexity function is defined on a compact oriented 3-manifold M , and part of of this complexity function involves listing all finite quotients of 1 .M /. The question as to whether the finite quotients of the fundamental group of a compact orientable irreducible 3-manifold M determine M arises naturally here (see Remark 3.7 of [9]).…”