• DOI:10.1090/S0894-0347-09-00642-0
• Corpus ID: 8553648

Positivity of the universal pairing in 3 dimensions

@article{Calegari2008PositivityOT,
  title={Positivity of the universal pairing in 3 dimensions},
  author={Danny Calegari and Michael H. Freedman and Kevin Walker},
  journal={Journal of the American Mathematical Society},
  year={2008},
  volume={23},
  pages={107-188}
}

• Danny Calegari, M. Freedman, K. Walker
• Published 21 February 2008
• Mathematics
• Journal of the American Mathematical Society

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.