Calegari, Danny and Freedman, Michael H. and Walker, Kevin (2010) Positivity of the Universal Pairing in 3 Dimensions. Journal of the American Mathematical Society, 23 (1). pp. 107-188. ISSN 0894-0347 http://resolver.caltech.edu/CaltechAUTHORS:20100201-095353095
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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.
|Additional Information:||© 2009 American Mathematical Society. Reverts to public domain 28 years from publication. Received by editor(s): February 29, 2008. Posted: August 7, 2009. We would like to thank Ian Agol, Daryl Cooper, Nathan Dunfield, Cameron Gordon, Alexei Kitaev, Sadayoshi Kojima, Marc Lackenby, Darren Long, Shigenori Matsumoto, John Morgan, Marty Scharlemann, and the anonymous referee for numerous helpful comments, suggestions and corrections. We would especially like to single out Alexei Kitaev for thanks for posing questions which were the initial stimulus for this work. The first author was partially funded by NSF grants DMS 0405491 and DMS 0707130.|
|Classification Code:||2000 Mathematics Subject Classification: Primary 57R56; Secondary 57M50.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Joy Painter|
|Deposited On:||11 Feb 2010 16:43|
|Last Modified:||26 Dec 2012 11:44|
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