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Positivity of the universal pairing in $ 3$ dimensions

Authors: Danny Calegari, Michael H. Freedman and Kevin Walker
Journal: J. Amer. Math. Soc. 23 (2010), 107-188
MSC (2000): Primary 57R56; Secondary 57M50
Published electronically: August 7, 2009
MathSciNet review: 2552250
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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) \le \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.

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Additional Information

Danny Calegari
Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125

Michael H. Freedman
Affiliation: Microsoft Station Q, University of California, Santa Barbara, California 93106

Kevin Walker
Affiliation: Microsoft Station Q, University of California, Santa Barbara, California 93106

Received by editor(s): February 29, 2008
Published electronically: August 7, 2009
Additional Notes: The first author was partially funded by NSF grants DMS 0405491 and DMS 0707130.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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