## Positivity of the universal pairing in $3$ dimensions

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- by Danny Calegari, Michael H. Freedman and Kevin Walker
- J. Amer. Math. Soc.
**23**(2010), 107-188 - DOI: https://doi.org/10.1090/S0894-0347-09-00642-0
- Published electronically: August 7, 2009
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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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## Bibliographic Information

**Danny Calegari**- Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
- MR Author ID: 605373
- Email: dannyc@its.caltech.edu
**Michael H. Freedman**- Affiliation: Microsoft Station Q, University of California, Santa Barbara, California 93106
- Email: michaelf@microsoft.com
**Kevin Walker**- Affiliation: Microsoft Station Q, University of California, Santa Barbara, California 93106
- Email: kevin@canyon23.net
- 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.
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 107-188 - MSC (2000): Primary 57R56; Secondary 57M50
- DOI: https://doi.org/10.1090/S0894-0347-09-00642-0
- MathSciNet review: 2552250