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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Positivity of the universal pairing in $3$ dimensions
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by Danny Calegari, Michael H. Freedman and Kevin Walker PDF
J. Amer. Math. Soc. 23 (2010), 107-188 Request permission

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
  • 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