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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 | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Ian Agol, Peter A. Storm, and William P. Thurston.
    Lower bounds on volumes of hyperbolic Haken 3-manifolds.
    J. Amer. Math. Soc., 20(4):1053-1077 (electronic), 2007.
    With an appendix by Nathan Dunfield. MR 2328715 (2008i:53086)
  • 2. Michael Atiyah.
    The geometry and physics of knots.
    Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge, 1990. MR 1078014 (92b:57008)
  • 3. C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel.
    Topological quantum field theories derived from the Kauffman bracket.
    Topology, 34(4):883-927, 1995. MR 1362791 (96i:57015)
  • 4. F. Bonahon and L. C. Siebenmann.
    The characteristic toric splitting of irreducible compact $ 3$-orbifolds.
    Math. Ann., 278(1-4):441-479, 1987. MR 909236 (90a:57017)
  • 5. Hubert L. Bray.
    Proof of the Riemannian Penrose inequality using the positive mass theorem.
    J. Differential Geom., 59(2):177-267, 2001. MR 1908823 (2004j:53046)
  • 6. A. J. Casson and C. McA. Gordon.
    Reducing Heegaard splittings.
    Topology Appl., 27(3):275-283, 1987. MR 918537 (89c:57020)
  • 7. Bennett Chow and Dan Knopf.
    The Ricci flow: An introduction, volume 110 of Mathematical Surveys and Monographs.
    American Mathematical Society, Providence, RI, 2004. MR 2061425 (2005e:53101)
  • 8. Tobias H. Colding and William P. Minicozzi, II.
    Minimal surfaces, volume 4 of Courant Lecture Notes in Mathematics.
    New York University Courant Institute of Mathematical Sciences, New York, 1999. MR 1683966 (2002b:49072)
  • 9. Dennis M. DeTurck.
    Deforming metrics in the direction of their Ricci tensors.
    J. Differential Geom., 18(1):157-162, 1983. MR 697987 (85j:53050)
  • 10. Robbert Dijkgraaf and Edward Witten.
    Topological gauge theories and group cohomology.
    Comm. Math. Phys., 129(2):393-429, 1990. MR 1048699 (91g:81133)
  • 11. Theodore Frankel.
    Applications of Duschek's formula to cosmology and minimal surfaces.
    Bull. Amer. Math. Soc., 81:579-582, 1975. MR 0362166 (50:14608)
  • 12. Daniel S. Freed.
    Higher algebraic structures and quantization.
    Comm. Math. Phys., 159(2):343-398, 1994. MR 1256993 (95c:58034)
  • 13. Daniel S. Freed and Frank Quinn.
    Chern-Simons theory with finite gauge group.
    Comm. Math. Phys., 156(3):435-472, 1993. MR 1240583 (94k:58023)
  • 14. Michael Freedman, Joel Hass, and Peter Scott.
    Least area incompressible surfaces in $ 3$-manifolds.
    Invent. Math., 71(3):609-642, 1983. MR 695910 (85e:57012)
  • 15. Michael H. Freedman, Alexei Kitaev, Chetan Nayak, Johannes K. Slingerland, Kevin Walker, and Zhenghan Wang.
    Universal manifold pairings and positivity.
    Geom. Topol., 9:2303-2317 (electronic), 2005. MR 2209373 (2006k:57080)
  • 16. Mikhael Gromov.
    Groups of polynomial growth and expanding maps.
    Inst. Hautes Études Sci. Publ. Math., (53):53-73, 1981. MR 623534 (83b:53041)
  • 17. Richard S. Hamilton.
    The formation of singularities in the Ricci flow.
    In Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), pages 7-136. Int. Press, Cambridge, MA, 1995. MR 1375255 (97e:53075)
  • 18. Richard S. Hamilton.
    Non-singular solutions of the Ricci flow on three-manifolds.
    Comm. Anal. Geom., 7(4):695-729, 1999. MR 1714939 (2000g:53034)
  • 19. Joel Hass and Peter Scott.
    The existence of least area surfaces in $ 3$-manifolds.
    Trans. Amer. Math. Soc., 310(1):87-114, 1988. MR 965747 (90c:53022)
  • 20. Allen Hatcher.
    Notes on Basic $ 3$-Manifold Topology.
    Available from the author's website, 2000.
  • 21. John Hempel.
    $ 3$-Manifolds.

    Ann. of Math. Studies, No. 86. Princeton University Press, Princeton, N. J., 1976. MR 0415619 (54:3702)
  • 22. John Hempel.
    Residual finiteness for $ 3$-manifolds.
    In Combinatorial group theory and topology (Alta, Utah, 1984), volume 111 of Ann. of Math. Stud., pages 379-396. Princeton Univ. Press, Princeton, NJ, 1987. MR 895623 (89b:57002)
  • 23. William H. Jaco and Peter B. Shalen.
    Seifert fibered spaces in $ 3$-manifolds.
    Mem. Amer. Math. Soc., 21(220):viii+192, 1979. MR 539411 (81c:57010)
  • 24. Klaus Johannson.
    Homotopy equivalences of $ 3$-manifolds with boundaries, volume 761 of Lecture Notes in Mathematics.
    Springer, Berlin, 1979. MR 551744 (82c:57005)
  • 25. Matthias Kreck and Peter Teichner.
    Positivity of topological field theories in dimension at least 5.
    J. Topol., 1(3):663-670, 2008. MR 2417448 (2009d:81332)
  • 26. Marc Lackenby.
    Heegaard splittings, the virtually Haken conjecture and property $ (\tau)$.
    Invent. Math., 164(2):317-359, 2006. MR 2218779 (2007c:57030)
  • 27. F. Laudenbach.
    Sur les $ 2$-sphères d'une variété de dimension $ 3$.
    Ann. of Math. (2), 97:57-81, 1973. MR 0314054 (47:2606)
  • 28. Alexander Lubotzky.
    Discrete groups, expanding graphs and invariant measures, volume 125 of Progress in Mathematics.
    Birkhäuser Verlag, Basel, 1994.
    With an appendix by Jonathan D. Rogawski. MR 1308046 (96g:22018)
  • 29. William Meeks, III, Leon Simon, and Shing Tung Yau.
    Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature.
    Ann. of Math. (2), 116(3):621-659, 1982. MR 678484 (84f:53053)
  • 30. William H. Meeks, III and Shing Tung Yau.
    The equivariant Dehn's lemma and loop theorem.
    Comment. Math. Helv., 56(2):225-239, 1981. MR 630952 (83b:57006)
  • 31. Pengzi Miao.
    Positive mass theorem on manifolds admitting corners along a hypersurface.
    Adv. Theor. Math. Phys., 6(6):1163-1182 (2003), 2002. MR 1982695 (2005a:53065)
  • 32. Jean-Pierre Otal.
    Thurston's hyperbolization of Haken manifolds.
    In Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), pages 77-194. Int. Press, Boston, MA, 1998. MR 1677888 (2000b:57025)
  • 33. Roger Penrose.
    Structure of space-time; in Battelle rencontres. 1967 lectures in mathematics and physics.
    Edited by Cecile M. DeWitt and John A. Wheeler. W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0353936 (50:6418)
  • 34. Grisha Perelman.
    The entropy formula for the Ricci flow and its geometric applications, 2002.
  • 35. Grisha Perelman.
    Ricci flow with surgery on three-manifolds, 2003.
  • 36. Frank Quinn.
    Lectures on axiomatic topological quantum field theory.
    In Geometry and quantum field theory (Park City, UT, 1991), volume 1 of IAS/Park City Math. Ser., pages 323-453. Amer. Math. Soc., Providence, RI, 1995. MR 1338394 (96e:57021)
  • 37. M. S. Raghunathan.
    Discrete subgroups of Lie groups.
    Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68.
    Springer-Verlag, New York, 1972. MR 0507234 (58:22394a)
  • 38. N. Reshetikhin and V. G. Turaev.
    Invariants of $ 3$-manifolds via link polynomials and quantum groups.
    Invent. Math., 103(3):547-597, 1991. MR 1091619 (92b:57024)
  • 39. Justin Roberts.
    Irreducibility of some quantum representations of mapping class groups.
    J. Knot Theory Ramifications, 10(5):763-767, 2001.
    Knots in Hellas '98, Vol. 3 (Delphi). MR 1839700 (2002f:57065)
  • 40. Richard Schoen.
    Estimates for stable minimal surfaces in three-dimensional manifolds.
    In Seminar on minimal submanifolds, volume 103 of Ann. of Math. Stud., pages 111-126. Princeton Univ. Press, Princeton, NJ, 1983. MR 795231 (86j:53094)
  • 41. Jennifer Schultens.
    Heegaard genus formula for Haken manifolds.
    Geom. Dedicata, 119:49-68, 2006. MR 2247647 (2007e:57020)
  • 42. Peter Scott.
    The geometries of $ 3$-manifolds.
    Bull. London Math. Soc., 15(5):401-487, 1983. MR 705527 (84m:57009)
  • 43. Miles Simon.
    Deformation of $ C\sp 0$ Riemannian metrics in the direction of their Ricci curvature.
    Comm. Anal. Geom., 10(5):1033-1074, 2002. MR 1957662 (2003j:53107)
  • 44. J. Singer.
    Three-dimensional manifolds and their Heegaard diagrams.
    Trans. Amer. Math. Soc., 35:88-111, 1933. MR 1501673
  • 45. William P. Thurston.
    Geometry and topology of $ 3$-manifolds (a.k.a. Thurston's notes), 1979.
  • 46. Friedhelm Waldhausen.
    Eine Klasse von $ 3$-dimensionalen Mannigfaltigkeiten. I, II.
    Invent. Math. 3 (1967), 308-333; ibid., 4:87-117, 1967. MR 0235576 (38:3880)
  • 47. Kevin Walker.
    Preprint, available at

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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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