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.
- Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053–1077. With an appendix by Nathan Dunfield. MR 2328715, DOI 10.1090/S0894-0347-07-00564-4
- Michael Atiyah, The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1990. MR 1078014, DOI 10.1017/CBO9780511623868
- C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no. 4, 883–927. MR 1362791, DOI 10.1016/0040-9383(94)00051-4
- F. Bonahon and L. C. Siebenmann, The characteristic toric splitting of irreducible compact $3$-orbifolds, Math. Ann. 278 (1987), no. 1-4, 441–479. MR 909236, DOI 10.1007/BF01458079
- Hubert L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), no. 2, 177–267. MR 1908823
- A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275–283. MR 918537, DOI 10.1016/0166-8641(87)90092-7
- Bennett Chow and Dan Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. MR 2061425, DOI 10.1090/surv/110
- Tobias H. Colding and William P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics, vol. 4, New York University, Courant Institute of Mathematical Sciences, New York, 1999. MR 1683966
- Dennis M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1, 157–162. MR 697987
- Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699, DOI 10.1007/BF02096988
- Theodore Frankel, Applications of Duschek’s formula to cosmology and minimal surfaces, Bull. Amer. Math. Soc. 81 (1975), 579–582. MR 362166, DOI 10.1090/S0002-9904-1975-13745-1
- Daniel S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994), no. 2, 343–398. MR 1256993, DOI 10.1007/BF02102643
- Daniel S. Freed and Frank Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no. 3, 435–472. MR 1240583, DOI 10.1007/BF02096860
- Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
- Michael H. Freedman, Alexei Kitaev, Chetan Nayak, Johannes K. Slingerland, Kevin Walker, and Zhenghan Wang, Universal manifold pairings and positivity, Geom. Topol. 9 (2005), 2303–2317. MR 2209373, DOI 10.2140/gt.2005.9.2303
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534, DOI 10.1007/BF02698687
- Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
- Richard S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), no. 4, 695–729. MR 1714939, DOI 10.4310/CAG.1999.v7.n4.a2
- Joel Hass and Peter Scott, The existence of least area surfaces in $3$-manifolds, Trans. Amer. Math. Soc. 310 (1988), no. 1, 87–114. MR 965747, DOI 10.1090/S0002-9947-1988-0965747-6
- Allen Hatcher. Notes on Basic $3$-Manifold Topology. Available from the author’s website, 2000.
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- John Hempel, Residual finiteness for $3$-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 379–396. MR 895623
- William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
- Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
- Matthias Kreck and Peter Teichner, Positivity of topological field theories in dimension at least 5, J. Topol. 1 (2008), no. 3, 663–670. MR 2417448, DOI 10.1112/jtopol/jtn016
- Marc Lackenby, Heegaard splittings, the virtually Haken conjecture and property $(\tau )$, Invent. Math. 164 (2006), no. 2, 317–359. MR 2218779, DOI 10.1007/s00222-005-0480-x
- F. Laudenbach, Sur les $2$-sphères d’une variété de dimension $3$, Ann. of Math. (2) 97 (1973), 57–81 (French). MR 314054, DOI 10.2307/1970877
- Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski. MR 1308046, DOI 10.1007/978-3-0346-0332-4
- 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 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
- William H. Meeks III and Shing Tung Yau, The equivariant Dehn’s lemma and loop theorem, Comment. Math. Helv. 56 (1981), no. 2, 225–239. MR 630952, DOI 10.1007/BF02566211
- Pengzi Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, Adv. Theor. Math. Phys. 6 (2002), no. 6, 1163–1182 (2003). MR 1982695, DOI 10.4310/ATMP.2002.v6.n6.a4
- Jean-Pierre Otal, Thurston’s hyperbolization of Haken manifolds, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 77–194. MR 1677888
- R. Penrouz and R. Penrouz, Struktura prostranstva-vremeni, Izdat. “Mir”, Moscow, 1972 (Russian). Translated from the English by L. P. Griščuk and N. V. Mickevič; Edited by Ja. B. Zel′dovič and I. D. Novikov; With an epilogue by the author. MR 0353936
- Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications, 2002.
- Grisha Perelman. Ricci flow with surgery on three-manifolds, 2003.
- Frank Quinn, Lectures on axiomatic topological quantum field theory, Geometry and quantum field theory (Park City, UT, 1991) IAS/Park City Math. Ser., vol. 1, Amer. Math. Soc., Providence, RI, 1995, pp. 323–453. MR 1338394, DOI 10.1090/pcms/001/05
- M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
- Justin Roberts, Irreducibility of some quantum representations of mapping class groups, J. Knot Theory Ramifications 10 (2001), no. 5, 763–767. Knots in Hellas ’98, Vol. 3 (Delphi). MR 1839700, DOI 10.1142/S021821650100113X
- Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR 795231
- Jennifer Schultens, Heegaard genus formula for Haken manifolds, Geom. Dedicata 119 (2006), 49–68. MR 2247647, DOI 10.1007/s10711-006-9045-4
- Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
- Miles Simon, Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074. MR 1957662, DOI 10.4310/CAG.2002.v10.n5.a7
- James Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), no. 1, 88–111. MR 1501673, DOI 10.1090/S0002-9947-1933-1501673-5
- William P. Thurston. Geometry and topology of $3$–manifolds (a.k.a. Thurston’s notes), 1979.
- Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
- Kevin Walker. TQFTs. Preprint, available at http://canyon23.net/math/.
- Danny Calegari
- Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
- MR Author ID: 605373
- Email: firstname.lastname@example.org
- Michael H. Freedman
- Affiliation: Microsoft Station Q, University of California, Santa Barbara, California 93106
- Email: email@example.com
- Kevin Walker
- Affiliation: Microsoft Station Q, University of California, Santa Barbara, California 93106
- Email: firstname.lastname@example.org
- 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