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

HTML articles powered by AMS MathViewer

- 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.

## References

- 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/.

## 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