Positivity of the universal pairing in 3 dimensions

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) ≤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.

[1]  J. Singer.  Three-dimensional manifolds and their Heegaard diagrams , 1933 .

[2]  Friedhelm Waldhausen,et al.  Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I , 1967 .

[3]  F. Waldhausen.  Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. II , 1967 .

[4]  H. Schöpf.  Battelle Rencontres ‐ 1967 Lectures in Mathematics and Physics (Ed. by C. M. DeWitt and J. A. Wheeler). XVII + 557 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. , 1970 .

[5]  M. Raghunathan.  Discrete subgroups of Lie groups , 1972 .

[6]  F. Laudenbach.  Sur Les 2-Spheres D'une Variete de Dimension 3 , 1973 .

[7]  Theodore Frankel,et al.  Applications of Duschek's formula to cosmology and minimal surfaces , 1975 .

[8]  William Jaco,et al.  Seifert fibered spaces in 3-manifolds , 1979 .

[9]  Klaus Johannson,et al.  Homotopy Equivalences of 3-Manifolds with Boundaries , 1979 .

[10]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[11]  S. Yau,et al.  The equivariant Dehn's lemma and loop theorem , 1981 .

[12]  S. Yau,et al.  Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature , 1982 .

[13]  P. Scott,et al.  The geometries of 3-manifolds , 1983 .

[14]  M. Freedman,et al.  Least area incompressible surfaces in 3-manifolds , 1983 .

[15]  D. DeTurck.  Deforming metrics in the direction of their Ricci tensors , 1983 .

[16]  Richard Schoen,et al.  Estimates for stable minimal surfaces in three dimensional manifolds , 1984 .

[17]  C. McA. Gordon,et al.  Reducing heegaard splittings , 1987 .

[18]  John Hempel,et al.  RESIDUAL FINITENESS FOR 3-MANIFOLDS , 1987 .

[19]  F. Bonahon,et al.  The characteristic toric splitting of irreducible compact 3-orbifolds , 1987 .

[20]  J. Hass,et al.  The existence of least area surfaces in 3-manifolds , 1988 .

[21]  M. Atiyah.  The geometry and physics of knots: Frontmatter , 1990 .

[22]  M. Atiyah.  The Geometry and Physics of Knots , 1990 .

[23]  E. Witten,et al.  Topological gauge theories and group cohomology , 1990 .

[24]  Vladimir Turaev,et al.  Invariants of 3-manifolds via link polynomials and quantum groups , 1991 .

[25]  F. Quinn,et al.  Lectures on axiomatic topological quantum field theory , 1991 .

[26]  D. Freed,et al.  Chern-Simons theory with finite gauge group , 1991, hep-th/9111004.

[27]  R. Hamilton,et al.  The formations of singularities in the Ricci Flow , 1993 .

[28]  Daniel S. Freed,et al.  Higher algebraic structures and quantization , 1992, hep-th/9212115.

[29]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[30]  N. Habegger,et al.  Topological Auantum Field Theories derived from the Kauffman bracket , 1995 .

[31]  J. Otal.  Thurston's Hyperbolization of Haken Manifolds , 1996 .

[32]  Irreducibility of some quantum representations of mapping class groups , 1999, math/9909128.

[33]  R. Hamilton.  Non-singular solutions of the Ricci flow on three-manifolds , 1999 .

[34]  A. Hatcher,et al.  Notes on Basic 3-Manifold Topology , 2001 .

[35]  J. Schultens.  Heegaard Genus Formula for Haken Manifolds , 2001, math/0108028.

[36]  H. Bray.  PROOF OF THE RIEMANNIAN PENROSE INEQUALITY USING THE POSITIVE MASS THEOREM , 2001 .

[37]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[38]  Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface , 2002, math-ph/0212025.

[39]  Marc Lackenby.  Heegaard splittings, the virtually Haken conjecture and Property (τ) , 2002 .

[40]  M. Simon,et al.  Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature , 2002 .

[41]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[42]  B. Chow,et al.  The Ricci Flow : An Introduction I , 2013 .

[43]  Sergey Nikitin.  On 3-manifolds , 2005 .

[44]  W. Thurston,et al.  Lower bounds on volumes of hyperbolic Haken 3-manifolds , 1999, math/0506338.

[45]  Kevin Walker,et al.  Universal manifold pairings and positivity , 2005, math/0503054.

[46]  P. Teichner,et al.  Positive topological field theories and manifolds of dimension 5 , 2007 .

[47]  Positivity of topological field theories in dimension at least 5 , 2008 .