# Positivity of the universal pairing in 3 dimensions

@article{Calegari2008PositivityOT, title={Positivity of the universal pairing in 3 dimensions}, author={Danny Calegari and Michael H. Freedman and Kevin Walker}, journal={Journal of the American Mathematical Society}, year={2008}, volume={23}, pages={107-188} }

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…

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## References

SHOWING 1-10 OF 63 REFERENCES

Universal manifold pairings and positivity

- Mathematics
- 2005

Gluing two manifolds M1 and M2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x = �aiMi yields a sesquilinear pairing p = h , i with values in (formal…

Invariants of 3-manifolds via link polynomials and quantum groups

- Mathematics
- 1991

The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a…

Positivity of topological field theories in dimension at least 5

- Mathematics
- 2008

In this paper we answer a question of Mike Freedman, regarding the efficiency of positive topological field theories as invariants of smooth manifolds in dimensions greater than 4. We show that…

Seifert fibered spaces in 3-manifolds

- Mathematics
- 1979

Publisher Summary This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy…

Three-dimensional manifolds and their Heegaard diagrams

- Mathematics
- 1933

One of the outstanding problems in topology today is the classification of n-dimensional manifolds, n >3. Poincare, the founder of modern analysis situs, devoted several papers to it and allied…

Positive topological field theories and manifolds of dimension 5

- Mathematics
- 2007

AbstractInthis paperwe answeraquestionofMikeFreedman, regardingtheeﬃciencyofpositive topological ﬁeld theories as invariants of smooth manifolds in dimensions >4. We show that simply connected closed…

On 3-manifolds

- Mathematics
- 2005

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P…

Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature

- Mathematics
- 1982

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some…

The geometries of 3-manifolds

- Mathematics
- 1983

The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of…

The entropy formula for the Ricci flow and its geometric applications

- Mathematics
- 2002

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric…