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Positivity of the universal pairing in 3 dimensions. (English) Zbl 1201.57024

Gluing two manifolds along a common boundary yields a closed manifold. Extending to formal linear combinations gives rise to the universal pairing \(p=\langle-,-\rangle\) with values in formal combinations of closed manifolds. In [M. H. Freedman, A. Kitaev, C. Nayak, J. K. Slingerland, K. Walker and Z. Wang, Geom. Topol. 9, 2303–2317 (2005; Zbl 1129.57035)], it is suggested that a fruitful way to think about TQFT is as a representation of this universal pairing onto a finite dimensional complex-valued quotient pairing. A long-standing question is whether TQFT separates \(d\) dimensional manifolds when \(d=3\). A necessary condition would be that \(\langle x,x\rangle\neq 0\) whenever \(x\neq 0\), i.e. that \(p\) be positive. This happens when \(d<3\) but not when \(d>3\) [M. Kreck and P. Teichner, J. Topol. 1, No. 3, 663–670 (2008; Zbl 1151.57035)].
Using a great variety of \(3\)-dimensional techniques, and studying non-trivial JSJ decompositions, the main theorem is that \(p\) is indeed positive in dimension \(3\). The proof involves constructing a complexity function on \(3\)-manifolds which satisfies a gluing axiom which the authors suggest we think of as kind of a topological Cauchy-Schwatz inequality. Along the way, a number of results of independent interest are proven, including a generalization of a result in [I. Agol, P. A. Storm, W. P. Thurston and N. Dunfield, J. Am. Math. Soc. 20, No. 4, 1053–1077 (2007; Zbl 1155.58016)].
This paper is a joy to read.

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
57M50 General geometric structures on low-dimensional manifolds
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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