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Positivity of topological field theories in dimension at least 5. (English) Zbl 1151.57035

Universal manifold pairings were introduced by the research group at Microsoft Station Q to study which aspects of manifold topology are visible to unitary topological field theories [M. H. Freedman, A. Kitaev, C. Nayak, J. K. Slingerland, K. Walker and Z. Wang, Geom. Topol. 9, 2303-2317 (2005; Zbl 1129.57035)]. In physically relevant cases the pairing is positive definite, and the associated topological field theories are called PTFT’s in this paper. Freedman et al. showed that PTFT’s distinguish manifolds in dimensions \(0\), \(1\), and \(2\), but that conversely in dimension \(4\) they can distinguish neither homotopy-equivalent simply connected 4-manifolds nor smoothly \(s\)-cobordant \(4\)–manifolds. This leaves open the question for dimension \(3\), where it is surely most interesting, and for dimension at least \(5\).
The present paper tackles the question of how efficient PTFT’s are in distinguishing manifolds in dimension at least \(5\). For dimension at least \(6\), the authors construct simply connected examples which PTFT’s cannot distinguish. In dimension \(5\) they prove that simply connected manifolds can be distinguished by PTFT’s, but construct non-simply-connected examples which PTFT’s cannot distinguish. The examples have boundary \(S^3\times S^n\) for \(n>1\) and \(S^{4k-1}\) for \(k>1\), the latter case being equivalent to the case of closed \(4k\)–manifolds.

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)

Citations:

Zbl 1129.57035
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References:

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