A quantum obstruction to embedding (Q2760482)

The authors describe the approach to topological quantum field theory given by \textit{J. D. Roberts} [Quantum invariants via skein theory, Ph.D. thesis, University of Cambridge 1994] and develop quantum invariant obstructions to embedding one (compact, oriented) 3-manifold into another. They introduce three ideals in \({\mathbb Z}[u]\) (where \(u\) is a primitive \(8r\)th root of unity for an odd prime \(r\)), associated to a compact, oriented 3-manifold (with boundary). NEWLINENEWLINENEWLINEThe first two, \(I_r(M)\) and \(I_r^{\text{TV}}(M)\), are invariants of oriented and unoriented homeomorphisms, resp.. \(I_r(M)\) is the ideal generated by the Witten-Reshetikhin-Turaev invariants of all closed manifolds containing \(M\) and \(I_r^{\text{TV}}(M)\) is the ideal generated by the Turaev-Viro invariants of all closed manifolds containing \(M\). It is shown that if the 3-manifold \(M\) embeds in the 3-manifold \(N\), then \(I_r(N)\subset I_r(M)\) and \(I_r^{\text{TV}}(N)\subset I_r^{\text{TV}}(M)\). As a corollary they obtain a generalization of the observation that a manifold containing a punctured lens space does not embed in \(S^3\) as follows: \(N\) does not embed in \(S^3\) if \(N\) contains a submanifold \(M\) such that \(I_r(M)\) or \(I_r^{\text{TV}}(M)\) is not all of \({\mathbb Z}[u]\). An example is given to show that the invariants are not merely homological. The third invariant depends on the marked boundary of \(M\) and is used only as a tool to estimate \(I_r^{\text{TV}}(M)\).