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Abelian link invariants and homology. (English) Zbl 1311.81178

Summary: We consider the link invariants defined by the quantum Chern-Simons field theory with compact gauge group \(\operatorname{U}(1)\) in a closed oriented 3-manifold \(M\). The relation of the Abelian link invariants with the homology group of the complement of the links is discussed. We prove that, when \(M\) is a homology sphere or when a link–in a generic manifold \(M\)–is homologically trivial, the associated observables coincide with the observables of the sphere \(S^3\). Finally, we show that the \(\operatorname{U}(1)\) Reshetikhin-Turaev surgery invariant of the manifold \(M\) is not a function of the homology group only, nor a function of the homotopy type of \(M\) alone.{
©2010 American Institute of Physics}

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
58J28 Eta-invariants, Chern-Simons invariants
81T20 Quantum field theory on curved space or space-time backgrounds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
55N33 Intersection homology and cohomology in algebraic topology
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