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Gauge theoretic invariants of Dehn surgeries on knots. (English) Zbl 1065.57009

The paper describes new methods for computing gauge theoretic invariants for homology 3-spheres, such as the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2)-representations. The ultimate goal is to develop a method for computing the SU(3) Casson invariant. Recall that the SU(3) Casson invariant is given by a sum of two terms: the signed count of conjugacy classes of irreducible SU(3)-representations and a correction term involving only conjugacy classes of irreducible SU(2)-representations. The paper shows how to compute the latter term, which is done in terms of rho invariants for flat SU(2) connections. The paper is self-contained. The first sections familiarize the reader with the gauge theoretic notions and results needed in the sequel. Section 3 proves an essential result, Theorem 3.9, which describes the splitting of the spectral flow under Dehn surgery. Section 4 contains a detailed analysis of connections on the solid torus and shows how to compute the spectral flow between two nontrivial connections on the solid torus. These results are then used in Section 5 to develop formulas for some gauge theoretic invariants of flat connections on Dehn surgeries, namely the \({\mathbb C}^2\)-spectral flow, the Chern-Simons invariant, and the rho invariants. These are then applied to Dehn surgeries on torus knots, with \((2,q)\)-torus knots treated in more detail. In the end of the paper, the authors compute the SU(3) gauge theoretic Casson invariant for Dehn surgeries on \((2,3)\), \((2,5)\), \((2,7)\) and \((2,9)\) torus knots.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
53D12 Lagrangian submanifolds; Maslov index
58J28 Eta-invariants, Chern-Simons invariants
58J30 Spectral flows
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References:

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