×

Chern-Simons invariants of 3-manifolds and representation spaces of knot groups. (English) Zbl 0681.57006

A formula is proven expressing the Chern-Simons invariant of a flat SU(2)-connection on a 3-manifold as an integral along a path (joining the holonomy of the given connection to the trivial representation) of SL(2,C)-representations of the complement of a knot in a 3-manifold. The integrand is a simple function of the values of these representations on the peripheral subgroup of the knot. This formula is then applied to compute Chern-Simons invariants of Seifert-fibered homology spheres (previously computed using a different method by Stern and Fintushel) and to compute numerically the Chern-Simons invariants of several manifolds obtained from Dehn surgery on the figure 8 knot. Finally, a formula is proven which explicitly gives the SU(2)-representations and Chern-Simons invariants of torus bundles over \(S^ 1\) in terms of their monodromy matrices.
Reviewer: P.A.Kirk

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57R20 Characteristic classes and numbers in differential topology
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [CS] Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math.99, 48-69 (1974) · Zbl 0283.53036 · doi:10.2307/1971013
[2] [F] Floer, A.: An instanton invariant for 3-manifolds. Commun. Math. Phys.118, 215-240 (1988) · Zbl 0684.53027 · doi:10.1007/BF01218578
[3] [FS] Fintushel, R., Stern, R.: Instanton homology of Seifert fibered homology spheres. Preprint 1988
[4] [KK] Kirk, P., Klassen, E.: Representation spaces of Seifert fibered homology spheres. To appear in Topology · Zbl 0721.57007
[5] [K] Klassen, E.: Representations of knot groups inSU(2). To appear in Trans. A.M.S. · Zbl 0743.57003
[6] [R] Riley, R.: Nonabelian representations of 2-bridge knot groups. Quart. J. Math. Oxf.35, 191-208 (1984) · Zbl 0549.57005 · doi:10.1093/qmath/35.2.191
[7] [T] Taubes, C.: Casson’s invariant and gauge theory. Preprint 1988 · Zbl 0702.53017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.