This article studies the asymptotic linking pairing lk on the space of exact 2-forms B²(S³) on the 3-sphere S³ through the geometry of Hopf fibrations. Mitsumatsu [7] tried to apply this pairing to 3-dimensional contact topology. He considered a positive definite subspace P(ξ) in (B²(M),lk) associated with a contact structure ξ on a closed 3-manifold M. Further he introduced an invariant of ξ, called the analytic torsion. We investigate the case of the standard contact structure on S³ and construct a positive definite subspace of arbitrary large dimension in the lk-orthogonal complement of P(ξ). This shows that the analytic torsion is infinite. Also we show that it is infinite even for any closed contact 3-manifold.

This article cannot obtain the latest cited-by information.