Linking pairing and Hopf fibrations on $S^{3}$

Abstract

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