Linking pairing and Hopf fibrations on \(S^{3}\) (Q2339660)

For a closed oriented \(3\)-manifold \(M\) and a volume form \(d\text{vol}\) on \(M\) there is a one to one correspondence between the space \(\Omega^2(M)\) of differential 2-forms and the space \(\mathfrak{X}\) of \(C^\infty\)-vector fields. The space \(\mathfrak{X}_d(M)\) of divergence-free vector fields is isomorphic to the space \(\mathbb Z^2(M)\) of closed \(2\)-forms and the space \(\mathfrak{X}_h(M)\) corresponds to the space \(B^2(M)\) of exact \(2\)-forms. The pairing \(lk\) on \(B^2(M)\) defined as \(lk(d\alpha_1,d\alpha_2)=\int_M\alpha_1\wedge d\alpha_2\) is called the linking pairing. The space \(P(\xi)\) of exact \(2\)-forms \(d(\varphi\alpha)\) is a positive definite subspace in \((B^2(M),lk)\) for any smooth function \(\varphi\), where \(\alpha\) determines a positive contact structure \(\xi\) on \(M\). The analytic torsion \(\text{Tor}^{\text{an}}(M,\xi)\) of a positive contact structure \(\xi\) is defined by the supremum of the dimensions of positive definite subspaces in the \(lk\)-orthogonal complement \(P(\xi)^{\perp_{lk}}\) of \(P(\xi)\) in \((B^2(M),lk)\). In this paper, the author studies the linking pairing on the three-sphere \(\mathbb S^3\) through the geometry of Hopf fibrations and proves that on the \(3\)-sphere \(\mathbb S^3\) and the standard positive contact structure \(\xi_{\text{st}}\) there exists a positive definite subspace in \(P(\xi)^{\perp_{lk}}\) with respect to \(lk\) of arbitrary large dimension, which means that the analytic torsion of \((\mathbb S^3,\xi_{\text{st}})\) is infinite. Also, it is proven that the analytic torsion is infinite for any closed contact \(3\)-manifold.