Journal of the Mathematical Society of Japan

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

Noboru OGAWA

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.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 1 (2015), 419-432.

Dates
First available in Project Euclid: 22 January 2015

https://projecteuclid.org/euclid.jmsj/1421936559

Digital Object Identifier
doi:10.2969/jmsj/06710419

Mathematical Reviews number (MathSciNet)
MR3304028

Zentralblatt MATH identifier
1320.57030

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Citation

OGAWA, Noboru. Linking pairing and Hopf fibrations on $S^{3}$. J. Math. Soc. Japan 67 (2015), no. 1, 419--432. doi:10.2969/jmsj/06710419. https://projecteuclid.org/euclid.jmsj/1421936559

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