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January, 2015 Linking pairing and Hopf fibrations on $S^{3}$
Noboru OGAWA
J. Math. Soc. Japan 67(1): 419-432 (January, 2015). DOI: 10.2969/jmsj/06710419


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.


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Noboru OGAWA. "Linking pairing and Hopf fibrations on $S^{3}$." J. Math. Soc. Japan 67 (1) 419 - 432, January, 2015.


Published: January, 2015
First available in Project Euclid: 22 January 2015

zbMATH: 1320.57030
MathSciNet: MR3304028
Digital Object Identifier: 10.2969/jmsj/06710419

Primary: 57R17
Secondary: 57M50

Keywords: contact structures , Hopf fibrations , linking pairing

Rights: Copyright © 2015 Mathematical Society of Japan


Vol.67 • No. 1 • January, 2015
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