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The diffeotopy group of S1×S2 via contact topology

Published online by Cambridge University Press:  18 March 2010

Fan Ding
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, PR China (email: [email protected])
Hansjörg Geiges
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany (email: [email protected])
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Abstract

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As shown by Gluck in 1962, the diffeotopy group of S1×S2 is isomorphic to ℤ2⊕ℤ2 ⊕ℤ2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S1×S2, based at the standard tight contact structure, is isomorphic to ℤ; (ii) inspired by previous work of Fraser, an example is given of an integer family of Legendrian knots in S1×S2#S1×S2 (with its standard tight contact structure) that can be distinguished with the help of contact surgery, but not by the classical invariants (topological knot type, Thurston–Bennequin invariant, and rotation number).

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2010

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