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The diffeotopy group of S1×S2 via contact topology

Part of: Differential topology Symplectic geometry, contact geometry Low-dimensional topology

Published online by Cambridge University Press:  18 March 2010

Fan Ding  and
Hansjörg Geiges
Fan Ding
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, PR China (email: dingfan@math.pku.edu.cn)
Hansjörg Geiges
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany (email: geiges@math.uni-koeln.de)
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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).

Keywords

diffeotopy groupcontact topologyLegendrian knot

MSC classification

Secondary: 57R50: Diffeomorphisms 53D10: Contact manifolds, general 57M25: Knots and links in $S^3$ 57R52: Isotopy
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 1096 - 1112
Copyright
Copyright © Foundation Compositio Mathematica 2010

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