Compositio Mathematica: Volume 146 -

The diffeotopy group of S1×S2 via contact topology
Part of:
Fan Ding, Hansjörg Geiges
Published online by Cambridge University Press:

18 March 2010, pp. 1096-1112
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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).

Compositio Mathematica