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As shown by H. Gluck in 1962, the diffeotopy group of S^1 \times S^2 isisomorphic to Z_2 + Z_2 + Z_2. Here an alternative proof of this result isgiven, relying on contact topology. We then discuss two applications to contacttopology: (i) it is shown that the fundamental group of the space of contactstructures on S^1 \times S^2, based at the standard tight contact structure, isisomorphic to the integers; (ii) inspired by previous work of M. Fraser, anexample is given of an integer family of Legendrian knots in S^1 \times S^2 #S^1 \times S^2 (with its standard tight contact structure) that can bedistinguished with the help of contact surgery, but not by the classicalinvariants (topological knot type, Thurston-Bennequin invariant, and rotationnumber).

The diffeotopy group of S^1 \times S^2 via contact topology