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

TL;DR

This paper provides an alternative contact topology proof of the diffeotopy group of S^1 imes S^2, explores the topology of contact structures, and presents new Legendrian knot examples distinguishable by contact surgery.

Contribution

It offers a contact topology-based proof of the diffeotopy group of S^1 imes S^2 and introduces novel Legendrian knots distinguishable via contact surgery.

Findings

Diffeotopy group of S^1 imes S^2 is Z_2 + Z_2 + Z_2.

Fundamental group of the space of contact structures is Z.

Existence of Legendrian knots distinguishable by contact surgery.

Abstract

As shown by H. Gluck in 1962, the diffeotopy group of S^1 \times S^2 is isomorphic to Z_2 + Z_2 + Z_2. 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 S^1 \times S^2, based at the standard tight contact structure, is isomorphic to the integers; (ii) inspired by previous work of M. Fraser, an example 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 be distinguished with the help of contact surgery, but not by the classical invariants (topological knot type, Thurston-Bennequin invariant, and rotation number).