Barbell diffeomorphisms and the Hatcher-Wagoner 2nd obstruction.

In their seminal paper, Budney and Gabai introduced the notion of a Barbell diffeomorphism and used it to show the existence of knotted 3-balls in the 4-sphere as well as the existence of infinitely many homotopic but nonisotopic diffeomorphisms of S^1 x S^3. In this talk, I will discuss a certain class of barbell diffeomorphisms of (S^1 x S^2 x I). As shown by Singh, the diffeomorphism group of (S^1 x S^2 x I) admits an infinitely generated subgroup of diffeomorphisms that are pseudo-isotopic but not isotopic. The obstruction used to distinguish these diffeomorphisms is Hatcher and Wagoner's second obstruction. The main goal of this talk is to show that the class of barbell diffeomorphism realizes these same obstruction classes as Singh and are thus pseudo-isotopic but not isotopic.