Gluck twisting 4-manifolds with odd intersection form (Q395067)

The Gluck twist operation is a surgery along an embedded \(S^2\) in a 4-manifold with trivial regular neighborhood. That is, the operation consists in removing the trivial neighborhood and regluing the neighborhood by a non-trivial map \(S^2\times S^1\to S^2\times S^1\). This is just regarded as the 4-dimensional version of Dehn surgery in the 3-dimensionial setting. Unlike Dehn surgery, this operation can produce 4-dimensional exotic smooth structures. For example, in [Topology 27, No. 2, 239--243 (1988; Zbl 0649.57011)], \textit{S. Akbulut} constructed a fake structure over \(S^3\tilde{\times}S^1\#S^2\times S^2\) by a Gluck twist.NEWLINENEWLINEThe Cappell-Shaneson spheres are obtained from surgeries on 3-torus bundles over \(S^1\). The way to do the surgery has two choices. The pair of resulting Cappell-Shaneson spheres for a 3-torus bundle is related to a Gluck twist. As another application of the Gluck twist, in [Ann. Math. (2) 149, No. 2, 497--510 (1999; Zbl 0931.57016)], \textit{S. Akbulut} essentially used Gluck's technique to prove that Scharlemann's manifold has standard smooth structure.NEWLINENEWLINEThe Gluck twist along embedded spheres in the 4-sphere looks interesting. Any Gluck twist of any 2-knot in the 4-sphere gives a potential counterexample to the 4-dimensional smooth Poincaré conjecture. There exist some partial results for Gluck twists on the 4-sphere. So far, nobody has constructed any exotic 4-sphere by Gluck twists or any other technique.NEWLINENEWLINEIn this paper, the authors give a sufficient condition for a Gluck twist not to change the diffeomorphism type. Their condition considers a sphere class with odd self-intersection. The deformation of a handle diagram by a Gluck twist is undone because of the existence of the sphere class.NEWLINENEWLINEAnother interesting point is that they rewrite the Gluck twist as a plug twist \((W_{2m+1,0},f_{2m+1,0})\) defined by the authors in [\textit{S. Akbulut} and \textit{K. Yasui}, J. Gökova Geom. Topol. GGT 2, 40--82 (2008; Zbl 1214.57027)].