Gluck surgery and framed links in 4-manifolds (Q2725540)

A 2-knot \(K\) is meant to be an embedded 2-sphere in the 4-sphere or in a 4-manifold \(X\) such that a tubular neighborhood of it is diffeomorphic to \(S^2\times D^2\). Gluck surgery is the operation of cutting out this tubular neighborhood from \(X\) and pasting it back via a map \(\tau\) called twisting [\textit{H. Gluck}, Trans. Am. Math. Soc. 104, 308-333 (1962; Zbl 0111.18804)]. Let \(\Sigma_X (K)\) be a 4-manifold obtained from \(X\) by Gluck surgery along \(K\). If \(K\) is null-homotopic in \(X\), then Gluck surgery along \(K\) does not change the homotopy type of the manifold, but may change the diffeomorphism type. A problem is to construct a pair of manifolds obtained by Gluck surgery that are homotopy equivalent but non-diffeomorphic (fake 4-manifolds). In the non-orientable case the problem was solved by \textit{S. Akbulut} [Contemp. Math. 44, 281-286 (1985; Zbl 0577.57004); Topology 27, No. 2, 239-243 (1988; Zbl 0649.57011)]. The authors give an alternative proof of a theorem of \textit{P. Melvin} [Blowing up and down in 4-manifold, Ph. D. Thesis, UC Berkeley, 1977] which states that Gluck surgery along a banded 2-knot is independent of the bands. Then they study Gluck surgeries along a 2-knot obtained by ribbon moves from another 2-knot, and prove that the resulting manifolds are always diffeomorphic.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].