A plug with infinite order and some exotic 4-manifolds (Q2803995)

Corks and plugs are fundamental objects in smooth 4-manifolds, and allow to study their smooth structures. These objects were defined by \textit{S. Akbulut} [J. Differ. Geom. 33, No. 2, 335--356 (1991; Zbl 0839.57015)] and \textit{S. Akbulut} and \textit{K. Yasui} [J. Gökova Geom. Topol. GGT 2, 40--82 (2008; Zbl 1214.57027)]. Twisting the cork (or plug) in the 4-manifold \(X\) by an involution, one can obtain an exotic copy of \(X\). Conversely, it was shown in [\textit{S. Akbulut} and \textit{R. Matveyev}, Int. Math. Res. Not. 1998, No. 7, 371--381 (1998; Zbl 0911.57025); \textit{C. L. Curtis} et al., Invent. Math. 123, No. 2, 343--348 (1996; Zbl 0843.57020); \textit{R. Matveyev}, J. Differ. Geom. 44, No. 3, 571--582 (1996; Zbl 0885.57016)] that any simply connected exotic pair can be obtained by the contractible cork twisting.NEWLINENEWLINEIn this article, the author relaxes the definition of cork and plug to allow infinite order. He defines such corks and plugs, and proves the following: (i) there exists a plug \((P, \varphi)\) with infinite order, where \(P\) is a simply connected, compact, Stein \(4\)-manifold with \(b_2 = 2\), (ii) the square \((P, \varphi^{2})\) of the plug twist is a non-contractible cork with infinite order.NEWLINENEWLINEThe author also shows how (in a special case of \(X\)) the plug twist \((P, \varphi)\) changes \(X\) to a knot surgery manifold \(X_K\), where \(K\) is any knot with unknotting number \(1\). Furthermore, he shows that a \((P, \varphi^{2})\)-twist on \(X\) yields \(X_{K_n}\), where \(K_n\) is a knot obtained by \(n\) times iteration of the knotting operation from the unknot to \(K\).NEWLINENEWLINEIn the last section, the author discusses the enlargements of the manifold \(P\) by attaching \(-1\)-framed \(2\)-handles and also gives nice applications of his results.