Elliptic surfaces and some simple exotic \(\mathbb{R}^ 4\)'s
The main purpose of the present paper is to exhibit some exotic \(R^4\)'s which admit simple, explicit description. The authors focus their attention on those examples of exotic \(R^4\)'s which arise from the \(h\)-cobordism theorem. They use a generalization of \textit{S. Akbulut}'s construction [J. Differ. Geom. 33, 335-356 (1991; Zbl 0839.57015)] to obtain a sequence of non-product \(h\)-cobordisms. The 4-manifolds involved in these constructions are minimal elliptic surfaces, \(E(n)\), and their logarithmic 0-transforms, \(E(n;0)\). Akbulut's \(h\)-cobordism is between \(X\#k \overline {CP}^2\) and \(E(2;0) \#k \overline {CP}^2\), where \(X\) is a homotopy \(K3\) surface and \(k\) is an arbitrary positive integer. In the present paper the authors show that the homotopy \(K3\) surface used by Akbulut is the standard \(K3\) surface, that is \(X=E(2)\). Furthermore, Akbulut had described the surgery induced by his \(h\)-cobordism as the result of deleting a Mazur manifold from \(X\# \overline{CP}^2\) and then gluing it back by an identification of the boundaries that differs from the initial one by an involution. The authors generalize Akbulut's construction to obtain an explicit \(h\)-cobordism between \(E(n)\#2 \overline {CP}^2\) and \(E(n;0) \# \overline{CP}^2\), for any integer \(n\geq 2\).
scientific article; zbMATH DE number 955317