Nonlocally smoothable topological symmetries of four-manifolds (Q1099852)

The authors provide first examples of closed manifolds which do not admit locally smoothable involutions but do admit topological involutions. More specifically, they show that there exist infinitely many closed, simply connected four-dimensional manifolds which do not admit locally smoothable involutions but which do admit topological involutions. Their examples include manifolds of even intersection form, as well as of odd intersection form. However, the even intersection form manifolds with topological involutions are constructed in a more systematic way. The authors use their earlier result [ibid. 53, 759-764 (1986)] to conclude that the constructed manifolds do not admit (nontrivial) locally smoothable involutions. As a consequence, they note that the manifold \(| E_ 8|\) has no (nontrivial) locally smoothable involution but it admits a topological involution.