Asymmetric four-dimensional manifolds
The main result of this paper is a proof of the existence of simply connected closed manifolds which admit no locally smooth involutions. The principal example is the 4-manifold \(E_ 8\). (This is the closed simply connected 4-manifold with intersection form \(E_ 8\) constructed by \textit{M. H. Freedman} [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)].) The nonexistence of an involution depends on the fact that the Kirby- Siebenmann obstruction to smoothability is nontrivial for \(E_ 8\). In fact the authors' proof shows that any closed 4-manifold admitting a locally smooth involution must have vanishing Kirby-Siebenmann obstruction. The authors construct an infinite number of examples by considering the connected sum of \(E_ 8\) with copies of \(S^ 2\times S^ 2\). The arguments work as well if \(E_ 8\) is replaced with the Chern manifold, the nonsmoothable 4-manifold which is homotopy equivalent to \({\mathbb{C}}P^ 2\). The existence of this manifold is also due to Freedman.
simply connected closed manifolds which admit no locally smooth involutions
simply connected 4-manifold with intersection form \(E_ 8\)
Kirby-Siebenmann obstruction to smoothability
scientific article; zbMATH DE number 4096402