Smooth structures on nonorientable four-manifolds and free involutions

In this paper, we investigate existence of inequivalent smooth structures on closed smooth nonorientable 4-manifolds building upon results of Akbulut, Cappell–Shaneson, Fintushel–Stern, Gompf and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the orientation 2-covers yields existence results of orientation-reversing exotic free involutions.