Smooth structures on nonorientable 4-manifolds via twisting operations

Five observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere (Formula presented.) inside (Formula presented.) such that performing a Gluck twist on (Formula presented.) produces a manifold (Formula presented.) that is homeomorphic but not diffeomorphic to the total space of the nontrivial 2-sphere bundle over the real projective plane (Formula presented.). The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold (Formula presented.) and a mapping torus that was used by Cappell–Shaneson to construct an exotic (Formula presented.). This construction of (Formula presented.) is similar to the one of the Cappell–Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside (Formula presented.) produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of (Formula presented.). The fourth observation records new examples of pairs of homeomorphic but not diffeomorphic closed 4-manifolds with Euler characteristic one. These include the total space of the nontrivial (Formula presented.) -bundle over (Formula presented.). Knotting phenomena of 2-spheres in nonorientable 4-manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fifth observation.