Smooth structures on nonorientable 4‐manifolds via twisting operations

Abstract Five observations compose the main results of this note. The first records the existence of a smoothly embedded 2‐sphere inside such that performing a Gluck twist on produces a manifold that is homeomorphic but not diffeomorphic to the total space of the nontrivial 2‐sphere bundle over the real projective plane . The second observation is that there is a 5‐dimensional cobordism with a single 2‐handle between the 4‐manifold and a mapping torus that was used by Cappell–Shaneson to construct an exotic . This construction of is similar to the one of the Cappell–Shaneson homotopy 4‐spheres. The third observation is that twisting an embedded real projective plane inside produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of . 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 ‐bundle over . 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.