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 $S$ inside $R{P}^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is homeomorphic but not diffeomorphic to the total space of the nontrivial 2-sphere bundle over the real projective plane $S\left(2\gamma \oplus R\right)$. The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold $Y$ and a mapping torus that was used by Cappell–Shaneson to construct an exotic $R{P}^4$. This construction of $Y$ is similar to the one of the Cappell–Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside $Y$ produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of $R{P}^4$. 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 $R{P}^2$-bundle over $R{P}^2$. 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.