Smooth structures on non-orientable $4$-manifolds via twisting operations

TL;DR

This paper constructs and analyzes non-orientable 4-manifolds using twisting operations, revealing exotic smooth structures, cobordisms, and knotting phenomena that contrast with orientable cases.

Contribution

It demonstrates the existence of exotic smooth structures on non-orientable 4-manifolds via twisting operations and explores their relationships with known constructions like Cappell-Shaneson spheres.

Findings

Existence of a sphere in $ P^2 imes S^2$ producing a manifold homeomorphic but not diffeomorphic to a known bundle.

Construction of a 5D cobordism linking the new manifold to a Cappell-Shaneson type manifold.

Twisting an embedded $ P^2$ yields manifolds homeomorphic but not diffeomorphic to a circle sum of $ P^4$.

Abstract

Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is homeomorphic but not diffeomorphic to the total space of the non-trivial 2-sphere bundle over the real projective plane $S(2\gamma \oplus \mathbb{R})$. 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 $\mathbb{R} P^4$. 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 $\mathbb{R}P^4$. Knotting phenomena of 2-spheres in non-orientable 4-manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fourth observation.