Closed $4$--Manifolds Foliated by Hyperplanes

Abstract

Let $M^4$ be a closed, orientable $4$--manifold carrying a transversely oriented $C^2$ codimension--one foliation whose leaves are diffeomorphic to ${\mathbb{R}}^3$. We prove that $M^4$ is homeomorphic to the $4$--torus ${\mathbb{T}}^4$. We also show that, whenever the original smooth structure on $M$ admits a smooth defining $1$--form, the conclusion sharpens to a diffeomorphism $M\cong {\mathbb{T}}^4$.