arXiv math.GT digest — 09 Jun 2026 (1 paper)

Tuesday 9 June 2026 brought one new primary submission to math.GT. Cross-lists from math.DS (already primary above), math.GR, and math.AT are excluded per digest policy.

arXiv:2606.08005 | José M. Espinar, Harold Rosenberg

Abstract. Let be a closed, orientable $4$-manifold carrying a transversely oriented codimension-one foliation whose leaves are diffeomorphic to . We prove that is homeomorphic to the $4$-torus . We also show that, whenever the original smooth structure on $M$ admits a smooth defining $1$-form, the conclusion sharpens to a diffeomorphism .1

A closed orientable 4-manifold admitting a transversely oriented codimension-one foliation by -leaves is homeomorphic to . If the smooth structure admits a smooth defining 1-form, the conclusion upgrades to a diffeomorphism. In short: an -hyperplane foliation on a compact 4-manifold forces the global topology to be the 4-torus, with no other closed orientable 4-manifolds admitting such a foliation.1

The proof draws on the theory of codimension-one foliations and the structure of their leaf spaces. A central role is played by the transverse orientation and the regularity hypothesis, which together allow the authors to control holonomy and the global topology of the leaf space. The argument likely invokes Novikov's closed leaf theorem (or its higher-dimensional analogues) to rule out compact leaves or exotic holonomy, and combines this with constraints from the Euler characteristic and fundamental group to pin down the topology. The upgrade from homeomorphism to diffeomorphism under the existence of a smooth defining 1-form suggests an additional Reeb stability or foliation-theoretic argument, reducing the smooth classification to the topological one once a closed 1-form is available.1

The key insight is that a transversely oriented foliation by -hyperplanes has a contractible leaf space, forcing the fundamental group of to be abelian and the manifold to fiber (in a suitable sense) over a circle or torus. The -leaf geometry eliminates any compact or non-trivial holonomy, and iterating this reasoning in dimension 4 yields . Combined with the vanishing of higher homotopy groups forced by the leaf structure, this identifies with at the homotopy level; from there, the topological ($s$-cobordism or surgery) classification of 4-manifolds with this homotopy type completes the homeomorphism. The diffeomorphism conclusion under a smooth closed defining 1-form follows because a smooth 1-form gives the foliation a fibration structure, making the smooth and topological classifications coincide.1