Differentiable classification of 4-manifolds with singular Riemannian foliations

TL;DR

This paper classifies certain 4-manifolds with singular Riemannian foliations, showing they are diffeomorphic to standard elliptic manifolds or their connected sums, and identifies specific foliations of codimension 1.

Contribution

It provides a classification of closed simply connected 4-manifolds with singular Riemannian foliations, revealing their diffeomorphism types and enumerating non-homogeneous foliations.

Findings

Any such 4-manifold with a disk bundle decomposition is diffeomorphic to standard elliptic 4-manifolds.

Manifolds with nontrivial singular Riemannian foliations are connected sums of standard manifolds.

Exactly 3 non-homogeneous singular Riemannian foliations of codimension 1 exist on these manifolds.

Abstract

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard $\mathbb{S}^4$, $\pm\mathbb{CP}^2$ and $\mathbb{S}^2\times\mathbb{S}^2$. A classification of singular Riemannian foliations of codimension 1 on all closed simply connected 4-manifolds is obtained as a byproduct. In particular, there are exactly 3 non-homogeneous singular Riemannian foliations of codimension 1, complementing the list of cohomogeneity one 4-manifolds.