Foliations and compact leaves on 4-manifolds I. Realization and self-intersection of compact leaves

Abstract

We introduce an easily tractable cohomological criterion for the existence of 2-dimensional foliations with a prescribed compact leaf on a 4-manifold relying on standard methods, Milnor's inequality for the existence of a flat connection on an $\mathbb{R}^2$-bundle over a surface, and Thurston's $h$-principle. This is used to investigate the self-intersection numbers of compact leaves of foliations on the product of two surfaces, in particular the question whether these numbers are bounded on a given 4-manifold.