We present existence results for certain singular 2–dimensional foliations on 4–manifolds. The singularities can be chosen to be simple, for example the same as those that appear in Lefschetz pencils. There is a wealth of such creatures on most 4–manifolds, and they are rather flexible: in many cases, one can prescribe surfaces to be transverse or be leaves of these foliations.
The purpose of this paper is to offer objects, hoping for a future theory to be developed on them. For example, foliations that are taut might offer genus bounds for embedded surfaces (Kronheimer’s conjecture).
"Existence of foliations on 4–manifolds." Algebr. Geom. Topol. 3 (2) 1225 - 1256, 2003. https://doi.org/10.2140/agt.2003.3.1225