Existence of foliations on 4-manifolds.pdfVIP

ISSN 1472-2739 (on-line) 1472-2747 (printed) 1225 Algebraic Geometric Topology ATGVolume 3 (2003) 1225–1256Published: 13 December 2003 Existence of foliations on 4–manifolds Alexandru Scorpan Abstract 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). AMS Classification 57R30; 57N13, 32Q60 Keywords Foliation, four-manifold, almost-complex 1 Introduction Foliations play a very important role in the study of 3–manifolds, but almost none so far in the study of 4–manifolds. There are hints, though, that they should play an important role here as well. For example, for M4 = N3 × S1 , Kronheimer obtained genus bounds for embedded surfaces from certain taut foliations [11], which are sharper than the ones coming from Seiberg–Witten basic classes. He conjectured that such bounds might hold in general. 1.1 Summary (In this paper, all foliations will be 2–dimensional and oriented, all mani- folds will be 4–dimensional, closed and oriented; unless otherwise specified, of course.) For a foliation F to exist on a manifold M , the tangent bundle must split TM = TF ⊕ NF . Since in general that does not happen, one must allow for singularities of F . An important example is [6]: c? Geometry Topology Publications 1226 Alexandru Scorpan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .