Foliations transverse to fibers of a bundle

Abstract

Consider a fiber bundle where the base space and total space are compact, connected, oriented smooth manifolds and the projection map is smooth. It is shown that if the fiber is null-homologous in the total space, then the existence of a foliation of the total space which is transverse to each fiber and such that each leaf has the same dimension as the base implies that the fundamental group of the base space has exponential growth. Introduction. Let (E, p, B) be a locally trivial fiber bundle where /»:£-*-/? is the projection, E and B are compact, connected, oriented smooth manifolds and p is a smooth map. (By smooth we mean C for some r^l and, henceforth, all maps are assumed smooth.) Let b denote the dimension of B and k denote the dimension of the fiber (thus, dim E= b+k). By a section of the fibration we mean a smooth map a:B-+E such that p o a=idB. It is well known that if a section exists then the fiber over any point in B represents a nontrivial element in Hk(E; Z) since the image of a section is a compact orientable manifold which has intersection number one with the fiber over any point in B. The notion of a section may be generalized as follows. Definition. A polysection of (E,p,B) is a covering projection tr:B-*B together with a map C:B-*E such that the following diagram commutes. Note that if the covering projection is a diffeomorphism then the map f o 7T_1 is a section in the usual sense. One important situation in which polysections arise is the following. Suppose that F is a smooth foliation of E such that each leaf of F is Received by the editors April 4, 1973 and, in revised form, May 1, 1973. AMS (MOS) subject classifications (1970). Primary 57D30; Secondary 55F10.