  We prove that every Stein manifold X of dimension n admits [(n+1)/2]
holomorphic functions with pointwise independent differentials, and this number
is maximal for every n. In particular, X admits a holomorphic function without
critical points; this extends a result of Gunning and Narasimhan from 1967 who
constructed such functions on open Riemann surfaces. Furthermore, every
surjective complex vector bundle map from the tangent bundle TX onto the
trivial bundle of rank q < n=dim X is homotopic to the differential of a
holomorphic submersion of X to C^q. It follows that every complex subbundle E
in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the
tangent bundle of a holomorphic foliation of X. If X is parallelizable, it
admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any
dimension; the question whether such X also admits a submersion (=immersion) in
C^n remains open. Our proof involves a blend of techniques (holomorphic
automorphisms of Euclidean spaces, solvability of the di-bar equation with
uniform estimates, Thom's jet transversality theorem, Gromov's convex
integration method). A result of possible independent interest is a lemma on
compositional splitting of biholomorphic mappings close to the identity
(Theorem 4.1).
