The subject of this paper is the homotopy principle, also called the h-principle or the Oka-Grauert principle, concerning sections of certain holomorphic fiber bundles on Stein manifolds. We give a proof of a theorem of Gromov (1989) from sec. 2.9 in [Gro]; see theorems 1. 3 and 1.4 below. This result, which extends the work of H. Grauert from 1957 ([Gr3], [Gr4], [Car]), has been used in the proofs of the embedding theorem for Stein manifolds into Euclidean spaces of minimal dimension [EGr], [Sch].1.1 Definition. Let h: Z → X be a holomorphic mapping of complex manifolds. A section of h is any map f : X → Z such that h• f is the identity on X. We say that sections of h satisfy the h-principle (or the Oka-Grauert principle) if each continuous section f 0 : X → Z can be deformed to a holomorphic section f 1 : X → Z through a continuous one parameter family (a homotopy) of continuous sections f t : X → Z (0 ≤ t ≤ 1), and any two holomorphic sections f 0 , f 1 : X → Z which are homotopic through continuous sections are also homotopic through holomorphic sections. If this holds for a trivial bundle Z = X × F → X, we say that maps X → F satisfy the h-principle.1.2 Definition. (Gromov [Gro]) A (dominating) spray on a complex manifold F is a holomorphic vector bundle p: E → F , together with a holomorphic map s: E → F , such that s is the identity on the zero section F ⊂ E, and for each x ∈ F the derivative Ds(x) maps E x (which is a linear subspace of T x E) surjectively onto T x F .The following result can be found in sec. 2.9 of [Gro].1.3 Theorem. If F is a complex manifold which admits a spray, then the sections of any locally trivial holomorphic fiber bundle with fiber F over any Stein manifold satisfy the h-principle. In particular, mappings from Stein manifolds into F satisfy the h-principle.Stronger results are given in theorem 1.4 and corollary 1.5 below. In the sequel [FP] to this paper we give a proof of Gromov's Main Theorem ([Gro], sect. 4.5) to the effect that the h-principle holds for sections of holomorphic submersions h: Z → X, where X is Stein and each point x ∈ X has a neighborhood U ⊂ X such that Z|U = h −1 (U ) admits a fiber-spray (see def. 3.1 below).For non-specialists we recall that a complex manifold is called Stein (after Karl Stein, 1951 [Ste]) if it has 'plenty' of global holomorphic functions. For the precise definition

We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromov's result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201

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Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and a discrete sequence b_j in C^m where m > [3n/2], there exists a proper holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,.... This is the interpolation version of the embedding theorem due to Eliashberg, Gromov and Schurmann. The dimension m cannot be lowered in general due to an example of Forster

Abstract. This paper presents the proof of the relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces using conic neighbourhoods of holomorphic sections over 1-convex spaces. A proof of a version of Cartan's Theorem A for 1-convex spaces is also given.