Mapping cylinders and the Oka principle

Abstract

We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka is about fibrancy in suitable model structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three dierent model structures and use these factorizations to prove implications between ostensibly dierent Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a Oka saying that certain notions of cofibrancy for manifolds are equivalent to being Stein. Introduction. In this paper, we apply concepts and tools from abstract homotopy theory to complex analysis and geometry, based on the foundational work in (L2), continuing our development of the idea that the Oka is about fibrancy in suitable model structures. A mapping cylinder in a model category is an object through which a given map can be factored as a cofibration followed by an acyclic fibration (or sometimes merely an acyclic map). We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three dierent model structures. We apply these factorizations to the Oka Principle, mainly to prove implications between ostensibly dierent Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. We then move beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds (this involves the same sort of factorization through a mapping cylinder as before). Using the so-called Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are