Algebra and Topology Seminar

4pm Tuesday 2 September 2003

Finnur Larusson
University of Western Ontario

Complex geometry and abstract homotopy theory.

An abstract version of homotopy theory, based on the concept of a model structure on a category, was introduced by Quillen in the late 1960s. There has been much activity in recent years in abstract homotopy theory and its applications to various fields, perhaps the most notable example being Voevodsky's homotopy theory of schemes. I've been working on introducing and applying abstract homotopy theory in complex analysis and geometry. I embed the category of complex manifolds into model categories where new notions of holomorphic maps being fibrations or cofibrations emerge. These model categories provide a framework in which one can use the concepts and tools of homotopy theory to study continuous or holomorphic deformations of holomorphic maps, as well as lifting and extension properties of holomorphic maps, such as those given by Gromov's Oka Principle. The Oka Principle is a vague but fundamental maxim of analytic geometry with a long history and supported by many results, saying that on a Stein manifold (complex submanifold of affine space), analytic problems of a cohomological (or even homotopical) nature have only topological obstructions. I'll present results that shed new light on the Oka Principle and describe some novel questions that this project has raised on both the analytic and homotopic sides.




Return to list of seminars