A strong Oka principle for embeddings of some planar domains into CxC*

2010/11/18 by Tyson Ritter, Ritter, Tyson
Mathematics · #32E10 #32H02 #32H35 #32M17 #32M25 #32Q28 (Secondary) #32Q40 (Primary) #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32E10 #msc:32H02 #msc:32H35 #msc:32M17 #msc:32M25 #msc:32Q28 #msc:32Q40

paper · pdf · doi:10.48550/arxiv.1011.4116

25 pages

arxiv created 2010/11/18 · arxiv updated 2010/11/19

Abstract

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into CxC*, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalisation to CxC* of recent results of Wold and Forstneric on the long-standing problem of properly embedding open Riemann surfaces into C², with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Larusson's holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.

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