Brett Chenoweth, Deformation retractions from spaces of holomorphic maps onto spaces of continuous maps

Brett Chenoweth: Deformation retractions from spaces ofholomorphic maps onto paces of continuous maps

Abstract: A fundamental property of an Oka manifold Y is that every ontinuous map from a Stein manifold X to Y can be deformed to a holomorphic map. In a recent paper, Lárusson considers the natural question of whether it is possible to simultaneously deform all continuous maps f from X to Y to holomorphic maps, in a way that depends continuously on f and does not change f if f is holomorphic to begin with. In other words, is O(X, Y) a deformation retract of C (X, Y)? Lárusson provided a partial answer to this question. In this talk, we study this question primarily in the context of domains in C. The main tools we use come from complex analysis, Oka theory, algebraic topology and the theory of absolute neighbourhood retracts.

We further develop the work of Lárusson on the topological
relationship between spaces of continuous maps and spaces of holomorphic maps from Stein manifolds to Oka manifolds, mainly in the context of domains in C. The main tools we use come from complex analysis, Oka theory, algebraic topology and the theory of absolute neighbourhood retracts. One of our main results provides a large supply of infinitely connected domains X in C such that O(X, C∗) is a deformation retract of C (X, C∗).