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

Source: Topology seminar

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∗).