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The basic result of Oka theory, due to Gromov, states that every continuousmap $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformedto a holomorphic map. It is natural to ask whether this can be done for all $f$at once, in a way that depends continuously on $f$ and leaves $f$ fixed if itis holomorphic to begin with. In other words, is $\scrO(S,X)$ a deformationretract of $\scrC(S,X)$? We prove that it is if $S$ has a strictlyplurisubharmonic Morse exhaustion with finitely many critical points; inparticular, if $S$ is affine algebraic. The only property of $X$ used in theproof is the parametric Oka property with approximation with respect to finitepolyhedra, so our theorem holds under the weaker assumption that $X$ is an Okamanifold. Our main tool, apart from Oka theory itself, is the theory ofabsolute neighbourhood retracts. We also make use of the mixed model structureon the category of topological spaces.

Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds