Affine simplices in Oka manifolds

Abstract

We show that the homotopy type of a complex manifold $X$ satisfying the Oka property is captured by holomorphic maps from the affine spaces ${\mathbb{C}}^n,n\ge 0$, into $X$. Among such $X$ are all complex Lie groups and their homogeneous spaces. We present generalisations of this result, one of which states that the homotopy type of the space of continuous maps from any smooth manifold to $X$ is given by a simplicial set whose simplices are holomorphic maps into $X$.