Affine simplices in Oka manifolds

TL;DR

This paper demonstrates that the homotopy type of Oka manifolds can be characterized by holomorphic maps from affine spaces, extending to the space of continuous maps from smooth manifolds, with implications for complex Lie groups.

Contribution

It establishes a novel link between the homotopy type of Oka manifolds and holomorphic maps from affine spaces, generalizing previous results.

Findings

Homotopy type of Oka manifolds is captured by holomorphic maps from C^n.

The homotopy type of continuous maps from smooth manifolds is described by a simplicial set of holomorphic maps.

All complex Lie groups and their homogeneous spaces are examples of Oka manifolds.

Abstract

In this note, we show that the homotopy type of a complex manifold X satisfying the Oka property is captured by holomorphic maps from the affine spaces C^n, n\geq 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.