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We show that any element of the special linear group SL2(R) is a product of two exponentials if the ring R is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to SL2(C) is not surjective. Our result extends to the linear group GL2(R), where it is sharp as well. [ABSTRACT FROM AUTHOR]

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle E with a structure group bundle $${{\mathscr {G}}}$$ on a reduced Stein space X, such that the fibre of E is a homogeneous space of the fibre of $${{\mathscr {G}}}$$ , with the complexification $$K^{{\mathbb {C}}}$$ of a compact real Lie group K acting on X, $${{\mathscr {G}}}$$ , and E. Our main result is that the inclusion of the space of $$K^{{\mathbb {C}}} \hbox {-equivariant}$$ holomorphic sections of E over X into the space of $$K\hbox {-equivariant}$$ continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch's classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper. [ABSTRACT FROM AUTHOR]