  In this article we evaluate mass corrections up to $O((m^2/q^2)^6)$ to the
three-loop polarization function induced by an axial-vector current. Special
emphasis is put on the evaluation of the singlet diagram which is absent in the
vector case. As a physical application $O(\alpha_s^2)$ corrections to the
production of top quarks at future $e^+e^-$ colliders is considered. It is
demonstrated that for center of mass energies $\sqrt{s} >~ 500$ GeV the
inclusion of the first seven terms into the cross section leads to a reliable
description.
