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.