  We present the matching condition for the strong coupling contant alpha_s at
a heavy quark threshold to four loops in the modified minimal subtraction
scheme. Our results lead to further decrease of the theoretical uncertainty of
the evolution of the strong coupling constant through heavy quark thresholds.
Using a low energy theorem we furthermore derive the effective coupling of the
Higgs boson to gluons (induced by a virtual heavy quark) in four- and
(partially) through five-loop approximation.
