We present the matching condition for the strong coupling constant
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.
