  Global actions were introduced by Bak in order to have a homotopy theory in a
purely algebraic setting. In this paper we apply his techniques in a particular
case: the (single domain) unimodular row global action. More precisely, we
compute the the path connected component and fundamental group for the
unimodular row global action. An explicit computation of the fundamental group
of the (connected component of) unimodular row global action is closely related
to stability questions in K-theory. This will be shown by constructing an exact
sequence with the fundamental group functor as the middle term and having
surjective stability for the functor K_2 on the left and injective stability
for the functor K_1 on the right.
