  This is the third installment in our series of articles (dg-ga/9712005,
dg-ga/9710032) on the application of the PU(2) monopole equations to prove
Witten's conjecture (hep-th/9411102) concerning the relation between the
Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The moduli
space of solutions to the PU(2) monopole equations provides a noncompact
cobordism between links of compact moduli spaces of U(1) monopoles of
Seiberg-Witten type and the moduli space of anti-self-dual SO(3) connections,
which appear as singularities in this larger moduli space. In this paper we
prove the first part of a general gluing theorem for PU(2) monopoles, with the
proof of the second half to appear in a companion article. The ultimate purpose
of the gluing theorem is to provide topological models for neighborhoods of
ideal Seiberg-Witten moduli spaces appearing in lower levels of the Uhlenbeck
compactification of the moduli space of PU(2) monopoles and thus permit
calculations of their contributions to Donaldson invariants using the
PU(2)-monopole cobordism.
