  We prove the existence of perturbations for the PU(2) monopole equations,
yielding transversality on the complement of the anti-self-dual or reducible
solutions, and the existence of an Uhlenbeck compactification for the moduli
space of solutions to these perturbed PU(2) monopole equations. In December
1994, V. Pidstrigach and A. Tyurin and then others proposed a method to prove
Witten's conjecture concerning the relation between the Donaldson and
Seiberg-Witten invariants of smooth four-manifolds. Their proposal uses a
moduli space of solutions to the PU(2) monopole equations, which are a natural
generalization of the U(1) monopole equations of Seiberg and Witten and the
equation for anti-self-dual SO(3) connections, to construct a cobordism between
links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the
moduli space of anti-self-dual connections, which appear as singularities in
this larger moduli space. A basic requirement of this cobordism technique is
the existence of an Uhlenbeck compactification for the moduli space of PU(2)
monopoles and of generic-parameter transversality results for all the moduli
spaces of PU(2) monopoles which appear in this compactification, on the
complement of the anti-self-dual and U(1) solutions.

Differential Geometry;Algebraic Geometry;High Energy Physics - Theory;Algebraic Geometry;Differential Geometry