  We prove that the moduli space of solutions to the PU(2) monopole equations
is a smooth manifold of the expected dimension for simple, generic parameters
such as (and including) the Riemannian metric on the given four-manifold. In a
previous article, dg-ga/9710032, we proved transversality using an extension of
the holonomy-perturbation methods of Donaldson, Floer, and Taubes, together
with the existence of an Uhlenbeck compactification for the perturbed moduli
space. However, it remained an important and interesting question to see
whether there were simpler, more intrinsic alternatives to the holonomy
perturbations and this is the issue we settle here. The idea that PU(2)
monopoles might lead to a proof of Witten's conjecture (hep-th/9411102,
hep-th/9709193) concerning the relation between the two types of four-manifold
invariants was first proposed by Pidstrigach and Tyurin in 1994
(dg-ga/9507004): the space of PU(2) monopoles contains the moduli space of
anti-self-dual connections together with copies of the various Seiberg-Witten
moduli spaces, these forming singular loci in the higher-dimensional space of
PU(2) monopoles. Results in this direction, due to the author and Leness, are
surveyed in dg-ga/9709022, with a detailed account appearing in dg-ga/9712005.
Our transversality theorem ensures that the anti-self-dual and Seiberg-Witten
loci are the only singularities and that the PU(2) monopole moduli space forms
a smooth - though non-compact, because of bubbling - cobordism between the
links of the singularities.
