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We prove that the moduli space of solutions to the PU(2) monopole equationsis a smooth manifold of the expected dimension for simple, generic parameterssuch as (and including) the Riemannian metric on the given four-manifold. In aprevious article, dg-ga/9710032, we proved transversality using an extension ofthe holonomy-perturbation methods of Donaldson, Floer, and Taubes, togetherwith the existence of an Uhlenbeck compactification for the perturbed modulispace. However, it remained an important and interesting question to seewhether there were simpler, more intrinsic alternatives to the holonomyperturbations 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-manifoldinvariants was first proposed by Pidstrigach and Tyurin in 1994(dg-ga/9507004): the space of PU(2) monopoles contains the moduli space ofanti-self-dual connections together with copies of the various Seiberg-Wittenmoduli spaces, these forming singular loci in the higher-dimensional space ofPU(2) monopoles. Results in this direction, due to the author and Leness, aresurveyed in dg-ga/9709022, with a detailed account appearing in dg-ga/9712005.Our transversality theorem ensures that the anti-self-dual and Seiberg-Wittenloci are the only singularities and that the PU(2) monopole moduli space formsa smooth - though non-compact, because of bubbling - cobordism between thelinks of the singularities.Comment: Communications in Analysis and Geometry 8 (2000), to appear. 54  pages. References updated; published versio

Generic metrics, irreducible rank-one $\mathrm{PU}(2)$ monopoles, and transversality