Generic regularity and orientability for the sign-fixed self-dual moduli problem

Assume full AC. Let $X$ be a closed simply connected oriented Riemannian four-manifold with positive-definite intersection form, and let $P\to X$ be an $SU(2)$-bundle with $c_2(P)[X]=-1$. Complete the connections in $H^3$ and the gauge transformations in $H^4$, and use the unbased irreducible configuration quotient. The curvature section $\mathrm{\Phi }([A])=F_A^{-}$ takes values in its $H^2$ equation bundle. Choose an open subset $U$ covered by finitely many bounded Coulomb slice balls centered at smooth irreducible connections, and an open $V$ whose closure is contained in the union of strictly smaller closed balls in these charts, their images lying in $U$.

Allow smooth gauge-equivariant perturbation sections supported in $U$, whose slice representatives are smooth $H^3$-valued maps with finite uniform $H^3$ norm on the specified chart balls. Interpret them in the $H^2$ equation bundle by inclusion. Give this class the topology of the uniform $H^3$ section norm on those finitely many balls together with convergence of the section and its slice derivative $H^3\to H^2$ uniformly on compact subsets. The perturbations for which $\mathrm{\Phi }+\sigma$ is transverse to zero at every zero in $\bar{V}$ form an open dense subset. Their zero sets in $V$ are smooth orientable five-manifolds, and the real determinant line of the gauge-fixed deformation family is orientable on the irreducible quotient. The based quotient may be used to pull back this orientation; it carries an additional $SO(3)$ framing fiber and is not the unbased five-dimensional moduli space.

Donaldson, An Application of Gauge Theory to the Topology of 4-Manifolds, §§II.1–II.4

The complete source proof on Donaldson's printed pp.288–295 was read, including the $H^3\mathrm{/}{H}^4$ exponents, the perturbations valued in $H^3$ with equation target $H^2$, Lemma 9 and Lemma 10. This replaces a defective previous formalization, not the mathematical theorem by an easier special case. A nonempty open subset of an infinite-dimensional Hilbert manifold cannot have compact closure: in a contained coordinate ball, an orthonormal sequence at a fixed small radius has no convergent subsequence. Thus the previous phrase “relatively compact open set” did not express the finite bounded slice collars used in the source. Uniform control of the smoothing section value is the compactness mechanism for openness; arbitrary compact derivatives with only compact-open control do not supply that mechanism. The previous based-quotient wording also omitted its three framing directions. The three actual suppliers prove the corrected analytic operation and the original orientability conclusion, with full AC stated explicitly.