One-bubble neighborhoods in SO(3)-monopole moduli spaces

We prove an oriented neighborhood theorem at the first Uhlenbeck level of \(SO(3)\)-monopole moduli spaces, over a compact regular Seiberg–Witten moduli space of arbitrary dimension whose solutions have nonzero spinor. A finite-dimensional obstruction model admits an equivariant compactified embedding that maps the obstruction zero set homeomorphically onto an Uhlenbeck-open neighborhood of the ideal stratum; its restriction to that zero set at positive scale is a diffeomorphism onto its image. The principal analytic step is a strong approximation theorem in critical norms for arbitrary convergent one-bubble monopole sequences. Uniform gauge estimates and summable decay along the neck recover the gluing parameters, including the relative frame, and give reverse uniqueness. A global equivariant stabilization treats the background family without requiring a constant-rank spectral cutoff. Comparison with Donaldson's instanton-addition map identifies the induced orientation with the standard monopole orientation. Under the additional topological hypotheses of Feehan and Leness, the construction supplies the analytic neighborhood input to their level-one link pairing. We state the resulting polynomial formula and its low-degree consequences for Witten's conjecture on abundant, effective manifolds of Seiberg–Witten simple type.