SO(3) monopoles, level-one Seiberg–Witten moduli spaces, and Wittenʹs conjecture in low degrees

We prove Wittenʹs formula relating the Donaldson and Seiberg–Witten series modulo powers of degree c+2, with c=−14(7χ+11σ), for four-manifolds obeying some mild conditions, where χ and σ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg–Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg–Witten moduli spaces lying in the first (or ‘one-bubble’) level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles.