Orthogonal group sections via light bulb systems

For any oriented, smooth $4$–manifold $X$ diffeomorphic to $(S^2\times S^2)^{\# n}\# D^4$, this paper shows that the natural group homomorphism $\mathrm{Mod}(X,\partial X)\to \mathrm{O}(H_2(X;\mathbb{Z}))$ admits a section. In fact, every light bulb system in $X$ gives rise to a section, which can be characterized uniquely in terms of implanted barbell twists and implanted crossbell involutions. Moreover, all sections arising from different light bulb systems become identical after a single $S^2\times S^2$–stabilization of $X$.