Barbell twists are natural

For any oriented smooth $4$–manifold $X$ diffeomorphic to $(S^2\times D^2{)}^{\mathrm{♮}n}$ ($n\ge 0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{M}\mathrm{o}\mathrm{d}(X,\mathrm{\partial }X)\cong \mathrm{M}\mathrm{o}\mathrm{d}(D^4,\mathrm{\partial }{D}^4)\times {\wedge }^2{H}_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney–Gabai barbell twist $\phi \in \mathrm{M}\mathrm{o}\mathrm{d}(\mathcal{N},\mathrm{\partial }\mathcal{N})$ is identified with a generator of the factor subgroup ${\wedge }^2{H}_2(\mathcal{N};\mathbb{Z})\cong \mathbb{Z}$. Up to boundary-fixing diffeotopy, the barbell spines of $\mathcal{N}$ are completely classified by the bases of $H_2(\mathcal{N};\mathbb{Z})\cong {\mathbb{Z}}^2$, forming a homogeneous set modeled on the group $\mathrm{G}\mathrm{L}(H_2(\mathcal{N};\mathbb{Z}))\cong \mathrm{G}\mathrm{L}(2,\mathbb{Z})$. Any barbell spine of $\mathcal{N}$ gives rise to an implanted barbell twist equal to $\phi$ or ${\phi }^{-1}$ in $\mathrm{M}\mathrm{o}\mathrm{d}(\mathcal{N},\mathrm{\partial }\mathcal{N})$, according to the sign of the homological basis orientation.