Barbell twists are natural

Abstract

For any oriented smooth $4$--manifold $X$ diffeomorphic to $(S^2\times D^2{)}^{♮n}$ ($n\ge 0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times {\wedge }^2{H}_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney--Gabai barbell twist $\phi \in \mathrm{Mod}(\mathcal{N},\partial \mathcal{N})$ 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 $\mathrm{GL}(H_2(\mathcal{N};\mathbb{Z}))\cong \mathrm{GL}(2,\mathbb{Z})$. Any barbell spine of $\mathcal{N}$ gives rise to an implanted barbell twist equal to $\phi$ or ${\phi }^{-1}$ in $\mathrm{Mod}(\mathcal{N},\partial \mathcal{N})$, according to the sign of the homological basis orientation.

arXiv comment30 pages; a false assertion (Lemma 4.5 in Version 2) removed; Section 4 (Extendability for boundary diffeomorphisms of a 1-handlebody) added, replacing Section 4 (Embedding uniqueness for several 1-handles) and Section 5 (Embedding uniqueness for a 1-handlebody) in Version 2; Proof of Lemma 6.2 revised