Dual decompositions of 4-manifolds

Abstract:

This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index $\leq 2$. In dimensions $\geq 5$ results of Smale (trivial $\pi _{1}$) and Wall (general $\pi _{1}$) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ($H_{1}K\simeq H^{2}L$ and $H_{2}K\simeq H^{1}L$). (3.3) If $(N,\partial N)$ is 1-connected then there is a “pseudo” handle decomposition without 1-handles, in the sense that there is a pseudo collar $(M,\partial N)$ (a relative 2-handlebody with spine that 2-deforms to $\partial N$) and $N$ is obtained from this by attaching handles of index $\geq 2$.