Dual decompositions of 4-manifolds

This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index less than or equal to 2. In dimensions greater than or equal to 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 (H1K similar or equal to H-2 L and H2K similar or equal to H-1 L). (3.3) If (N, partial derivativeN) is 1-connected then there is a "pseudo" handle decomposition without 1-handles, in the sense that there is a pseudo collar (M, partial derivativeN) (a relative 2-handlebody with spine that 2-deforms to partial derivativeN) and N is obtained from this by attaching handles of index greater than or equal to 2.