Dual 2-complexes in 4-manifolds

This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index ≤ 2 (“Heegard” decompositions). In dimensions ≥ 5 results of Smale (trivial π1) and Wall (general π1) 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 may change. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) dual spines of a Heegard decomposition of the 4-sphere if and only if they satisfy the conclusions of the Alexander-Lefshetz duality theorem (H1K ≃ HL and H2K ≃ HL). (3.3) If (N, ∂N) is 1-connected then there is a “pseudo” handle decomposition without 1handles, in the sense that there is a pseudo collar (M, ∂N) (a relative 2-handlebody with spine that 2-deforms to ∂N) and N is obtained from this by attaching handles of index ≥ 2.