Heegaard diagrams for closed 4-manifolds

Abstract

Let W4=H0∪λH1∪μH2∪γH3∪H4 be a handle decomposition of a closed, orientable PL 4-manifold. Let M4=H0∪λH1∪μH2 and let N4=N4(γ)=γH3∪H4=γ#(S1×B3). Then W4 is M4∪N4, identified along ∂M4=∂N4=γ#(S1×S2). The first observation in this paper is that W4 does not depend upon the method of attaching N4, as a consequence of a theorem of F. Laudenbach and V. Poénaru [Bull. Soc. Math. France 100 (1972), 337–344;], who showed (implicitly) that the homotopy group of ∂N4 is generated by maps which extend to N4. Dually, W4 does not depend upon the method of attaching H0∪λH1≅N4(λ). Hence W4 depends only on the cobordism C(λ,γ) from λ#(S1×S2) to γ#(S1×S2) defined by the 2-handles. The author calls (W4,C(λ,γ)) a Heegaard splitting of W4. The associated Heegaard diagram is a pair (λ#S1×S2,w) where w is a framed link in λ#S1×S2. It is noted that an arbitrary pair (λ#S1×S2,w) need not be a Heegaard diagram for a 4-manifold. Two diagrams are equivalent if there is a homeomorphism of pairs which preserves the framings. Moves are given which relate any two Heegaard diagrams for the same 4-manifold. The completeness of these moves is proved in Theorem 3 (and also Theorem 3′). A concept of a dual diagram is introduced. It is not known whether each Heegaard diagram is geometrically realizable as the diagram for some closed 4-manifold.