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Montesinos Amilibia, José María
(1977)
*Heegaard diagrams for closed 4-manifolds.*
In
Geometric topology.
Academic Press, New York, pp. 219-237.
ISBN 0-12-158860-2

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## 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.

Item Type: | Book Section |
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Additional Information: | Proceedings of the Georgia Topology Conference held in Athens, Ga., August 1–12, 1977. |

Uncontrolled Keywords: | Topological manifolds |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 22037 |

Deposited On: | 21 Jun 2013 15:55 |

Last Modified: | 09 Sep 2020 08:17 |

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