Heegaard surfaces in Haken 3-manifolds
1990; American Mathematical Society; Volume: 23; Issue: 1 Linguagem: Inglês
10.1090/s0273-0979-1990-15910-5
ISSN1088-9485
Autores Tópico(s)Mathematics and Applications
ResumoThe purpose of this note is to announce, for the case of Haken 3-manifolds, a complete solution of Waldhausen's conjecture concerning finiteness for Heegaard surfaces.It is a classical result due to Moise [Moi] that all compact (orientable) 3-manifolds N have finite triangulations.Closely related to triangulations are Heegaard graphs, i.e., those finite, possibly disconnected, graphs r in N with TndN = dT and the property that (JV-UÇT))~~ is a handlebody (here £/(..) denotes the regular neighborhood in N).Indeed, the 1-skeleton of any triangulation of N is a Heegaard graph-but of course not vice versa.A surface F in N is a Heegaard surface, provided there is a Heegaard graph r in JV such that F is isotopic in N to dU(T\JdN)-dN.If F is a Heegaard surface, the pair (N, F) is called a Heegaard splitting for TV.Heegaard splittings are independent of triangulations, and yet closely connected to them.This suggests we should use them for a classification of 3-manifolds.In this context Reidemeister [Re] and Singer [Si] showed that the existence of a common subdivision for any two triangulations implies that Heegaard surfaces in 3-manifolds are stably isotopic, i.e., isotopic modulo finite connected sums with the standard torus in S .Based on this observation, Waldhausen [Wal] proved that any two Heegaard surfaces of S 3 are isotopic iff they have the same genus.Using different methods, this result has been extended to Heegaard surfaces of lens spaces [Bo, BO] and minimal (see below) Heegaard surfaces of the 3-torus [FH].However, in [BGM] explicit examples of irreducible 3-manifolds are constructed with at least two nonhomeomorphic Heegaard splittings.In fact, it is now clear [BRZ] that uniqueness of Heegaard surfaces fails drastically and is a rather special phenomenon for 3-manifolds.In
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