Summary. In this paper we give a classification theorem of genus two Heegaard splittings of Seifert fibered manifolds over S z with three exceptional fibers, except for when two of the exceptional fibers have the same invariants with opposite orientation.
w 1. IntroductionLet M be a closed connected 3-manifold. A Heegaard splitting of genus g for M is a decomposition M=HlWH2, Htc~H2=OH2=SH2 where Hi, i=1, 2, is a handlebody of genus g. Two such decompositions M = H I • H2 t = H 2 w H2 2 will be called homeomorphic (isotopic) (1) M is a Brieskorn manifold: show that any genus two Heegaard splitting of a Seifert fibered space over S 2 with three exceptional fibers is isotopic to a "vertical" one, except in the two special cases mentioned in (iii) above. They also give sufficient conditions for two such vertical Heegaard splittings to be isotopic. In order to obtain the classification theorem we must distinguish between vertical Heegaard splittings of these manifolds. This is done in Theorem 4.1 which is a consequence of Theorem 3.1 and 3.3. The idea is to consider Nielsen equivalence classes of systems of generators of the fundamental group determined by the different Heegaard splittings. The fundamental group of such a manifold has a representation onto a triangle group T(p, q, r) of isometries of the hyperbolic plane H 2. In this representation the commutator of the generators is represented by a hyperbolic isometry. If the two Heegaard splittings are isotopic, the two systems of generators must be Nielsen equivalent. If, however, the Heegaard splittings are homeomorphic, the two systems of generators must be Nielsen equivalent only after an automorphism of T(p, q, r) that is induced by an automorphism of nl(M). In both cases the induced isometries must be conjugate in T(p, q, r). This is shown not to be the case by distinguishing their translation lengths. This is done in Theorems 3.1 and 3.3. In the case when ~1, ct2, ct3 are not pairwise relatively prime the proof uses in a crucial way Lemma 3.2 which was proved by Larry Washington. Lemma 3.2 is of number theoretic interest, as it distinguishes between basic cyclotomic units in certain cyclotomic fields. It is of interest that the question of equivalence of Heegaard splittings of a 3-manifold is connected to the question of Nielsen equivalence of corresponding generators for the fundamental group, and in this particular case reduces to the question: When are two quotients of basic cyclotomic units of the form sin an sin c n P q equal ? sin b n' sin dn P q
if there is a homeomorphism h: M ~ M (isotopy h: M xI~M) such that h(H])=H 2 or H 2 (ttl(H~)=H 2 or H22In the case where the manifold M has fibers with invariants (ct, fl), (ct, -fl), (%, f13) where f13 $ + 1 mod ct 3, the corresponding generators of two of the vertical Heegaard splittings are, after an automorphism of T(p, q, r), in fact Nielsen equivalent by Theorem 4.5. Hence we cannot distinguish between them by the above techniques. Note that the fiber preserving homeomorphism, exchanging fibers with ...
