By John Harer, Arnold Kas, Robion Kirby
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Additional info for Handlebody Decompositions of Complex Surfaces
The resulting surface has singular fibers q 1 r of type III at each of the a's, of type I* at the bfs and of type III* at each of the regular if c*s. If s = p + 2q + 3r, then the fiber over s = 0 mod 4, is of type III*, I*, III for °° is s = 1, 2, 3 mod 4. 42 HARER, KAS, KIRBY Thus we can get any configuration involving singular fibers of types III* and where v I* with the condition III, v(III) + 2v(I*) + 3v(III*) E 0 mod 4, is the number of singular fibers of a given type. 14. gp(t) = 0 . 15. = S.
If s = p + 2q + 3r, then the fiber over s = 0 mod 4, is of type III*, I*, III for °° is s = 1, 2, 3 mod 4. 42 HARER, KAS, KIRBY Thus we can get any configuration involving singular fibers of types III* and where v I* with the condition III, v(III) + 2v(I*) + 3v(III*) E 0 mod 4, is the number of singular fibers of a given type. 14. gp(t) = 0 . 15. = S. mod 6 . The necessity of the conditions in Examples (1) and (2) above comes from considering the Euler characteristic surface 0 E(S) Notice that the Euler characteristic of of the elliptic is the sum of the « -l S, E(S) = I E(TT (t)) Euler characteristics of the singular fibers of S t where E(TT (t)) is zero if TT (t) is a regular fiber.
The first four types are described sepa- rately and consist of curves with multiplicity one (recall that the sume of the curves in an exceptional fiber, counted with multiplicity, is homologous to a nonsingular fiber). Regular neighborhoods of fibers of * type are plumb2 ing manifolds obtained by plumbing disk bundles over S with Euler number -2 according to the graphs; the multiplicities are indicated. Type I : The fiber here is a rational curve with one double point; £ 2 topologically an S with two points identified.