By John Harer, Arnold Kas, Robion Kirby
Read or Download Handlebody Decompositions of Complex Surfaces PDF
Similar algebraic geometry books
Introduction to modern number theory : fundamental problems, ideas and theories
This version has been known as ‘startlingly up-to-date’, and during this corrected moment printing you'll be certain that it’s much more contemporaneous. It surveys from a unified viewpoint either the trendy country and the tendencies of continuous improvement in numerous branches of quantity concept. Illuminated by way of basic difficulties, the important rules of contemporary theories are laid naked.
From the reports of the 1st printing of this publication, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My common impact is of a very great ebook, with a well-balanced bibliography, instructed! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors supply the following an up to the moment consultant to the subject and its major functions, together with a few new effects.
An introduction to ergodic theory
This article presents an advent to ergodic concept appropriate for readers understanding uncomplicated degree concept. The mathematical must haves are summarized in bankruptcy zero. it really is was hoping the reader could be able to take on study papers after examining the publication. the 1st a part of the textual content is worried with measure-preserving modifications of chance areas; recurrence homes, blending houses, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy thought are mentioned.
- Sheaves on manifolds
- Gesammelte Abhandlungen. Zahlentheorie
- Current Trends in Arithmetical Algebraic Geometry (Contemporary Mathematics)
- Homology of Local Rings
Additional info for Handlebody Decompositions of Complex Surfaces
Example text
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.