Monodromy calculus in low dimensional topology and "racks".
Project/Area Number  10440015 
Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  The University of Tokyo 
Principal Investigator 
MASUMOTO Yuki Grad. Sch. of Math. Sci. U. Tokyo, Prof., 大学院・数理科学研究科, 教授 (20011637)

CoInvestigator(Kenkyūbuntansha) 
KAMADA Seiichi Department of Math. Osaka City Univ. Lecturer, 理学部, 講師 (60254380)
KAWAUCHI Akio Department of Math. Osaka City Univ. Prof., 理学部, 教授 (00112524)
FUKUHARA Shinji Department of Math. Tsuda Coll, Prof., 学芸学部, 教授 (20011687)
OHTSUKI Tomotada Dept. Math. & Comp. Sci., Tokyo Inst. Tech., Assoc. Prof., 情報理工学研究科, 助教授 (50223871)

Project Period (FY) 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥4,500,000 (Direct Cost : ¥4,500,000)
Fiscal Year 1999 : ¥2,500,000 (Direct Cost : ¥2,500,000)
Fiscal Year 1998 : ¥2,000,000 (Direct Cost : ¥2,000,000)

Keywords  racks / Lefschetz fibration / monodromy / braid group 
Research Abstract 
We have obtained a structure theorem of the rack of cords based on n + 1 points on the 2shere. This theorem can be interpreted as a conjugation formula in the braid group of n + 1 strings of the 2shere We have also succeeded in determining the structure of the center of the associated group of the rack. A joint paper by S. Kamada and Y. Matsumoto describing these results has been completed. The relationship between the above results and the research of the monodromies of Lefschetz fibratoins has now become quite clear. Namely, if we consider hyperelliptic Lefschetz fibrations whose vanishing cycles are all nonseparating simple closed curves on a general fiber, its monodromy takes the value in the rack of cords based on n + I points on the 2sphere XィイD2n+1ィエD2(SィイD12ィエD1). Our new idea is to consider the "associated semigroup" of the rack XィイD2n+1ィエD2(SィイD12ィエD1), which is one step before the u5ual associated group. Then it is shown that elements of the center of the associated semi
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group are in one to one correspondence with the isomorphism classes of the Lefschetz fibrations over the 2sphere satisfying the above conditions. Thus our next objective is to determine the structure of the center of this semigroup. From our viepoint, Chakiris's thesis submitted to Columbia University about 20 years ago determined the generator of the center of the above semigroup in the holomorphic category. If we can synthesize our viepoint and his results, it might give us a new approach to the SiebertTian conjecture that a CィイD1∞ィエD1Lefschetz fibrtion satisfying the above conditions has a holomorphic structure. We had two symposiums on racks supported by this grantinaid. One is "Symposium on low dimendional topology and racks" held on September 2lst22nd, 1998, and the other is "Symposium on low dimensional topology and racks (II)" on October 2lst23rd, 1999 Both were at the Graduate School of Mathematical Sciences, the University of Tokyo, and were successful, each gathering more than 50 participants. Less

Report
(3results)
Research Output
(18results)