The logarithms of Dehn twists

被引:21
作者
Kawazumi, Nariya [1 ]
Kuno, Yusuke [2 ]
机构
[1] Univ Tokyo, Dept Math Sci, Meguro Ku, 3-8-1 Komaba, Tokyo 538914, Japan
[2] Tsuda Coll, Dept Math, Kodaira, Tokyo 1878577, Japan
关键词
mapping class group; Dehn twists; Goldman Lie algebra; Johnson homomorphisms;
D O I
10.4171/QT/54
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as "the logarithms" of Dehn twists. The formula generalizes the classical formula describing the action on the first homology of the surface, and Morita's explicit computations of the extended first and the second Johnson homomorphisms. For the proof we use a homological interpretation of the Goldman Lie algebra in the framework of Kontsevich's formal symplectic geometry. As an application, we prove that the action of the Dehn twist of a simple closed curve on the k-th nilpotent quotient of the fundamental group of the surface depends only on the conjugacy class of the curve in the k-th quotient.
引用
收藏
页码:347 / 423
页数:77
相关论文
共 26 条
[1]   Canonical extensions of the Johnson homomorphisms to the Torelli groupoid [J].
Bene, Alex James ;
Kawazumi, Nariya ;
Penner, R. C. .
ADVANCES IN MATHEMATICS, 2009, 221 (02) :627-659
[2]  
BOURBAKI N, 1972, GROUPES ALGEBRES LIE, pCH2
[3]  
BROWN KS, 1994, GRADUATE TEXTS MATH, V87
[4]  
Cartan H., 1956, HOMOLOGICAL ALGEBRA
[5]  
Chas M., 1999, PREPRINT
[6]   EXTENSION OF CINFINITY FUNCTION ALGEBRA BY INTEGRALS AND MALCEV COMPLETION OF PI1 [J].
CHEN, KT .
ADVANCES IN MATHEMATICS, 1977, 23 (02) :181-210
[7]   Extending Johnson's and Morita's homomorphisms to the mapping class group [J].
Day, Matthew B. .
ALGEBRAIC AND GEOMETRIC TOPOLOGY, 2007, 7 :1297-1326
[8]   EXTENSIONS OF JOHNSON'S AND MORITA'S HOMOMORPHISMS THAT MAP TO FINITELY GENERATED ABELIAN GROUPS [J].
Day, Matthew B. .
JOURNAL OF TOPOLOGY AND ANALYSIS, 2013, 5 (01) :57-85
[10]   Infinitesimal presentations of the Torelli groups [J].
Hain, R .
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 1997, 10 (03) :597-651