Study of the mapping class group using the intersections of curves on surfaces

2017 Fiscal Year Final Research Report

• Project/Area Number: 26800044
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tsuda University
• Principal Investigator: Kuno Yusuke 津田塾大学, 学芸学部, 准教授 (80632760)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 写像類群 / ゴールドマン括弧積 / テュラエフ余括弧積 / ジョンソン準同型

Outline of Final Research Achievements

The main object of this research project was the Turaev cobracket which is a topological operation measuring the self-intersections of curves on surfaces. In particular, we studied the relationship between the Kashiwara-Vergne problem in Lie theory and the Turaev cobracket. As a result, we established the formality property of the Turaev cobracket and obtained its application to the theory of the Johnson homomorphisms.
We also studied generalized Dehn twists, which is a certain algebraic construction associated to any closed curves on surfaces. In particular, we tried to clarify a geometric interpretation of generalized Dehn twists and partially succeeded in the sense that we obtained a first approximation of them in terms of homology cylinders.

Free Research Field

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