Homology cobordism groups of surfaces and Lie algebras of associated graphs

2014 Fiscal Year Final Research Report

• Project/Area Number: 24740040
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: SAKASAI Takuya 東京大学, 数理(科)学研究科(研究院), 准教授 (60451902)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: グラフホモロジー / ホモロジー同境 / モジュライ空間 / 位相幾何学

Outline of Final Research Achievements

We studied the structure of the Lie algebra of graphs (of Lie types) associated with homology cobordism groups of surfaces.
In joint works with Shigeyuki Morita and Masaaki Suzuki, we investigated the chain complex of the Lie algebra to compute its Euler characteristic by using (super) computers. Our computational results show that there exist many non-trivial odd dimensional rational homology classes of the outer automorphism groups of free groups of ranks up to 11. Some results on Lie algebras of similar types are also obtained. We clarified the structure of the Lie algebra itself up to degree 6 by using symplectic representation theory.
As for the structure of homology cobordism groups, we extended the definition of infinitesimal trace homomorphisms in a joint work with G. Massuyeau for surface cases. We also studied higher dimensional generalizations.

Free Research Field

位相幾何学