2006 Fiscal Year Final Research Report Summary
Topological toric theory and combinatorics
| Project/Area Number |
17540092
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka City University |
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Principal Investigator |
MASUDA Mikiya Osaka City University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (00143371)
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| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Yoshitake Osaka City University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (20271182)
HIBI Takayuki Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (80181113)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Toric manifold / fan / convex polytope / equivariant cohomology / Bott tower / GKM graph / topology / combinatorics |
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| Research Abstract |
I have beein developing the theory of toric varieties in algebraic geometry from a topological point of view for these years. At the same time, a research from the same viewpoint has been independently developed in England and Russia, and now this emerging subject is called Toric Topology. In this research, I mainly studied the relation between topology and Combinatorics jointly with Taras Panov, Zhi Lu etc, and have made effort for developing this new subject. In particular, I organized an international conference on Toric Topology for a week from the end of May 2006 at Osaka City University. The organizers are Megumi Harad (Toronto), Yael Karshon(Toronto) and Taras Panov(Moscow) except me. This conference was supported mainly by 21COE program "Constitution of wide-angle mathematical basis focused on knots", and other resources are research grants of Shigenori-Matsumoto, Taras Panov, Yoshitake Hashimoto and this my research grant. Fortunately, we had 140 participants and half of them
… More
are foreigners. We realized that many people are interested in this emerging area.
As for my own research, I got the following outcome.
(1) Gullemin-Zara have studied the relation between the topology of manifolds with torus actions and graph theory. This is a very interesting study because it introduces ideas of topology to the study of graphs. Motivated by their research, we have introduced the notion of torus graph and showed that its equivariant cohomology agrees with the face ring of a simplicial poset which was already introduced in commutative algebra. The paper of this research has appeared in Adv. Math. 212 (2007), 458-483.
(2) It is known that equivariant cohomology contains a rich geometrical of a toric manifold and fits very well to the study of toric manifolds. In this research I have proved that equivariant cohomology completely distinguishes toric manifolds.
(3) A Bott tower is an iterated CP^1 bundle and provides a good family of toric manifolds. I worked with Taras Panov on the topological classification of those Bott towers (the paper is submitted). Also, I worked with Dong Youp Suh on the topological classification of higher Bott towers which are higher dimensional generalization of Bott towers. Less
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