2004 Fiscal Year Final Research Report Summary
Construction of the topological toric theory
| Project/Area Number |
15540090
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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, 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, School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20271182)
KATO Shin Osaka City University, School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10243354)
HIBI Takayuki Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80181113)
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| Project Period (FY) |
2003 – 2004
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| Keywords | toric manifold / fan / convex polytope / combinatorics / topology / face ring / equivariant cohomology / graph theory |
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| Research Abstract |
We have developed the theory of toric varieties from a topological point of view. This is not only reconstruction of the theory of toric varieties in terms of topology but also provides new objected such as multi-fan and multi-polytope etc.which makes this research more interesting. The topological object corresponding to toric manifold is a torus manifold. The purpose of our research was to study geometrical properties of torus manifolds.
(1) Torus manifolds which have vanishing odd degree cohomology behave well among torus manifolds. In fact, toric manifolds have such property. I have characterized those torus manifolds in terms of orbit space in a joint work with Panov. Motivated by this research, I have characterized the numbers of simplices of simplicial cell decompositions of spheres. This solves a conjecture by Stanley.
(2) The relation between the topology of torus manifolds and graph theory is studied by Guillemin-Zara. This introduces an idea of equivariant topology in the theory of graph and is quite interesting. We have proved that the equivariant cohomology ring of a torus graph with axial function agrees with the face ring of a simplicial poset.
(3) The notion of small cover is in some sense a real version of toric theory. This has a lot of similarity to toric theory but there are some essential differences For instance, a small cever has a non-trivial fundamental group and most of small cavers are non-onientable while every toric manifold is simply connected and orientable. I tried to classify small covers over cubes as well as toric manifold over cubes.
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