2004 Fiscal Year Final Research Report Summary
Research on the higher homotopy commutativity and higher dimensional polytopes
| Project/Area Number |
15540083
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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 | Kochi University |
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Principal Investigator |
HAMMI Yutaka Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
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| Co-Investigator(Kenkyū-buntansha) |
SHIMOMURA Katsumi Kochi University, Faculty of Science, Professor, 理学部, 教授 (30206247)
KOMATSU Kazushi Kochi University, Faculty of Science, Associate Professor, 理学部, 助教授 (00253336)
MORISUGI Kaoru Wakayama University, Faculty of Education, Professor, 教育学部, 教授 (00031807)
TSUEIYAMA Kouzou Shimane University, Faculty of Education, Professor, 教育学部, 教授 (20093651)
KAWAMOTO Yusuke National Defense Academy in Japan, School of Liberal Arts and General Education, Lecturer, 総合教育学群, 講師
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| Project Period (FY) |
2003 – 2004
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| Keywords | H-space / higher homotopy commutativity / C_n-space / A_n-space / permutohedron / associahedron / permutoassociahedron / retraction |
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| Research Abstract |
The summary of research results is as follows.
1.We extend the definition of the higher homotopy commutativity of the multiplications given for associative H-spaces by Williams to the one for higher homotopy associative H-spaces. Then we gave the mod p torus theorem the H-spaces with finitely generated cohomology ring. The result appears in T.ran.Amer.Math.Soc.356.
2.We studied variety of higher homotopy commutativity related to the C_n-structure introduced by Williams. Then we considered polytopes representing the higher homotopy commutativity. In particular, we considered the subset of the n-th symmetric group consisting of the inverse of fixed shuffles, and showed that the polytope which represents the higher homotopy commutativity given by this subset is so-called shuffle polytope. Moreover, we showed that the decomposition used to show that a C_n-space of Williams is a quasi C_n-space by Hemmi is the same as the one by given by Kapranov and Voevodsky
3.We obtained a handle decomposition of a punctured Rieman sphere corresponding to the tessellation by Devados. The handle decomposition is given by extending the decomposition of associahedra by Boardman Vogt.
4.If a map between H-spaces is an H-map then it induces a map between projective spaces of the H-spaces. Converse also holds under the assumption that the target space of the map is homotopy associative. We showed that the assumption of the homotopy assocaativity of the target space is needed when we show the converse. We also extend this results for maps between A_n-spaces. The result appears in Hiroshima J Math 35.
5.We studied the action of the reduced power operations on the mod p cohomology of finite A_p-spaces with higher homotopy commutativity for an odd prime p. The result is to appear in Geometry and Topology Monographs.
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