2002 Fiscal Year Final Research Report Summary
Research on higher dimensional polyhedra and operads
| Project/Area Number |
13640080
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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 |
HEMMI Yutaka Faculty of Science, Professor, 理学部, 教授 (70181477)
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| Co-Investigator(Kenkyū-buntansha) |
MORISUGI Kaoru Wakayama University, Faculty of Education, Professor, 教育学部, 教授 (00031807)
KOMATSU Kazushi Faculty of Science, Associate Professor, 理学部, 助教授 (00253336)
SHIMOMURA Katsumi Faculty of Science, Professor, 理学部, 教授 (30206247)
KAWAMOTO Yusuke National Defense Academy in Japan, School of Liberal Arts and General Education, Research Associate, 総合教育学群, 助手
TSUKIYAMA Kouzou Shimane University, Faculty of Education, Professor, 教育学部, 教授 (20093651)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Hopf space / H-map / A_n-space / A_n-map / projective space / associahedron / homotopy group / Whitehead product |
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| Research Abstract |
The summary of reserch results is as follows.
1. We study conditions for mod p finite H-spaces to be quasi p-regular, and we give a generalization to the theorem of Kumpel on p-regulraity. Our result includes results by Harper, McClearly and Wilkerson.
2. We define unstable p-th order mod p cohomology operations for odd primes p. Then, by using the operations we show that there are no H-spaces with certain cohomology.
3. Williams gave a definition of higher order homotopy commutativity of the multiplication on associative H-spaces. We generalize his definition to higher homotopy assocative H-spaces. Then, by using this definition we give the mod p torus theorem for H-spaces with finitely generated cohomology.
4. We show that there is no even degree generators except for degree eight and twenty in the mod 3 cohomology of finite H-spaces. We also determine the cohomology strucre of them as algebras.
5. We consider the inbedding of an An-space X to its loop space of the projective n-1-space ΩP_<n-1>X. Then we study relations between an A_<n-1>-cohomology class x∈H^*(X;F) and its lift x^^^∈H^*(ΩP_<n-1>X;F), where F is a field.
6. It is well known that if a map f: X → Y between H-spaces is an H-map then the map induces a map between their projective planes. Moreover, it is also known that if Y is homotopy associative then the converse holds. We show that the assumption of homotopy associativity is necessary in showing the converse. We also generalize the result to the case of A_n-spaces.
7. It is proved that the moduli space of a punctured Rieman sphere has cellular decomposition. We consider a decomposition of the associahedron by using a homotopy theoretical property, and then by using the decomposition we construct a handle decomposition of the moduli space which is a dual decomposition of the original decomposition.
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