| Co-Investigator(Kenkyū-buntansha) |
HEMMI Yutaka Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
OSHIMA Hideaki Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70047372)
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| Research Abstract |
1. It is important to know what nilpotency the group [X, X] has for a Hopf space X. However another important thing is to study the composition structure of the group [X, X]. Morisugi determined the relationship between the composition structure and the group structure of [X, X] for X = SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.
Let M^n be the mod 2 Moore square. For n 【greater than or equal】 3, it is known that there exists a lift <η_n>^^^- of the suspension of the Hopf map, η_n: S^<n+1> → S^n. We investigated the order of the Whitehead product [ <η_n>^^^-, <η_n>^^^-] in π_<2n+1>(M^n).
Let G be the simple Lie group of classical type. Oshima showed that the group [G, G] is non-commutative for almost all cases of G mentioned above. And for some cases of G, he determined the nilpotency class of [G, G].
Let be a Hopf complex. In this case, [X, X] is, so called, an algebraic loop, that is, it has a binary operation with both left and right inverse. Oshima also investigated how they differed from each other.
3. Hemmi showed that the possible even dimensional generators of mod 3 cohomology rings of finite Hopf spaces occurred only in dimension 8 or 20. And he almost determined the structure of such mod 3 cohomology rings. He also showed that under some conditions, there was no Hopf space X with H (X; Z/p) ≡∧(x, p^1x, 【O!R】【O!R】【O!R】p^<p-2>x)
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