| Co-Investigator(Kenkyū-buntansha) |
ISHIHARA Kazuo Faculty of Science, Professor, 理学部, 教授 (90090563)
HANADA Noboru Faculty of Science, Professor, 理学部, 教授 (90033844)
WATANABE Yutaka Faculty of Science, Professor, 理学部, 教授 (60028131)
IRIYE Kouyemon Faculty of Science, Associate Professor, 理学部, 助教授 (40151691)
O'UCHI Moto Faculty of Science, Professor, 理学部, 教授 (70127885)
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| Research Abstract |
In the paper of the head investigator published in 1998, the ring structure of the integral cohomology of a homogeneous space EィイD26ィエD2/TィイD11ィエD1・SU(6), which is related with the irreducible symmetric space EII=EィイD26ィエD2/SィイD13ィエD1・SU(6), was determined by using a mathematics software Mathematical. Also, in his paper published in 1999, the relationship between the Churn character homomorphism of complex projective space CPィイD1nィエD1 and that of the special unitary group SU(n+1) was invest-gated. I described explicitly the Churn character homomorphism of the oriented Grassmannian SO(2m+4)/(SO(3)×SO(2m+1)), which is the irreducible symmetric space BDI, for m=1, 2, 3. In the integral cohomology ring of a compact orientable manifold M, the following two problems arise :
1) For each generator x∈HィイD1iィエD1(M ; Z), what is the smallest integer n such that xィイD1nィエD1=0?
2) For each product xィイD21ィエD2ィイD1kィイD21ィエD2ィエD1 … xィイD2lィエD2ィイD1kィイD2lィエD2ィエD1 of some generators xィイD2iィエD2 that has degree equal to the dimension of M, what is the integer n such that xィイD21ィエD2ィイD1kィイD21ィエD2ィエD1 … xィイD2lィエD2ィイD1kィイD2lィエD2ィエD1 = m[M]? Here [M] denote the fundamental cohomology class of M.
I solved these problems for a flag manifold M =G/T where G is a compact classical group and for the complex Grassmannian M =U(n+2)/(U(2)×U(n)).
From now on, I would like to study the topology of the inductive limit Sp(∞, R) constructed from the symplectic group Sp(n, R)⊂M(2n, R), in particular, the action of the modular group SL(n, Z)⊂SL(2, R)=Sp(1,R) on Sp(∞, R), together with invariant theory.
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