Topology of Lie groups and homogeneous spaces

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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.