Topology of Lie groups and homogeneous spaces

在1998年发表的头部研究员论文中，同源空间的整体同源性结构E y D26 y D2/T y D11 y D1·SU（6），其中与不可还原对称性空间EII=E y D26 y D2/S y D13 y D1·SU（6）有关，使用数学软件数学来确定。此外，在他1999年出版的论文中，复杂预测空间CP的圆字符同源性之间的关系D1 n YD 1，特别单元组SU（n+1）被投门。I described explicitly the Churn character homomorphism of the oriented Grassmannian SO（2 m +4）/（SO（3）×SO（2 m +1））， which is the irreducible symmetric space BDI， for m=1， 2， 3。在一个紧凑型定向流形M的整体同源性环中，以下两个问题引起：1）对于每个发生器x μ hiD 1 i yiD 1（M ; Z），那么x yiD 1 n yiD 1 =0的最小整数是什么？2） For each product xイD21イイD1kイイD21イエD2イイD1 ... xイイD2イイD1 of some generators xイイD2 that has Degree equal to the Dimension of M，what is the integer n such that x y y D21 y D1k y D21 y D2 y D1 ... x y D2l y D2 y D1 = m[M]？这里[M]拒绝M. I的基本同源学类为标志M =G/T解决了这些问题，其中G是一个紧凑的经典群体，而复杂的Grassmannian M =U（n+2）/（U（2）×U（n））。从现在开始，我想研究从符号群Sp（∞， R）构造的结构，M（2n， R），在特定的，模块化组SL（n， Z）〔SL（2， R）=Sp（1，R）与Sp（∞， R），与可变理论结合在一起。

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.