Abstract: Consider H^*(U_n/T_n, Z), the integral cohomology ring of homogeneous space U_n/T_n. It is a quotient ring of the polynomial ring in n variables. Particularly, the homogeneous part with degree d=dimU_n/T_n corresponds to the highest dimensional cohomology group Z, i.e., each degree d homogeneous polynomial f corresponds to an integer χ(f). The present article computes this correspondence explicitly. The connections with the intersection theory of the manifold U_n/T_n are also discussed. Similar result applies to the homogeneous space Sp_n/T_n.

Key words: homogeneous space, intersection theory, intersection number

摘要： 考虑齐性空间U_n/T_n的整系数上同调环H^*(U_n/T_n , Z)。它是n元多项式环的一个商环。特别地，商环的d=dimU_n/T_n次齐次部分对应于顶维上同调群Z，即每个d次齐次多项式 f都对应于一个整数 χ(f)。本文具体计算出了此对应。在几何上，这一对应给出了到U_n/T_n的连续映射的相交数的计算方法。同样的方法也适用于Sp_n/T_n上的相交理论的研究。

关键词: 齐性空间, 相交理论, 相交数