实Grassmann流形的上同调与KP流
Cohomology of real Grassmann manifold and KP flow
摘要：
我们考虑一个实现的真正的格拉斯曼流形Gr（k，n）的基础上定义的KP方程的相应的（奇异）的解决方案的一个特定的流。然后，我们表明，KP流可以提供一个明确的和简单的结构的关联图的积分上同调的Gr（k，n）。结果表明，Gr（k，n）中存在两类图，一类是平凡系数图，另一类是扭曲系数图，它们通过Poincare-Lefschetz对偶对应于Gr（k，n）的可定向和不可定向情形的同调群.我们还导出了Gr（k，n）的Poincare多项式的一个显式公式，并证明了Poincare多项式也与有限域$\F_q$上Gr（k，n）的一个适当版本上的点数有关，其中q是素数的幂.特别地，我们发现Gr（k，n）上的$\F_q$点的个数可以通过计算KP流上奇点的个数来计算.
英文摘要：
We consider a realization of the real Grassmann manifold Gr(k,n) based on a particular flow defined by the corresponding (singular) solution of the KP equation. Then we show that the KP flow can provide an explicit and simple construction of the incidence graph for the integral cohomology of Gr(k,n). It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they correspond to the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincare-Lefschetz duality. We also derive an explicit formula of the Poincare polynomial for Gr(k,n) and show that the Poincare polynomial is also related to the number of points on a suitable version of Gr(k,n) over a finite field $\F_q$ with q being a power of a prime. In particular, we find that the number of $\F_q$ points on Gr(k,n) can be computed by counting the number of singularities along the KP flow.
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