The Euler characteristic of orbit configuration space of moment-angle complex
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摘要: 设Im为m维标准方体, K'为单纯复形K的重心重分. 将K'上的锥形按一定规则逐片线性嵌入Im的典范单纯剖分中, 从而得到K对应的一类方体复形cc(K). 根据cc(K)的构造过程, 计算了cc(K)的f-向量, 即各个维数的胞腔个数. 通过投射(Dd)mIm的拉回, 可定义cc(K)上的moment-angle复形Z K,d. 将Z K,d放入轨道构型空间的框架中, 得到轨道构型空间FG(Z K,d,n). 由FG(Z K,d,n)的组合结构和著名的Inclusion-exclsion原理, 给出了轨道构型空间FG(Z K,d,n)的欧拉示性数利用f-向量表示的计算公式, 并且提供了一种计算Z K,d欧拉示性数的新方法.
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关键词:
- 轨道构型空间 /
- moment-angle复形 /
- 单纯复形 /
- 欧拉示性数
Abstract: Let Im be the m-dimensional standard cube and K the barycentric subdivision of simplicial complex K. There is a PL (piecewise linear) embedding of the cone over K to the canonical simplicial subdivision of Im by some rules. Then we obtain a kind of cubical complex cc(K) associated to K. According to the construction of cc(K), we calculate the f-vector of cc(K), i.e., the number of cells in every dimension. There is a definition of moment-angle complexZ K,d over cc(K) by the pullback of the projection(Dd)mIm. PuttingZ K,d into the framework of orbit configuration spaces, we get the orbit configuration spaceFG(Z K,d,n). By using the famous Inclusion-exclusion Principle and the combinatorial structure ofFG(Z K,d,n), we obtain the formula for the Euler characteristic of orbit configuration spaceFG(Z K,d,n) in terms of f-vector. In addition, we provided a new method of calculating the Euler characteristic of moment-angle complex Z K,d. -
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