Moment-angle复形轨道构型空间的欧拉示性数

The Euler characteristic of orbit configuration space of moment-angle complex

设I^m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I^m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(D^d)m→J^m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F_G(Z_(K,d,n)).由F_G(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.

Let I^m 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 I^m 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 complex Zg,d over cc（K） by the pullback of the projection （Dd）TM→ Im. Putting Zg,d into the framework of orbit configuration spaces, we get the orbit configuration space FG（ZK,d, n）. By using the famous Inclusion-exclusion Principle and the combinatorial structure of FG（K,d,n）, we obtain the formula for the Euler characteristic of orbit configuration space FG（ZK,d,n） in terms of f-vector. In addition,we provided a new method of calculating the Euler characteristic of moment-angle complex K,d.