多面体与多面体复形上的角和
Angle sums on polytopes and polytopal complexes
摘要：
我们将研究列在$α$ -向量中的多面体的角度和，努力利用每个维度中的面的f-向量与角度和的α-向量之间的类比。在$α$ -向量上的格拉姆关系和珀勒斯关系类似于在f-向量上的欧拉关系和德恩-萨默维尔关系。首先我们描述了某些多面体类的α-向量和$α$ -f-向量所张成的空间。家庭的多面体构造的角度和跨越空间的多面体定义的革兰氏和珀尔斯方程。这表明单形的角和空间的仿射跨度的维数是floor[（d-1）/2]，而单纯多面体和一般多面体的组合角和和面数的维数分别是d-1和2d-3。接下来我们考虑多面体复形的角和。我们类似于欧拉示性数定义α向量上的角特征。我们表明，在这两个对应的变化，并在某些奇维多面体复形的情况下，角特征是欧拉示性数的一半。最后，我们考虑球形和双曲多面体和多面体复形。已知格拉姆关系的球形和双曲类似物以及珀尔斯关系的球形类似物，并且我们在许多情况下展示了珀尔斯关系的双曲类似物。对于维数大于3的单形证明这个关系将完成这个结果的证明。此外，我们还展示了球面和双曲多面体的结构如何导致角特征和欧拉示性数的相应变化。
英文摘要：
We will study the angle sums of polytopes, listed in the $α$-vector, working to exploit the analogy between the f-vector of faces in each dimension and the alpha-vector of angle sums. The Gram and Perles relations on the $α$-vector are analogous to the Euler and Dehn-Sommerville relations on the f-vector. First we describe the spaces spanned by the the alpha-vector and the $α$-f-vectors of certain classes of polytopes. Families of polytopes are constructed whose angle sums span the spaces of polytopes defined by the Gram and Perles equations. This shows that the dimension of the affine span of the space of angle sums of simplices is floor[(d-1)/2], and that of the combined angle sums and face numbers of simplicial polytopes and general polytopes are d-1 and 2d-3, respectively. Next we consider angle sums of polytopal complexes. We define the angle characteristic on the alpha-vector in analogy to the Euler characteristic. We show that the changes in the two correspond and that, in the case of certain odd-dimensional polytopal complexes, the angle characteristic is half the Euler characteristic. Finally, we consider spherical and hyperbolic polytopes and polytopal complexes. Spherical and hyperbolic analogs of the Gram relation and a spherical analog of the Perles relation are known, and we show the hyperbolic analog of the Perles relations in a number of cases. Proving this relation for simplices of dimension greater than 3 would finish the proof of this result. Also, we show how constructions on spherical and hyperbolic polytopes lead to corresponding changes in the angle characteristic and Euler characteristic.
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