Chen B
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
Proc Natl Acad Sci U S A. 1998 Aug 4;95(16):9093-8. doi: 10.1073/pnas.95.16.9093.
We extend the concept of manifold with boundary to weight and boundary weight functions. With the new concept, we obtained the double reciprocity laws for simplicial complexes, cubical complexes, and lattice polyhedra with weight functions. For a polyhedral manifold with boundary, if the weight function has the constant value 1, then the boundary weight function has the constant value 1 on the boundary and 0 elsewhere. In particular, for a lattice polyhedral manifold with boundary, our double reciprocity law with a special parameter reduces to the functional equation of Macdonald; for a lattice polytope especially, the double reciprocity law with a special parameter reduces to the reciprocity law of Ehrhart. Several volume formulas for lattice polyhedra are obtained from the properties of the double reciprocity law. Moreover, the idea of weight and boundary weight leads to a new homology that is not homotopy invariant, but only homeomorphic invariant.
我们将带边界流形的概念扩展到权重和边界权重函数。借助这一新概念，我们得到了具有权重函数的单纯复形、立方体复形和格多面体的双互反律。对于带边界的多面体流形，如果权重函数取值恒为1，那么边界权重函数在边界上取值恒为1，在其他地方取值为0。特别地，对于带边界的格多面体流形，我们带有特殊参数的双互反律简化为麦克唐纳的函数方程；对于格多胞形而言，带有特殊参数的双互反律简化为埃尔哈特互反律。从双互反律的性质中得到了几个格多面体的体积公式。此外，权重和边界权重的概念引出了一种新的同调，它不是同伦不变的，而只是同胚不变 的。
