Weight functions, double reciprocity laws, and volume formulas for lattice polyhedra.

Abstract

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.