Algebraic and Polytopic Formulation to Cohomology

Abstract

The polytopic definition introduced recently describing the topology of manifolds is used to formulate a generating function pertinent to its topological properties. In particular, a polynomial in terms of one variable and a tori underlying this polynomial may be defined that generates an individual cohomological count. This includes the de Rham complex for example, as well as various index theorems by definition such as homotopy. The degree of the polynomials depends on the volume used to define the region parameterizing the manifolds; its potentially complex form and L-series is not presented in this work. However, the polynomials and the relevant torii uniformize the topological properties in various dimensions; in various dimensions this is interesting in view of known topologies.