Exterior Calculus on (Abstract) Simplicial Complexes and Generalized Homology.

Let $V$ be a finite set and let $K$ be a simplicial complex with its vertices in $V$. In this paper, we discuss the differential algebra as well as the co-differential algebra of $V$. We use the elements in the differential algebra to construct generalized homology of $K$ and use the elements in the co-differential algebra to construct generalized cohomology of $K$. We prove that if the element in the differential algebra and the element in the co-differential algebra are adjoint, then the generalized homology of $K$ and the generalized cohomology of $K$ are isomorphic.