Discrete Differential Calculus on Simplicial Complexes and Constrained Homology*

Let V be a finite set. Let K be a simplicial complex with its vertices in V .In this paper, the author discusses some differential calculus on V . He constructs some constrained homology groups of K by using the differential calculus on V . Moreover, he defines an independence hypergraph to be the complement of a simplicial complex in the complete hypergraph on V . Let L be an independence hypergraph with its vertices in V .He constructs some constrained cohomology groups of L by using the differential calculus on V .