Cohomology theory on discrete sets and digraphs

Abstract

The classical differential calculus of a higher order is provided by the graded algebra of differential forms on a smooth manifold with the operation of exterior differentiation. Our interest to this notion was motivated by a desire to develop a calculus of differential forms on discrete objects. As a rule, the role of discrete objects is played by discrete sets with additional relations as in digraphs or graphs. The methods of study are those of classical algebraic topology. The first basic results in this direction were obtained in the paper Grigor'yan A. A., Lin Y., Muranov Yu. V., Yau Sh.-T. Cohomology of digraphs and (undirected) graphs. Asian Journal of Mathematics, 2015, Vo. 19. No. 5, 887–932