ABSTRACT
This chapter discusses the concept of cohomology to the de Rham cohomology using exterior calculus on smooth manifolds, then introduce the important Hodge decomposition theorem and the cohomology group of harmonic differential forms. The simplicial homology of a simplicial complex is the difference between the closed chains and the exact chains. The simplicial cohomology is the difference between the closed forms and the exact forms. A smooth map between manifolds can pull back a differential form defined on the target to a differential form on the source. A polyhedral surface has a flat Riemannian metric with cone singularities. The Voronoi diagram and the Delaunay triangulation of the vertices can be directly defined on the surface. For surfaces with boundaries, the above computational methods need to be modified in the neighborhoods of the boundaries.
