I would like to read more about this version of simplicial homology and cohomology. Any reference or explanation is welcome. I hope this description is correct. Triangulate a smooth manifold and assign an orientation to each simplex in the triangulation. Define the boundary operator with respect to this orientation. This defines the simplicial chain complex. for the cochain complex use exactly the same simplices but define the coboundary operator as the transpose of the boundary operator. It maps k chains to K + 1 chains while the boundary operator maps k chains to k-1 chains. Thanks in advance.