Singular Homology of Hypergestures

In this chapter we interpret the basic cubic chain spaces of singular homology in terms of hypergestures in a topological space over a series of copies of the arrow digraph ò. This interpretation allows for a generalized homological setup. The generalization is (1) to topological categories instead of topological spaces, and (2) to any sequence of digraph pΓnqnPZ instead of the constant series of Ò. We then define the corresponding chain complexes, and prove the core boundary operator equation B2 “ 0, enabling the associated homology modules over a commutative ring R. We discuss some geometric examples and a musical one, interpreting contrapuntal rules in terms of singular homology.