Singular Homology of Hypergestures (original) (raw)
Abstract
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 pΓ n q nPZ instead of the constant series of Ò. We then define the corresponding chain complexes, and prove the core boundary operator equation B 2 " 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.-Σ-63.1 An Introductory Example Let us give an introductory example of the Escher Theorem, which has musical relevance, before we embark on the homological theme. Let G be a topological group, X a topological space, and GˆX Ñ X a continuous group action. Denote by G X the topological category whose objects are the elements of X, and whose morphisms g : x Ñ y are the triples px, y, gq P X 2ˆG such that y " gx, the topology of G X being induced from the product topology on X 2ˆG. If the topologies are all indiscrete, a continuous curve F : ∇ Ñ G X is just a functor. A classical example for such a topological category G X from transformational music theory is the canonical action of the general affine group G " Ý Ñ GLpZ 12 q " T Z12¸Zˆo n the pitch class set X " Z 12 , together with the indiscrete topology. Recall that in Section 62.2.1, we have constructed special curves, so-called discrete gestures, OE pgq : ∇ Ñ G X for every morphism g : x Ñ y as follows:
Figures (2)
![Fig. 63.1. Singular cubes on the torus. A 0-cube is a point, a 1-cube is a continuous line map, a 2-cube is a continuous square surface map. And four 1-cubes circumscribing the torus hole. The present homological approach is deduced from the following observations. To begin with, sin. gular homology is based on continuous functions on standard objects, either n-dimensional simplexes 01 n-dimensional cubes [635]. It is well known that both, the simplicial and the cubical homology, yield the same homology groups. Our approach is based on cubic homology; see [999] for a good presentation of cubic homology. This one considers continuous functions s : J” — X on the n-dimensional cubes, n-fold cartesiar products 1” =I x Ix...I of the real unit interval J, with values in a topological space X. These functions are called singular n-cubes. Let us look at some singular cubes on the torus surface X = T?, see Figure 63.1. A singular 0-cube is a map og from the singleton J° to T?, a singular 1-cube is a continuous line may o,:1—T?, a 2-cube is a continuous square surface map 02 : I? > T?.](https://figures.academia-assets.com/90653294/figure_001.jpg)
](Fig. 63.1. Singular cubes on the torus. A 0-cube is a point, a 1-cube is a continuous line map, a 2-cube is a continuous square surface map. And four 1-cubes circumscribing the torus hole. The present homological approach is deduced from the following observations. To begin with, sin. gular homology is based on continuous functions on standard objects, either n-dimensional simplexes 01 n-dimensional cubes [635]. It is well known that both, the simplicial and the cubical homology, yield the same homology groups. Our approach is based on cubic homology; see [999] for a good presentation of cubic homology. This one considers continuous functions s : J” — X on the n-dimensional cubes, n-fold cartesiar products 1” =I x Ix...I of the real unit interval J, with values in a topological space X. These functions are called singular n-cubes. Let us look at some singular cubes on the torus surface X = T?, see Figure 63.1. A singular 0-cube is a map og from the singleton J° to T?, a singular 1-cube is a continuous line may o,:1—T?, a 2-cube is a continuous square surface map 02 : I? > T?.
Fig. 63.2. The two reductions of the skeleton digraph of a gesture. This situation is suitable in the simple case of one single arrow, but for general digraphs, we need a formula that takes care of all possible arrows (if any). The idea is this: i the digraph that results from (1) omitting the tail vertex ¢ and (2) taking I is to (1) select an arrow a € Ap in the arrow set Ap of I, then (2) take t to the digraph I'|a~ obtained from removing t, and all arrows connected arrows connected to this head. So, for our elementary situation, we have f we look at what happens in the above evaluation at head and tail, we recognize that the head value is the result of calculating the gesture or the digraph resulting from all that is left after removing the arrows that are connected to t. In other words, the general procedure for a digraph he tail ¢, of a, and then(3) restrict to tg. The analogue procedure fo1 the head of a would yield the restricted digraph I'|at, resulting from removing the head hg as well as al! la~ = {h}, t lat = {t}. This construction will work if it manages to exhaust all arrows. More precisely, we may define the fac of a gesture g € '@K ona discrete digraph I’ as the sum }),<y, g(v) of the values of g on the digraph’
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