The previous sections introduced combinatorial concepts for CW-complex representations and abstract classification mechanisms to manage scientific data. Compared to the theory of CW-complexes, this section focuses more on the details of algebraic properties of linear mappings by a procedure to associate a sequence of Abelian groups or modules with, e.g., a topological space. Only a few concepts are introduced in this section [55,22], because of the fact that computational topology [56,19] is a complex and emerging part of scientific computing in its own right.
The motivation for this section is to retain the structure of geometrical objects and even physical field approximations for computational mechanisms, because the recovery of lost structural information of objects has proven to be a very complex and difficult tasks.
To use the elements of a dimension of a cell complex, e.g., all edges, in a computational manner, a mapping of the -cells onto an algebraic structure is needed. An algebraic representation of the assembly of cells with a given orientation is thereby made available. Whereas the cell topology is concerned about the internal structure of a given cell, the chain concept acts on certain -cells. A formal definition for a -chain ( chain group) is given by:
Thus, two -chains can be added, or a -chain can be multiplied by a scalar. In addition, -chains support algebraic-topological operations, including the boundary and coboundary operations. Based on these concepts, a cell complex can be seen as a formal structure where cells can be added, subtracted, and multiplied. The cell complexes used in this work have a chain group in every dimension. Homological concepts are applied here for the first time, due to the fact that homology examines the connectivity between two immediate dimensions. A structure-relating map between sets of chains is therefore introduced1.2.
This can be seen as a boundary operator, that maps -chains onto the -chains in their boundary. It should not be confused with the geometric boundary of a point set. This algebraic-topological operation defines a -chain in terms of a -chain. It is compatible with the additive and the external multiplicative structure of chains and builds a linear transformation:
Therefore, the boundary operator can be used linearly
which means that the boundary operator can be used separately for each cell. The cell complex properties can be easily calculated by means of chains. 3-cells intersect on 2-cells or have an empty intersection. This operation can be described by the boundary operator of the cell complex and the corresponding orientation induced on them:
Figure 1.8 depicts two examples of 1-chains, 2-chains, and an example of the boundary operator. Applying the appropriate boundary operator to the 2-chain example read:
Using the boundary operator on a sequence of chains of different dimensions, a chain complex is obtained:
Before the introduction of the concept of chains only the simple structure of a cell complex was available. The cell complex only contains the set of cells and their connectivity. The introduction of the chain concept provides the concept of an assembly of cells and the corresponding algebraic structure. Chains can be seen as mappings from oriented cells as part of a cell complex to another space. This definition establishes the algebraic access of computational methods to handle the concept of a cell complex.
In addition to cell complexes, scientific computing requires the notation and access mechanisms to global quantities related to macroscopic -dimensional space-time domains, introduced in Section 1.2. This collection of possible quantities, which can be measured, can then be called a field, which permits the modeling of these measurements as a field function that can be integrated on arbitrary -dimensional (sub)domains. An important fact which has to be stated here is that all quantities which can be measured are always attached to a finite region of space. A field function can then be seen as the abstracted process of measurement of this quantity [31,35]. The concept of cochains allows the association of numbers not only to single cells, as chains do, but also to assemblies of cells. Briefly, the necessary requirements are that this mapping is not only orientation-dependent, but also linear with respect to the assembly of cells, modeled by chains. A cochain representation is now the global quantity association with subdomains of a cell complex, which can be arbitrarily built to discretize a domain. Physical fields therefore manifest on a linear assembly of cells. Based on cochains, topological laws can be given a discrete representation.
The space of all linear mappings on is denoted by , where the elements of are called cochains. Cochains express a representation for fields over a discretized domain . Addition and multiplication by a scalar are defined for the field functions and so for cochains. To extend the expression possibilities, coboundaries of cochains are introduced.
Then, the following sequence with is generated:
The concepts of chains can also be used to characterize properties of spaces, the homology and cohomology where it is only necessary to use . The algebraic structure of chains is an important concept, e.g., to detect a -dimensional hole that is not the boundary of a -chain, which is called a -cycle.
Figure 1.10 depicts the homology of the three-dimensional chain complex with the respective images and kernels, where the chain complex of is defined by . As can be seen, the boundary operator expression yields
The first homology group  is the set of closed 1-chains (curves) in a space, modulo the closed 1-chains which are also boundaries. In the remainder of this work the ring will be or , in which case the modules are vector spaces or Abelian groups, respectively, e.g., for the chain complex is given by or for the coefficient group the following cochain complex .
To give an example, the first homology group is the set of closed -chains (curves) modulo the closed -chains which are also boundaries. This group is denoted , where are cycles or closed 1-chains and are -boundaries. Another example is given in Figure 1.11.
The concepts of chains and cochains coincide on finite complexes . Geometrically, however, and are distinct  despite an isomorphism. An element of is a formal sum of -cells, where an element of is a linear function that maps elements of into a field. Chains are dimensionless multiplicities, whereas those associated with cochains are physical quantities . The extension of cochains from single cell weights to quantities associated with assemblies of cells is not trivial and makes cochains very different from chains, even on finite cell complexes. Nevertheless, there is an important duality between -chains and -cochains.
For a chain and a cochain , the integral of over is denoted by , and integration can be regarded as a mapping, where represents the corresponding dimension:
Integration in the context of cochains is a linear operation: given and , reads
Reversing the orientation of a chain means that integrals over that chain acquire the opposite sign