This section combines the fiber bundle concepts and the computational topological concepts. The fiber bundle approach and the separation properties related to a base space offers a distinct modularization for application development for scientific computing. A modular system of software components is directly induced which always communicates over a formal interface, the preimage property. Likewise, the concept of chains transforms the properties of a cell complex directly to a computationally manageable algebraic structure. The final objective is to introduce operators that can be formulated generically, independently from all dimensional attributes. First, the tangent bundle and cotangent bundle concept is illustrated in Figure 1.16.
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Figure 1.16: The identification of the concept of fiber bundles and the chain and cochain concept as dual spaces. |
Then, the approach of a fiber bundle data model, where a discrete
representation of a manifold is modeled by a CW-complex and the
corresponding
spaces, is given. These subspaces are modeled
by separate base spaces, where a fiber describes the relationship to
the remaining
spaces, with
. The base space of the
original problem is thereby constructed from these intrinsic
relationship fiber bundles on the skeletons. An example is sketched in
Figure 1.17, where a
three-dimensional cell complex is depicted. Here, the vertex-on-cell
and cell-on-vertex information is stored within the respective fiber
spaces.
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Figure 1.17: A representation of the intrinsic fiber bundles of the respective skeleton base spaces. |
Related to this formulation of the fiber bundle concept, the chain
concept uses the following concepts, where
represents an arbitrary cell complex, as given in Section
1.4:
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(1.50) |
So a chain can be described by the intrinsic relationship fiber bundle
of a
-skeleton, whereas the
part is mapped to a
linear form within
/
. The
additional weight property is normally degenerate, which means that
only the
information is stored, e.g., only storing the
cells of interest. These concepts can then be used in combination by
the following identifications:
The only difference for the cochain concept is the space under
consideration for the linear form, in this case the linear form on
. Then the linear forms on
and the
linear forms on
, which are identified by
multivectors and
-forms, can be transfered to the algebraic bodies
of chain and cochains complexes. The formal duality between vectors
and covectors is also transfered to the duality between chains and
cochains
.