Narrow Results

A framework of connections, gauge transformations, and more, is developed over cell complexes. The relationship between this and the continuum theory is explained. A 2-form over the space of flat connections over certain types of cell complexes is defined and its significant properties proved; this 2-form is induced by the Atiyah–Bott symplectic structure on the space of connections over closed, oriented surfaces.

Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, differential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foundation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions.

The paper shows relationships between relevant concepts from topology, differential geometry and computer aided geometric design that are not widely known in the CAD community. It also discusses semialgebraic, semianalytic and subanalytic sets as candidates for object representation. Stratifications of such sets are proposed for an object's cellular structure and new stratification concepts are introduced to support candidate applications.

An algorithm for automatic reducing of the topology of a h-genus three-dimensional polyhedron to the topology of a ball is proposed. The polyhedron is assumed to be represented by a three-dimensional cell complex. A special technique for separation of two-dimensional subcomplexes producing a cutting surfaces is described. This technique is based on aggregation of cells and calculation of Betti groups of the polyhedron.

On the boundary of a polyhedron, we construct a tessellation consisting of 8n pentagons (n ≥ 3). The tessellation produces a family of 3-dimensional orientable closed manifolds with a spine, which correspond to group presentations. We determine the group presentations and their homology groups.