A journey through simplicial complexes

Abstract

This paper reviews some aspects of simplicial complexes. The aspects reviewed in this paper are very key in building undoubtedly strong concept as reviewed on simplicial complexes. The aspects reviewed take into a consideration: how simplices are used to construct simplicial complexes. Key areas covered at the preliminary part of this paper are simplices, face of a simplex, more on facet of a simplex, orientation of simplices, boundary of simplices, interior of a simplex etc. Geometric objects such as points, edges, triangles and tetrahedrons were the materials used in constructing simplicial complexes. We also looked at the geometric realization theorem, skeleton of simplicial complexes, maximal elements and free face, theorems under simplicial complexes and creation of simplicial complexes.  At the results and discussion part, Betti numbers and Euler characteristic of simplicial complexes were computed and chimed in with real life examples. The paper was finalized by chipping in some applications of simplicial complexes. In all, the paper has opened up more and interesting study into simplicial complexes and algebraic topology at large.