Introduction:
A simplicial complex is a special type of graph wherein the notion of a vertex is replaced with a new higher dimensional analog, called a simplex. A simplex is, in simplest terms, an n-dimensional collection of vertices enclosed by faces.
Definitions:
Graph:
"A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints."1
Simplex:
"A simplex...is the generalization of a tetrahedral region of space to n dimensions."2
Simplicial Complex:
"...a simplicial complex K in Rnis a collection of simplices in Rn such that...every face of K is in K, and...the intersection of any two faces of K is a face of each of them."3
A graph is a thing made of point, some of which are linked by line segments. Generalize the idea to points that can also be grouped into triangles, or tetrahedra, etc.
For graphs we know Euler's formula E+2=V+F. Give this a geometric meaning.
Discuss (maybe in the 2-dimensional case) what might replace this formula compare a "triangulated" sphere to a "triangulated" doughnut.
1 Discrete Mathematics and Its Applications, Kenneth H. Rosen.
2 http://mathworld.wolfram.com/Simplex.html
3 Munkres, J. R. "Simplicial Complexes and Simplicial Maps." §1.2 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 7-14, 1993.
