One thing that comes up a lot in homology is the idea of simplices and simplicial complexes. They’re interesting in their own right, and they’re one more thing that we can talk about
that will help make understanding the homology and the homological chain complexes easier when we get to them.
A simplex is a member of an interesting family of filled geometric figures. Basically, a simplex is an N-dimensional analogue of a triangle. So a 1-simplex is a line-segment; a 2-simplex is a triangle; a three simplex is a tetrahedron; a four-simplex is a pentachoron. (That cool image to the right is a projection of a rotating pentachoron from wikipedia.) If the lengths of the sides of the simplex are equal, it’s called a regular simplex.
There are some neat things about simplices. For any N≥2, an N-simplex is a figure with N+1 faces, each of which is an N-1 simplex. So a tetrahedron – a 3-simplex – has four faces, each of which is a triangle; a pentachoron (a 4-simplex) has 5 tetrahedral faces. Also, for any N, an N-simplex is the convex hull of N+1 linearly independent points embedded in ?N.
A simplicial complex is where simplices start meeting up with topology. A simplicial complex is a topological space formed from a set of simplices. Basically, a topological T space is a simplicial complex K if/f it can be decomposed into a collection of simplices where:
- For every simplex S in K, every face of S is also in K.
- Every intersection between 2 simplices is a face of both of the intersecting simplices.
There’s one annoying part of that second requirement, which is that you always consider intersections using the lowest-dimension simplices that can include the intersection as a face. So you can have two tetrahedrons intersecting along an edge – they’re still a simplicial complex, because the intersection is a 1-simplex, so you consider it using the 2-simplices in K – and the line segment is a face of all of the triangles that meet at that edge.
A simplicial complex where the largest dimension of any simplex in the complex is N is called a simplicial N-complex. It’s called a pure N-complex if every simplex of dimension <N is a face of an N-simplex in the complex.
Every N-simplex is homeomorphic to an N-ball in ?N; and for N≤3, a manifold which is a subspace of ?N is homeomorphic to
a pure simplicial complex. The simplicial complex is sometimes called a triangulation of the space. Many of the early results about manifolds in topology were done using triangulations of
manifolds; the division of the manifold into simplexes was a major tool used to make topological proofs tractable.