Lecture 8a. A primer on simplicial complexes and collapsibility

Before we can apply more advanced fixed point theorems to the Evasiveness Conjecture, we need a little background on simplicial complexes, and everything starts with simplices.

Simplices

It’s most intuitive to start with the geometric viewpoint, in which case an -simplex is defined to be the convex hull of affinely independent points in .  These points are called the vertices of the simplex.  Here are examples for

A simplicial complex is then a collection of simplices glued together along lower-dimensional simplices.  More formally, if is a (geometric) simplex, then a face of is a subset formed by taking the convex hull of a subset of the vertices of .

Finally, a (geometric) simplicial complex is a collection of simplices such that

1. If and is a face of , then , and
2. If and , then is a face of both and .

Property (1) gives us downward closure, and property (2) specifies how simplices can be glued together (only along faces).  For instance, the first picture depicts a simplicial complex.  The second does not.