Simplicial Complexes

Abstract

Combinatorics is the slums of topology. — J. H. C. Whitehead (attr.) 3 Simplicial Complexes In the first lecture, we looked at concepts from point set topology, the branch of topology that studies continuity from an analytical point of view. This view does not have a computational nature: we cannot represent infinite point sets or their associated infinite open sets on a computer. Starting with this lecture, we will look at concepts from another major branch of topology: combinatorial topology. This branch also studies connectivity, but does so by examining constructing complicated objects out of simple blocks, and deducing the properties of the constructed objects from the blocks. While our view of the world–our ontology–will be mostly combinatorial in nature, we will see concepts from point set topology reemerging under disguise, and we will be careful to expose them! In this lecture, we begin by learning about simple building blocks from which we may construct complicated spaces. Simplicial complexes are combinatorial objects that represent topological spaces. With simplicial complexes, we separate the topology of a space from its geometry, much like the separation of syntax and semantics in logic. Given the finite combinatorial description of a space, we are able to count, and the miracle of combinatorial topology is that counting alone enables us to make statements about the connectivity of a space. We shall experience a first instance of this marvelous theory in the Euler characteristic. This topological invariant gives a simple constructive procedure for classifying 2-manifolds, completing our treatment from the last lecture. 3.1 Geometric Definition We begin with a definition of simplicial complexes that seems to mix geometry and topology. Combinations allow us to represent regions of space with very few points. In other words, allow us to describe simple cells which become our building blocks later. Definition 3.1 (combinations) Let S = {p 0 , p 1 , . . . , p k } ⊆ R d . A linear combination is x = k i=0 λ i p i , for some λ i ∈ R. An affine combination is a linear combination with k i=0 λ i = 1. A convex combination is a an affine combination with λ i ≥ 0, for all i. The set of all convex combinations is the convex hull. You may have seen the concept of independence in studying linear algebra.