Homology

Using Algebra to Help us characterize topological properties

of spaces

Not to be confused with:

• Homeomorphism• Homomorphism• Homotopy• Oh, and it is NOT the mixing of the cream and

skim milk

Application and Theory

• For the purpose of this project we are trying to use Homology to analyze granular flow, specifically the force chains. One way to do this is by computing the Betti numbers

• We also want to be able to understand the inner workings of CHomP and how the Betti numbers are computed.

• Homology is a branch of math in Algebraic Topology

• It uses Algebra to find topological features (invariants) of topological spaces specifically we will be dealing with cubical sets

• “…Allows one to draw conclusions about global properties of spaces and maps from local computations.” (Mischaikow)

Defining the Homology group

• The vector spaces are called the k-chains for X

• Boundary operator: This is a map defined as follows

• We call an element of the k-chains a cycle if for and

• An element is called a boundary if there exists such that

Defining the Homology group

• So finally we define the homology group as

• (a.k.a )

• Example:

Explanation of what Betti Numbers are

Corollary: Any finitely generated abelian group G is isomorphic to a group of the form

Where r is a nonnegative integer, denotes the group of integers modulo b, , provided k > 0, and divides . The numbers r and the b’s are uniquely determined by G . And the number r is the rank of subgroup and is called the BETTI NUMBER of G.

• - in 0 dimensions the holes are connected components

• - in 1 dimension we get tunnels• - in 2 dimensions we get cavities

What do the homology groups tell us about Topological Spaces ?

The Homology groups measure the number of k-dimensional “holes”

Examples

• What are the Betti numbers of this space?

Homology

= 8

= 5

= 0

More Examples

= 12

= 10

= 0

A tire (not so simple)

= 1

= 2

= 1

What the Betti numbers???

= I don’t know

We Need an Algorithm Which Can Compute the Complex Structures of the

previous Example: CHomP• There is an algorithm which uses Smith-

Normal Form but it is too inefficient• So we turn to using this idea of reducing

before computing the Homology• Namely Elementary Collapses and Acyclic

Reduction

Elementary Collapses

Acyclic Reduction: Chomp

• An set is acyclic if is isomorphic to if k=0, and otherwise zero.

• Namely this means it has trivial homology • The Main Idea is to compute the homology of

the reduced set X, which is called the relative homology with an acyclic subset of X

Shaving Process

• This is accomplished by “Shaving” where in removable cubes are removed from the original set X

• Algorithms using Shaving are fast, so it should be used first as an initial reduction

Showing the Algorithm in Action

mv

mv

mv

mv

mv m

v

mv

mv

mv

Reducing Further the Cubical Set

Questions?