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Ho m o l ogy

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Not to be confused with: Homeomorphism Homomorphism Homotopy Oh, and it is NOT the mixing of the cream and skim milk

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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.

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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)

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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

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Defining the Homology group So finally we define the homology group as (a.k.a ) Example:

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Explanation of what Betti Numbers are

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- 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”

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Examples What are the Betti numbers of this space? Homology = 8 = 5 = 0

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More Examples = 12 = 10 = 0

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A tire (not so simple) = 1 = 2 = 1

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What the Betti numbers??? = I don’t know

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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

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Elementary Collapses

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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

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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

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Showing the Algorithm in Action

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Reducing Further the Cubical Set

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Questions?

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