Download Topology Group

Homology
Not to be confused with:
•
•
•
•
Homeomorphism
Homomorphism
Homotopy
Or any other “homo-type” math concept I
have left out
• 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 kchains 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 is the rank
of subgroup
and is called the
of G.
What do the homology groups tell us about
Topological Spaces ?
The Homology groups measure the number of
k-dimensional “holes”
•
- in 0 dimensions the holes are connected
components
•
- in 1 dimension we get tunnels
•
- in 2 dimensions we get cavities
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:
• There is an algorithm which uses SmithNormal 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
Reducing Further the Cubical Set
Questions?