Homology

If you have not already done so I suggest you read the page about Simplicial Complexes first and also general introduction here.

Homology is an equivalence between cycles in a topological space. On this page we look at code to test this equivalence.

When we looked at the delta complex we got a chain of 'face maps' between each dimension and the next lower one. homology face maps

In homology we treat this as a chain of abelian groups.

The homology at each step in the chain measures the connectivity of the space. homology chain

Matrix Form of Face Maps

We can represent these face maps by matrix representations, this allows us to relate to linear algebra methods.

We can then generate the homology by putting them in Smith normal form (which is already implemented in Axiom/FriCAS).

Example for Tetrahedron

Here are the face maps as discussed on the delta complex page here. indexed table

We can convert this to a matrices as follows:

Face map Matrix
δ3 = tetrahedron to triangles
  from 1 tetrahedron
to triangle 1 to 4
1
-1
1
-1
δ2 = triangles to edges
  from triangle 1 to 4
to edge 1 to 6
1 1 0 0
-1 0 1 0
0 -1 -1 0
1 0 0 1
0 1 0 -1
0 0 -1 1
δ1 = edges to vertices
  from edges 1 to 6
to vertex 1 to 4
1 1 1 0 0 0
-1 0 0 1 1 0
0 -1 0 -1 0 1
0 0 -1 0 -1 -1

For these face maps to be valid chain they must be:

Orientation and Notation

Getting a consistent orientation naming/numbering is very important. We therefore need to take some time to define it precisely

Lets start with a 2-dimensional face, that is, a triangle:

Here it is denoted [a,b,c].

It is oriented, that is, as it is drawn here it has a clockwise orientation. An anti-clockwise triangle such as [a,c,b] can be thought of as an inverse.

Here we will denote the edges as bc, ca and ab but it is easier if the nodes are alphabetically ordered, so instead we denote the edges:

bc, -ac and ab

So now each edge is designated in alphabetical order and the middle edge is negative.

homology

We can subdivide the triangle by adding another node 'x' as shown in the diagram:

This allows us to create 3 triangles. Provided that each of these has a clockwise orientation then,

The external edges will have the same direction as the original triangle.

The internal edges will each have two edges in opposite directions, so we can think of them as canceling out, leaving us with the original triangle. So,

[a,b,c] = [b,c] - [a,c] + [a,b]

Designating the triangles in terms of their edges we have:

[bc, -ac , ab ] =
[bc, -xc , xb ] - [ac, -xc , xa ] + [ab, -xb , xa ]

homology

In theory we could divide each triangle up into 4, by splitting each edge, like this:

However we don't tend to use this method because:

  • Dividing external edges would force us to do the same to adjoining triangles which would force us to subdivide the whole surface.
  • This method does not necessarily work for dividing higher dimensional faces.
homology

Interpreting SmithNormalForm

Smith normal form is calculated from the face maps in linear algebra (matrix) form. The rows and columns are added and subtracted by multiples to get as much as possible of the matrix in leading diagonal form.

b1

0 0   0
0 b2 0
0 0 b3  
⊗
    ⊗
0
0

Once in this form we interpret it as follows:

H = Z b1⊕Z b2⊕Z b3⊕

Calculated Examples

Here are FriCAS examples for common topologies:

If this is different from other CAS programs it may be because they use 'reduced homology' where component of dimension 0 is n - 1 copies of Z. The code here does not use reduced homology.

Disc (filled in circle) an 2-sphere examples

Gives H0=Z, H1= 0, H2 = 0 which we represent as: [Z,0,0]

(1) -> DCF := DeltaComplexFactory(Integer)

   (1)  DeltaComplexFactory(Integer)
                                     Type: Type
(2) -> SCF := SimplicialComplexFactory(Integer)

   (2)  SimplicialComplexFactory(Integer)
                                     Type: Type
(3) -> S := sphereSolid(2)$SCF

   (3)  points 1..3
          (1,2,3)
         Type: FiniteSimplicialComplex(Integer)
(4) -> homology(S)

   (4)  [Z,0,0]
                           Type: List(Homology)

Sphere and Disc Surface (Boundary) Example

youtube example

gives H0=Z, H1= Z, Hn = 0 {n>1}

[Z,Z]

(5) -> cD := circle()$DCF

   (5)
         1D:[[1,- 1]]
           0D:[[0]]
                    Type: DeltaComplex(Integer)
(6) -> cS := sphereSurface(2)$SCF

   (6)  points 1..3
           (1,2)
          -(1,3)
           (2,3)
         Type: FiniteSimplicialComplex(Integer)
(7) -> homology(cD)

   (7)  [Z,Z]
                           Type: List(Homology)
(8) -> homology(cS)

   (8)  [Z,Z]
                           Type: List(Homology)

Dunce Hat Example

[Z,0,0]

(9) -> dhD := dunceHat()$DCF

   (9)
         2D:[[1,1,- 1]]
          1D:[[1,- 1]]
            0D:[[0]]
                    Type: DeltaComplex(Integer)
(10) -> dhS := dunceHat()$SCF

   (10)  points 1..8
           (1,2,8)
           (2,3,8)
           (3,7,8)
           (1,3,7)
           (1,2,7)
           (1,6,8)
           (1,2,6)
           (6,7,8)
           (2,4,6)
           (5,6,7)
           (2,5,7)
           (4,5,6)
           (2,3,4)
           (2,3,5)
           (1,3,4)
           (1,4,5)
           (1,3,5)
         Type: FiniteSimplicialComplex(Integer)
(11) -> homology(dhD)

   (11)  [Z,0,0]
                           Type: List(Homology)
(12) -> homology(dhS)

   (12)  [Z,0,0]
                           Type: List(Homology)

Torus example

[Z,Z*2,Z]

(13) -> tD := torusSurface()$DCF

   (13)
           2D:[[1,2,- 1,- 2]]
          1D:[[1,- 1],[1,- 1]]
                0D:[[0]]
                    Type: DeltaComplex(Integer)
(14) -> tS := torusSurface()$SCF

   (14)  points 1..7
           (1,2,3)
           (2,3,5)
           (2,4,5)
           (2,4,7)
           (1,2,6)
           (2,6,7)
           (3,4,6)
           (3,5,6)
           (3,4,7)
           (1,3,7)
           (1,4,5)
           (1,4,6)
           (5,6,7)
           (1,5,7)
         Type: FiniteSimplicialComplex(Integer)
(15) -> homology(tD)

   (15)  [Z,Z*2,Z]
                           Type: List(Homology)
(16) -> homology(tS)

   (16)  [Z,Z*2,Z]
                           Type: List(Homology)

Real Projective Space Example

[Z,C2,0]

(17) -> ppD := projectiveSpace(2)$DCF

   (17)
           2D:[[1,1]]
          1D:[[1,- 1]]
            0D:[[0]]
                    Type: DeltaComplex(Integer)
(18) -> ppS := projectiveSpace(2)$SCF

   (18)  points 1..6
           (1,2,3)
           (1,3,4)
           (1,2,6)
           (1,5,6)
           (1,4,5)
           (2,3,5)
           (2,4,5)
           (2,4,6)
           (3,4,6)
           (3,5,6)
         Type: FiniteSimplicialComplex(Integer)
(19) -> homology(ppD)

   (19)  [Z,C2,0]
                           Type: List(Homology)
(20) -> homology(ppS)

   (20)  [Z,C2,0]
                           Type: List(Homology)

Klein bottle example

[Z,Z+C2,0]

(21) -> kbD := kleinBottle()$DCF

   (21)
            2D:[[1,2,1,- 2]]
          1D:[[1,- 1],[1,- 1]]
                0D:[[0]]
                    Type: DeltaComplex(Integer)
(22) -> kbS := kleinBottle()$SCF

   (22)  points 1..8
           (3,4,8)
           (2,3,4)
           (2,4,6)
           (2,6,8)
           (2,5,8)
           (3,5,7)
           (2,3,7)
           (1,2,7)
           (1,2,5)
           (1,3,5)
           (4,5,8)
           (4,5,7)
           (4,6,7)
           (1,6,7)
           (1,3,6)
           (3,6,8)
         Type: FiniteSimplicialComplex(Integer)
(23) -> homology(kbD)

   (23)  [Z,Z+C2,0]
                           Type: List(Homology)
(24) -> homology(kbS)

   (24)  [Z,Z+C2,0]
                           Type: List(Homology)

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Correspondence about this page

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flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

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