| When we looked at the delta complex we got a chain of 'face maps' between each dimension and the next lower one. |  |
In homology we treat this as a chain of Abelian groups (more detail on the page here).
For implementation of co-chain see page here.
 |
FriCAS Implementation
In our FriCAS Implementation 'ChainComplex' represents, not the faces at each dimension, but the maps between them. Each map is represented by a (not necessarily square) matrix. These matrices may have a width or a height of zero.
Since the chain starts and ends with zero (trivial group) then:
- The first matrix has a height of zero. A matrix with a height of zero is shown with zeroes like this: [0 , 0 , 0] because it is hard to graphically depict a zero height matrix.
- The last matrix has a width of zero.
Creating Chain Complexes
Here are two ways to construct a chain complex:
Usually it is easier to construct a simplicial or delta complex first and then call chain function. Here we construct a simplicial complex then call chain function: |
(1) -> b1 := sphereSolid(2)$SimplicialComplexFactory(Integer)
(1) points 1..3
(1,2,3)
Type: FiniteSimplicialComplex(Integer)
(2) -> chain(b1)
+ 1 1 0 + + 1 +
| | | |
(2) [0 0 0],|- 1 0 1 |,|- 1|,[]
| | | |
+ 0 - 1 - 1+ + 1 +
Type: ChainComplex |
| We could alternatively construct from a list of matrices. |
Example 1 - Solid 2D circle.
(1) -> )expose DeltaComplex
DeltaComplex is now explicitly exposed in frame frame1
(1) -> b1 := sphereSolid(2)$SimplicialComplexFactory(Integer)
(1) points 1..3
(1,2,3)
Type: FiniteSimplicialComplex(Integer)
(2) -> d1 := deltaComplex(b1)
(2)
2D:[[1,- 2,3]]
1D:[[1,- 2],[1,- 3],[2,- 3]]
0D:[[0],[0],[0]]
Type: DeltaComplex(Integer)
(3) -> c1 := chain(d1)
+ 1 1 0 + + 1 +
(3) [0 0 0],|- 1 0 1 |,|- 1|,[]
+ 0 - 1 - 1+ + 1 +
Type: ChainComplex |
Example2 - Solid 3D Sphere.
(4) -> b2 := sphereSolid(3)$SimplicialComplexFactory(Integer)
(4) points 1..4
(1,2,3,4)
Type: FiniteSimplicialComplex(Integer)
(5) -> d2 := deltaComplex(b2)
(5)
3D:[[1,- 2,3,- 4]]
2D:[[1,- 2,4],[1,- 3,5],[2,- 3,6],[4,- 5,6]]
1D:[[1,- 2],[1,- 3],[1,- 4],[2,- 3],[2,- 4],[3,- 4]]
0D:[[0],[0],[0],[0]]
Type: DeltaComplex(Integer)
(6) -> c2 := chain(d2)
(6)
+ 1 1 0 0 +
+ 1 1 1 0 0 0 + |- 1 0 1 0 | + 1 +
|- 1 0 0 1 1 0 | | 0 - 1 - 1 0 | |- 1|
[0 0 0 0],| |,| |,| |,[]
| 0 - 1 0 - 1 0 1 | | 1 0 0 1 | | 1 |
+ 0 0 - 1 0 - 1 - 1+ | 0 1 0 - 1| +- 1+
+ 0 0 1 1 +
Type: ChainComplex |
Example 3 - Hollow 2D circle.
(7) -> b3 := sphereSurface(2)$SimplicialComplexFactory(Integer)
(7) points 1..3
(1,2)
-(1,3)
(2,3)
Type: FiniteSimplicialComplex(Integer)
(8) -> d3 := deltaComplex(b3)
(8)
1D:[[1,- 2],[- 1,3],[2,- 3]]
0D:[[0],[0],[0]]
Type: DeltaComplex(Integer)
(9) -> c3 := chain(d3)
+ 1 - 1 0 + ++
(9) [0 0 0],|- 1 0 1 |,||
+ 0 1 - 1+ ++
Type: ChainComplex |
Example 4 - Hollow 3D Sphere.
(10) -> b4 := sphereSurface(3)$SimplicialComplexFactory(Integer)
(10) points 1..4
(1,2,3)
-(1,2,4)
(1,3,4)
-(2,3,4)
Type: FiniteSimplicialComplex(Integer)
(11) -> d4 := deltaComplex(b4)
(11)
2D:[[1,- 2,4],[- 1,3,- 5],[2,- 3,6],[- 4,5,- 6]]
1D:[[1,- 2],[1,- 3],[1,- 4],[2,- 3],[2,- 4],[3,- 4]]
0D:[[0],[0],[0],[0]]
Type: DeltaComplex(Integer)
(12) -> c4 := chain(d4)
+ 1 - 1 0 0 +
+ 1 1 1 0 0 0 + |- 1 0 1 0 | ++
|- 1 0 0 1 1 0 | | 0 1 - 1 0 | ||
(12) [0 0 0 0],| |,| |,||
| 0 - 1 0 - 1 0 1 | | 1 0 0 - 1| ||
+ 0 0 - 1 0 - 1 - 1+ | 0 - 1 0 1 | ++
+ 0 0 1 - 1+
Type: ChainComplex |
Example 5 - Torus Surface.
(13) -> b5 := torusSurface()$SimplicialComplexFactory(Integer)
(13) 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)
(14) -> d5 := deltaComplex(b5)
(14)
VCONCAT
VCONCAT
VCONCAT
,
2D:
[[1,- 2,7], [1,- 5,10], [2,- 6,15], [3,- 4,16], [3,- 5,17],
[4,- 6,20], [7,- 9,13], [8,- 9,16], [8,- 11,18], [10,- 11,21],
[12,- 14,17], [12,- 15,18], [13,- 14,19], [19,- 20,21]]
,
1D:
[[1,- 2], [1,- 3], [1,- 4], [1,- 5], [1,- 6], [1,- 7], [2,- 3],
[2,- 4], [2,- 5], [2,- 6], [2,- 7], [3,- 4], [3,- 5], [3,- 6],
[3,- 7], [4,- 5], [4,- 6], [4,- 7], [5,- 6], [5,- 7], [6,- 7]]
,
0D:[[0],[0],[0],[0],[0],[0],[0]]
Type: DeltaComplex(Integer)
(15) -> c5 := chain(d5)
(15)
[0 0 0 0 0 0 0]
,
[[1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[- 1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0],
[0,- 1,0,0,0,0,- 1,0,0,0,0,1,1,1,1,0,0,0,0,0,0],
[0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,0,0,1,1,1,0,0,0],
[0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,0,- 1,0,0,1,1,0],
[0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,0,- 1,0,- 1,0,1],
[0,0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,0,- 1,0,- 1,- 1]]
,
+ 1 1 0 0 0 0 0 0 0 0 0 0 0 0 +
|- 1 0 1 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 1 1 0 0 0 0 0 0 0 0 0 |
| 0 0 0 - 1 0 1 0 0 0 0 0 0 0 0 |
| | ++
| 0 - 1 0 0 - 1 0 0 0 0 0 0 0 0 0 | ||
| 0 0 - 1 0 0 - 1 0 0 0 0 0 0 0 0 | ||
| 1 0 0 0 0 0 1 0 0 0 0 0 0 0 | ||
| 0 0 0 0 0 0 0 1 1 0 0 0 0 0 | ||
| 0 0 0 0 0 0 - 1 - 1 0 0 0 0 0 0 | ||
| 0 1 0 0 0 0 0 0 0 1 0 0 0 0 | ||
| 0 0 0 0 0 0 0 0 - 1 - 1 0 0 0 0 |,||
| 0 0 0 0 0 0 0 0 0 0 1 1 0 0 | ||
| 0 0 0 0 0 0 1 0 0 0 0 0 1 0 | ||
| 0 0 0 0 0 0 0 0 0 0 - 1 0 - 1 0 | ||
| 0 0 1 0 0 0 0 0 0 0 0 - 1 0 0 | ||
| 0 0 0 1 0 0 0 1 0 0 0 0 0 0 | ||
| 0 0 0 0 1 0 0 0 0 0 1 0 0 0 | ||
| | ++
| 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 1 1 |
| 0 0 0 0 0 1 0 0 0 0 0 0 0 - 1|
+ 0 0 0 0 0 0 0 0 0 1 0 0 0 1 +
Type: ChainComplex |
Example 6 - Projective Plane.
(16) -> b6 := projectiveSpace(2)$SimplicialComplexFactory(Integer)
(16) points 1..6
(1,2,3)
(1,3,4)
(1,2,6)
(1,5,6)
(1,4,5)
(2,3,5)
(2,4,5)
(2,4,6)
(3,4,6)
(3,5,6)
Type: FiniteSimplicialComplex(Integer)
(17) -> d6 := deltaComplex(b6)
(17)
VCONCAT
VCONCAT
VCONCAT
,
2D:
[[1,- 2,6], [1,- 5,9], [2,- 3,10], [3,- 4,13], [4,- 5,15],
[6,- 8,11], [7,- 8,13], [7,- 9,14], [10,- 12,14], [11,- 12,15]]
,
1D:
[[1,- 2], [1,- 3], [1,- 4], [1,- 5], [1,- 6], [2,- 3], [2,- 4],
[2,- 5], [2,- 6], [3,- 4], [3,- 5], [3,- 6], [4,- 5], [4,- 6],
[5,- 6]]
,
0D:[[0],[0],[0],[0],[0],[0]]
Type: DeltaComplex(Integer)
(18) -> c6 := chain(d6)
(18)
[0 0 0 0 0 0]
,
+ 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 +
|- 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 |
| 0 - 1 0 0 0 - 1 0 0 0 1 1 1 0 0 0 |
| 0 0 - 1 0 0 0 - 1 0 0 - 1 0 0 1 1 0 |
| 0 0 0 - 1 0 0 0 - 1 0 0 - 1 0 - 1 0 1 |
+ 0 0 0 0 - 1 0 0 0 - 1 0 0 - 1 0 - 1 - 1+
,
+ 1 1 0 0 0 0 0 0 0 0 +
|- 1 0 1 0 0 0 0 0 0 0 |
| 0 0 - 1 1 0 0 0 0 0 0 |
| | ++
| 0 0 0 - 1 1 0 0 0 0 0 | ||
| 0 - 1 0 0 - 1 0 0 0 0 0 | ||
| 1 0 0 0 0 1 0 0 0 0 | ||
| 0 0 0 0 0 0 1 1 0 0 | ||
| 0 0 0 0 0 - 1 - 1 0 0 0 |,||
| 0 1 0 0 0 0 0 - 1 0 0 | ||
| 0 0 1 0 0 0 0 0 1 0 | ||
| 0 0 0 0 0 1 0 0 0 1 | ||
| 0 0 0 0 0 0 0 0 - 1 - 1| ||
| | ++
| 0 0 0 1 0 0 1 0 0 0 |
| 0 0 0 0 0 0 0 1 1 0 |
+ 0 0 0 0 1 0 0 0 0 1 +
Type: ChainComplex |
Example 7 - Klein Bottle.
(19) -> b7 := kleinBottle()$SimplicialComplexFactory(Integer)
(19) 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)
(20) -> d7 := deltaComplex(b7)
(20)
VCONCAT
VCONCAT
VCONCAT
,
2D:
[[1,- 3,8], [1,- 5,10], [2,- 3,13], [2,- 4,14], [4,- 5,23],
[6,- 7,12], [6,- 10,15], [7,- 9,18], [8,- 11,22], [9,- 11,24],
[12,- 16,20], [13,- 15,21], [14,- 16,24], [17,- 19,21],
[17,- 20,22], [18,- 19,23]]
,
1D:
[[1,- 2], [1,- 3], [1,- 5], [1,- 6], [1,- 7], [2,- 3], [2,- 4],
[2,- 5], [2,- 6], [2,- 7], [2,- 8], [3,- 4], [3,- 5], [3,- 6],
[3,- 7], [3,- 8], [4,- 5], [4,- 6], [4,- 7], [4,- 8], [5,- 7],
[5,- 8], [6,- 7], [6,- 8]]
,
0D:[[0],[0],[0],[0],[0],[0],[0],[0]]
Type: DeltaComplex(Integer)
(21) -> c7 := chain(d7)
(21)
[0 0 0 0 0 0 0 0]
,
[[1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[- 1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,- 1,0,0,0,- 1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,0,1,1,1,1,0,0,0,0],
[0,0,- 1,0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,0,0,1,1,0,0],
[0,0,0,- 1,0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,0,0,0,1,1],
[0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,- 1,0,- 1,0],
[0,0,0,0,0,0,0,0,0,0,- 1,0,0,0,0,- 1,0,0,0,- 1,0,- 1,0,- 1]]
,
[[1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0], [0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0],
[- 1,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0,0], [0,0,0,- 1,1,0,0,0,0,0,0,0,0,0,0,0],
[0,- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0], [0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,- 1,0,1,0,0,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,- 1,0,1,0,0,0,0,0,0], [0,1,0,0,0,0,- 1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,- 1,- 1,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0],
[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0], [0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],
[0,0,0,0,0,0,1,0,0,0,0,- 1,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,- 1,0,- 1,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0], [0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,- 1,0,- 1], [0,0,0,0,0,0,0,0,0,0,1,0,0,0,- 1,0],
[0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0], [0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],
[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0]]
,
++
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++
Type: ChainComplex
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Next
To go on to implementation of co-chain see page here.