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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
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<td
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style="font-weight: bold;">About HAP: Simplicial and cubical complexes<br>
</span></big></td>
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href="aboutTDA.html"><small style="color: rgb(0, 0, 102);">next</small></a><br>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><big
style="font-weight: bold;"><br>
1. Simplicial Complexes<br>
<br>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">A
finite simplicial complex can be created in HAP by specifying its
maximal simplices. For instance, the following commmands construct the
simplicial projective plane <br>
<br>
<div style="text-align: center;"><img
style="width: 316px; height: 323px;" alt="" src="projectiveplane.jpg"><br>
</div>
<br>
and then calculate its integral homologies from the associated cellular
chain complex.<br>
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<td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
L:=[[1,2,6],[2,6,9],[2,3,9],[3,8,9],[3,4,8],[4,5,8],<br>
> [5,6,9],[5,9,10],[8,9,10],[7,8,10],[5,7,8],[5,6,7],<br>
> [4,5,10],[3,4,10],[3,7,10],[2,3,7],[2,6,7],[1,2,6]];;<br>
<br>
gap> ProjPlane:=MaximalSimplicesToSimplicialComplex(L);<br>
Simplicial complex of dimension 2.<br>
<br>
gap> C:=ChainComplex(ProjPlane);<br>
Chain complex of length 2 in characteristic 0 .<br>
<br>
gap> Homology(C,0);<br>
[ 0 ]<br>
gap> Homology(C,1);<br>
[ 2 ]<br>
gap> Homology(C,2);<br>
[ ]<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands compute the low-dimensional integral homologies of a
10-dimensional simplicial sphere.<br>
</td>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
n:=10;;<br>
gap>
n_Sphere:=MaximalSimplicesToSimplicialComplex(Combinations([0..n+1],n+1));<br>
Simplicial complex of dimension 10.<br>
<br>
gap> C:=ChainComplex(n_Sphere);<br>
Chain complex of length 10 in characteristic 0 .<br>
<br>
gap> List([0..n],m->Homology(C,m));<br>
[ [ 0 ], [ ], [ ], [ ], [ ], [ ], [
], [ ], [ ], [ ], [ 0 ], [ ] ]<br>
</td>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
simplicial complex arising as the order complex of the poset of
non-trivial elementary abelian p-subgroups of a finite group G has been
studied by D. Quillen and others. The following commands contruct this
simplicial complex for the Sylow 2-subgroup of the Mathieu group M<sub>12</sub>
(with p=2), and then verify that in this case the simplicial complex is
contractible.<br>
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Q:=QuillenComplex(SylowSubgroup(MathieuGroup(12),2),2);<br>
Simplicial complex of dimension 2.<br>
<br>
gap> C:=ChainComplex(Q);<br>
Chain complex of length 2 in characteristic 0 .<br>
<br>
gap> Homology(C,0);<br>
[ 0 ]<br>
gap> Homology(C,1);<br>
[ ]<br>
gap> Homology(C,2);<br>
[ ]<br>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
<div style="text-align: center;"><big style="font-weight: bold;"><br>
2. Cubical Complexes<br>
<br>
</big></div>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">In
HAP we us the term "pure cubical complex" to mean a subspace of
d-dimensional euclidian space arising as a union of finitely many
d-dimensional unit cubes whose vertices have integral coordinates. A
pure cubical space can be created by specifying a d-dimensional array
of 0s and 1s. For instance, the following commands construct a
3-dimensional cubical 2-sphere and verify its homology in low
dimensions. <br>
</td>
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
a:=[[1,1,1],[1,1,1],[1,1,1]];;<br>
gap> b:=[[1,1,1],[1,0,1],[1,1,1]];;<br>
gap> c:=[[1,1,1],[1,1,1],[1,1,1]];;<br>
gap> array:=[a,b,c];;<br>
gap> 2_sphere:=PureCubicalComplex(array);<br>
Pure cubical complex of dimension 3.<br>
<br>
gap> C:=ChainComplex(2_sphere);<br>
Chain complex of length 3 in characteristic 0 .<br>
<br>
gap> Homology(C,0);<br>
[ 0 ]<br>
gap> Homology(C,1);<br>
[ ]<br>
gap> Homology(C,2);<br>
[ 0 ]<br>
gap> Homology(C,3);<br>
[ ]<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
Cech complex of a pure cubical complex is a simplicial complex
such that the geometric realizations of the two complexes are homotopy
equivalent. The following commands compute the Cech complex of the
cubical 2-sphere and (again) verify the low dimensional homology of the
2-sphere.<br>
</td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
S:=CechComplexOfPureCubicalComplex(2_sphere);<br>
Simplicial complex of dimension 3.<br>
<br>
gap> C:=ChainComplex(S);<br>
Chain complex of length 3 in characteristic 0 .<br>
<br>
gap> Homology(C,0);<br>
[ 0 ]<br>
gap> Homology(C,1);<br>
[ ]<br>
gap> Homology(C,2);<br>
[ 0 ]<br>
gap> Homology(C,3);<br>
[ ]<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
above cubical 2-sphere has twenty-six 3-cells. The following commands
compute a homotopy retract with just six 3-cells.<br>
</td>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
new_sphere:=ContractedComplex(2_sphere);<br>
Pure cubical complex of dimension 3.<br>
<br>
gap> C:=ChainComplex(2_sphere);;<br>
gap> D:=ChainComplex(new_sphere);;<br>
gap> C!.dimension(3);<br>
26<br>
gap> D!.dimension(3);<br>
6<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following command shows that the Cech complex of this smaller
3-dimensional 2-sphere is actually a 2-dimensional simplicial complex.<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
S:=CechComplexOfPureCubicalComplex(new_sphere);<br>
Simplicial complex of dimension 2.<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands compute a 3-dimensional simplicial complex which is
homotopic to a torus.<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Circle:=PureCubicalComplex([[1,1,1],[1,0,1],[1,1,1]]);<br>
Pure cubical complex of dimension 2.<br>
<br>
gap> Torus:=DirectProductOfPureCubicalComplexes(Circle,Circle);<br>
Pure cubical complex of dimension 4.<br>
<br>
gap> ContractPureCubicalComplex(Torus);;<br>
<br>
gap> SimplicialTorus:=CechComplexOfPureCubicalComplex(Torus);<br>
Simplicial complex of dimension 3.<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
digital photograph can be represented as a 2-dimensional pure cubical
complex. This is done by choosing an integer threshold and including a
2-cell in the pure cubical complex for each pixel where the sum of the
three RGB values iis less than the threshold.<br>
<br>
The following commands use a threshold of 400 to represent the image<br>
<br>
<div style="text-align: center;"><img
style="width: 247px; height: 287px;" alt="" src="bw_image.bmp"><br>
<div style="text-align: left;"><br>
as a pure cubical complex. The complex has 40949 2-dimensional cells.<br>
</div>
</div>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
image:=ReadImageAsPureCubicalComplex("bw_image.bmp",400);<br>
Pure cubical complex of dimension 2.<br>
<br>
gap> C:=ChainComplex(image);<br>
Chain complex of length 2 in characteristic 0 .<br>
<br>
gap> C!.dimension(0);<br>
45664<br>
gap> C!.dimension(1);<br>
86630<br>
gap> C!.dimension(2);<br>
40949<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
number of cells in the above cubical complex makes it difficult to
compute the homology of the associated cellular chain complex. One way
around the difficulty is to:<br>
<ul>
</ul>
<ol>
<li>Find a homotopy retract R of the pure cubical complex.</li>
<li>Find a large contractible subcomplex S in R.</li>
<li>Construct the quotient C(R)/C(S) of the cellular
chain complexes.</li>
<li>Use the fact that H<sub>n</sub>(R) = H<sub>n</sub>(
C(R)/C(S) ) for n>0 and that H<sub>0</sub>(R) is isomorphic to the
direct sum H<sub>0</sub>(C(R)/C(S))+H<sub>0</sub>(S).</li>
</ol>
<ul>
</ul>
The following commands apply Steps 1-4 in order to calculate that the
above image has 3 path components and 20 1-cycles.<br>
</td>
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<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
image:=ReadImageAsPureCubicalComplex("bw_image.bmp",400);<br>
Pure cubical complex of dimension 2.<br>
<br>
gap> R:=ContractedComplex(image);<br>
Pure cubical complex of dimension 2.<br>
<br>
gap> S:=ContractibleSubcomplexOfPureCubicalComplex(R);<br>
Pure cubical complex of dimension 2.<br>
<br>
gap> C:=ChainComplexOfPair(R,S);<br>
Chain complex of length 2 in characteristic 0 .<br>
<br>
gap> Homology(C,0);<br>
[ 0, 0 ]<br>
<br>
gap> Homology(C,1);<br>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]<br>
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