Martin Baker wrote:
>
> Does anyone have any validated examples that I could use to check my
> homology (using Waldeks method) and homotopy code?
For fundamental group classic is n-loops, that is complex with
n+1 nodes in dimension 0 and 2n simplices of dimension 1 of
form [0 n] (each of them appears twice). If you slightly
generalize notion of simplicial complex to allow simplices
with equal vertices, then you may use just one node and n
copies of 1 dimensional simplex [0 0]. Of course the fundamental
group is free group on n generators.
Let us try first generalised version:
(5) -> vs1 := vertexSeta(1)$VertexSetAbstract
(5) 1
Type: VertexSetAbstract
(6) -> l2 := simplicialComplex(vs1, [[1,1], [1,1]])
(6)
(1,1)
(1,1)
Type: FiniteSimplicialComplex(VertexSetAbstract)
(7) -> fundamentalGroup(l2)
(7) <a b | >
Type: GroupPresentation
Looks fine. Now one with different vertices:
(8) -> vs3 := vertexSeta(3)$VertexSetAbstract
(8) 3
Type: VertexSetAbstract
(9) -> l2a := simplicialComplex(vs3, [[1,2], [1,2], [1, 3], [1,3]])
(9)
(1,2)
(1,2)
(1,3)
(1,3)
Type: FiniteSimplicialComplex(VertexSetAbstract)
(10) -> fundamentalGroup(l2a)
(10) < | >
Type: GroupPresentation
Hmm, something is wrong :(
Homotopically the same thing, but using interval instead of cental node:
(13) -> vs4 := vertexSeta(4)$VertexSetAbstract
(13) 4
Type: VertexSetAbstract
(14) -> l2c := simplicialComplex(vs4, [[1, 2], [2, 3], [2, 4], [1,3], [3,4]])
(14)
(1,2)
(2,3)
(2,4)
(1,3)
(3,4)
Type: FiniteSimplicialComplex(VertexSetAbstract)
(15) -> fundamentalGroup(l2c)
(15) <c f | >
Type: GroupPresentation
Looks fine.
> Below are some examples of homology and homotopy from simplicial and
> cubical complexes. I think the results are starting to look quite
> promising. There is obviously a problem with the homotopy since
> fundamental group is giving a different result for triangle and square.
It seems that there is problem with fundamental group of
FiniteCubicalComplex. AFAICS for FiniteSimplicialComplex results
are OK.
> Looks like there is also a problem with projective plane since this
> gives [Z,C2,0] but sage gives:
> RP4 = simplicial_complexes.RealProjectiveSpace(2)
> RP4.homology()
> {0: 0, 1: C2, 2: 0}
I think this is probably due to different devinitions. For
classical homology component of dimension 0 is n copies of Z,
where n is number of connected components. Since your examples
are connected the correct result is Z which perfectly agree
with your computations. People sometimes define so called
reduced homology where component of dimension 0 is n - 1
copies of Z. It looks like sage may be using this definition.
--
Waldek Hebisch
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