# Re: [fricas-devel] Algebraic Topology Issue 1

Martin,

after a look into your code I think there was a deep misunderstanding -
my fault :(

Kenzo indeed works slightly different, so I tacitly assumed one needs
the full base of the triangulation w.r.t the bdry homomorphism. Your
method instead requires properly oriented input. Now I also conceive
that you find the idea of vertex sets rather odd. So, actually, your UI
is expecting a formal sum (a Z2-Module over a set of cells if you like):

dunceHat =
[1,2,8]+[2,3,8]+[3,7,8]+[3,1,7]+[1,2,7]+[1,8,6]+[1,6,2]+[8,7,6]+[2,6,4]+[6,7,5]+[7,2,5]+[6,5,4]+[2,4,3]+[2,3,5]+[4,1,3]+[4,5,1]+[5,3,1]

=
[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]

=> now we get the correct result (cmp fig. on p.67):

ASIMP := FiniteSimplicialComplex(VertexSetAbstract)

(1) -> v1:List(List(NNI)) :=[[1,2,8],[2,3,8],[3,7,8],[3,1,7],[1,2,7], _
[1,8,6],[1,6,2],[8,7,6],[2,6,4],[6,7,5],[7,2,5],[6,5,4], _
[2,4,3],[2,3,5],[4,1,3],[4,5,1],[5,3,1]]

(1)

[[1,2,8], [2,3,8], [3,7,8], [3,1,7], [1,2,7], [1,8,6],
[1,6,2], [8,7,6], [2,6,4], [6,7,5], [7,2,5], [6,5,4], [2,4,3],
[2,3,5], [4,1,3], [4,5,1], [5,3,1]]

Type: List(List(NonNegativeInteger))

(2) -> dunceHat := simplicialComplex(vertexSeta(8::NNI),v1)\$ASIMP

(2)
(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(VertexSetAbstract)

(3) -> homology dunceHat

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

Sorry for any inconvenience caused.
Kurt

> AFAICS we need to take orientation into account and orientation requires
> a consistent set of notation standards. Could it be that other programs
> are using different conventions?
>
> (1) -> triangle := sphereSolid(2)\$SimplicialComplexFactory
>
>    (1)
>         (1,2,3)
>                     Type: FiniteSimplicialComplex(VertexSetAbstract)
> (2) -> delta(triangle)
>
> (2)
>         (1,2)
>        -(1,3)
>         (2,3)
>                  Type: FiniteSimplicialComplex(VertexSetAbstract)
>
> So [(1,2),-(1,3), (2,3)] represents a circular chain (boundary of a
> triangle).
>
> Which can be notated [(1,2),(3,1), (2,3)]
>
> OTOH [(1,2), (1,3), (2,3)] represents two parallel chains, starting and
> ending at the same vertices. This will have different homology.
>
> These conventions make sense to me because all we need to do is order
> the facets (left entry is most significant) and then alternate the sign.
>
> This is what concerns me about your proposal, for changing the VertexSet
> structure, the user interface could then loose this ordering information.
>
> So is this Cyclic?
> [(a,b),-(a,c), (b,c)]
>
> I would say yes, but only if a<b<c or a>b>c, but the user can't know
> this because they can't see the underlying index ordering.
>
> Martin B
>

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