Martin Baker wrote:
>
> On 02/07/16 21:15, Waldek Hebisch wrote:
> > 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.
>
> Do you think it would be reasonable to allow this in DeltaComplex but
> not in FiniteSimplicialComplex or FiniteCubicalComplex?
Yes. Classical definition of simplicial complex requires
different vertices and does not allow duplicate simplices.
> As usual I guess we have to clarify definitions. Hatcher (p102) gives
> delta complexes as a "mild generalization of the more classical notion
> of a simplical complex".
>
> I get the impression that this "mild generalization" is allowing
> multiple copies of facets? For instance, instead of the minimal
> triangulation of a projective plane having 10 triangles it can be
> represented as a square (2 triangles) where the opposite edges are the
> same (but with opposite orientation).
>
> The other difference in the code between delta and simplicial complexes
> is that in simplicial complexes everything is defined (indexed) directly
> in terms of vertices but in DeltaComplex indexing is in terms of next
> lower dimension (differential form???).
Once you allow different faces to have the same vertices
you need some way to specify which of faces appear on boundary
of higher dimensional face. This naturally leads to "indexing
in terms of next lower dimension".
--
Waldek Hebisch
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