Hi Waldek,
I don't think the DeltaComplex to SimplicialComplex coercion is crucial
so if you are not happy with it then it could be omitted.
The main reason I wrote it is to use in a regression test for
SimplicialComplex to DeltaComplex which is much more important.
so:
SimplicialComplex --> DeltaComplex --> SimplicialComplex
always gets back to the same place for all the test cases I can find.
However I accept that:
DeltaComplex --> SimplicialComplex --> DeltaComplex
does not get back to the same place in general and will fail to work for
the type of cases that you mention.
I read somewhere, probably Hatcher, that SimplicialComplex and
DeltaComplex are not isomorphic but that DeltaComplex is very marginally
more general. However I get the impression that, in most cases, there is
a triangulation in a higher dimension that does not repeat node and
edges in this way. I have not come across an algorithm to do this and
the approach that you suggest looks very interesting.
It seems to me that algebraic topology is a big area of mathematics and
requires a lot of code. Is it realistic for it to appear, fully formed,
in FriCAS in one step? I would really help me to get the basics in
FriCAS which would give me the confidence to spend more time on the details.
I really appreciate the work you are doing on this.
Martin
PS I think the above comments also apply to DeltaComplex --> CubicalComplex
On 29/08/16 19:18, Waldek Hebisch wrote:
AFAICS there is serious confusion in in the code. I have fixed
several problems, but some go beyond "fixing". In particular
DeltaComplex contains coerce to SimplicialComplex. AFAICS
DeltaComplex contains too little information to build
simplicial complex from it. The problem is already visible
at level of fundamental group (however here can be worked
around). Namely, consider delta complex having one 0-dimensonal
simplex (point), two 1 dimensional simplices (edges) and one
two dimensional simplex. The two edges form two loops with
a common point. Boundary of two dimensional is mapped
is mapped to the loops. Using your code it can be
build from the follwing face map:
[[[1, -1],[1, -1]], [1 2 1]]
Now, this fully specifies boundary of two dimensional simplex
as an oriented set. However, we need a mapping. In this case
we can resolve problem because orientation gives us only
one choice of linear map between edges. But in higher
dimensional case to define map we need correspondence
between vertices and all we have is orientation. Note
that knowing boundary as an oriented set is enough to
compute homology, so in fact there are pretty strong
global restrictions on possible maps. But it is
not clear if the map is uniquely determined (probably not),
and even if it is determined it looks hard to compute
it.
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