>
> On 30/08/16 16:43, Waldek Hebisch wrote:
> > Concening future developement, we need to rethink
> > some basics. First, current DeltaComplex looks
> > fine for homology (except for lack of vertices in
> > representation). But it is underspecified for
> > homotopy. As I wrote, there should be enough
> > information to compute fundamental group, but
> > not enough for higher homotopy. And even
> > fundamental group seem to give wrong results
> > sometimes. So one way to go forward is
> > to declare that DeltaComplex is only for
> > homolgy (and by extension cohomology).
> > Alternatively one may work out how to represent
> > maps, say piecewise linear map and use this
> > representation in DeltaComplex. At some
> > moment we will like to represent maps anyway
> > (they are needed for functorial properties and
> > for higher homotopy), so IMO resonable way
> > is to say that just now DeltaComplex is for
> > homology (and maybe fundamental group) and
> > implement similar but more topological domain
> > later.
>
> To help analyse this, I think torus shows the issues most clearly.
I think you are over-relaying on examples.
> So we can calculate the fundamental group:
> 1) In the case of a fully triangulated DeltaComplex from the mapping
> from 1D to 0D cells (the graph).
> 2) In the case where it has been reduced down to 1 vertex, we then use
> the mapping from 2D to 1D cells.
>
> But what about the situation inbetween these two extremes? Somehow I
> think we need to use both mappings.
Situation with fundamental group is actually relatively simple:
1) We need to find basis of loops. This is done via spanning
tree -- edges not is spanning tree generate loops. The only
complication is that if we have parallel edges only one
can be in spanning tree. If there are paralles edges not in
the tree each generates separate loop.
2) We need to find relations. This depends only on two-dimensonal
cells. Of course first we need to add implicit faces.
Boundary of each two-dimensonal face gives a loop. Here
we need to assume that boundary edges are given in order.
If an edge is a loop (that is both ends coincide) than orientation
tells us in which direction we travel along the loop. If
the edge joins two distint vertices, then we have a little
redundancy: orientation tells us which end is starting one
and which is finishing one. But starting point must coincide
with end point of previuosly build part of the loop. And
end point of final edge must coincide with starting point
of the loop. For example consider the following non-minimal
representation of torus:
two point: 1, 2
three edges: (1, 1), (1, 2), (2, 1) = -(1, 2)
one 2D cell: [1, 2, 3, -1, -3, -2]
the edge (1, 2) gives us spanning tree. (1, 1) and second
copy of (1, 2) generate loops. Explicitely, we have two
generators:
g1 = (1, 1), g2 = (1, 2) + -(1, 2)
or using cell numbers: g1 = 1, g2 = 2 + 3
Single 2D cell gives us single relation, in terms of generators:
g1 g2 g1^(-1) g2^(-1)
which means that g1 and g2 commute. Changing orientation of
edges we may get equivalent description -- the effect is that
signs of edges in the boundary of 2D cell my change so also
signs in final relation my change.
Note: in general fundamental group uses loops whith starting
point and end point beeing fixed (that is using a base point).
The abstract group does not depend on base point, unless
space is disconnected and you choose base point in different
connected component. But concrete realization depends on
base point. Above, if we choose point 1 as a base point
it is OK: all loops start and end at 1. In general we
may be forced to artificially enlarge a loop so that is
starts and ends at base point.
Things are getting more tricky if we look at higher homology
groups. Kenzo authors and Wikipedia claim that all homology
groups are uniquely determined by combinatorial data if we
work with simplicial sets. IIUC this is a bit like our
delta complex, but:
- all cells are simplices
- faces are given in standard order
- faces always use standard orientation (we can not specify
opposite orientation)
- some faces may be degenerate and there is explicit
notation to specify degenerate faces.
Degenerate means that we treat lower dimensional simplex
as higher dimensional one. For example, we can treat
point as degenerate edge. In particular simplest description
of n-dimensional sphere uses two real simplices: point and
one simplex in dimension n. Boundary of simplex of dimension
n consists only of degenerate simplices (they all degenerate
to the unique point).
AFAICS restrictions on simplicial set means that it may be
bigger than delta complex. OTOH simplicial complex in
strightforward way gives simplicial set, so in worst case
simplicial set is as big as simplicial complex and frequently
much smaller.
--
Waldek Hebisch
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