# we start with a 0-skeleton consisting of 3 points
c0:=ChainComplex(0,[1..3]);
dim 0:[ 1 .. 3 ]
# for the 1 skeleton we first connect 1 and 2:
c1:=ChainComplex(1,c0,[[1,2]]);
dim 0:[ 1 .. 3 ]
dim 1:[ [ 1, -1, 0 ] ]
Basis of 2-cells: [ ]
# remark that the arguments are now: the dimension, a 0-skeleton to
build upon
# and a set (with one member) of edges on which to build the 1-skeleton
# remark how the edges (1-cells) are kept in memory
# there seem to be no 1-cycles, so no 2 cells can be formed. What do we
know about this object?
HomologyGroups(c1);
[ [ 0, 0 ], [ ] ]
SingularFaces(c1);
[ 1, 2 ]
# Points 1 and 2 are border points so c1 is not a manifold. Let's
connect 2 and 3 too:
c1:=ChainComplex(1,c0,[[1,2],[2,3]]);
dim 0:[ 1 .. 3 ]
dim 1:[ [ 1, -1, 0 ], [ 0, 1, -1 ] ]
Basis of 2-cells: [ ]
# looking at the homology groups gives:
HomologyGroups(c1);
[ [ 0 ], [ ] ]
# the space is now connected, but still no cycles.
SingularFaces(c1);
[ 1, 3 ]
# 1 and 3 are still boundary points, let's connect them:
c1:=ChainComplex(1,c0,[[1,2],[2,3],[3,1]]);
dim 0:[ 1 .. 3 ]
dim 1:[ [ 1, -1, 0 ], [ 0, 1, -1 ], [ -1, 0, 1 ] ]
Basis of 2-cells: [ [ 1, 1, 1 ] ]
HomologyGroups(c1);
[ [ 0 ], [ 0 ] ]
SingularFaces(c1);
[ ]
# The homology groups are that of a circle and there are no border
points, so we probably represent a circle
# since the software proposes a 2 cell to be constructed we use it in a
2 skeleton:
c2:=ChainComplex(2,c1,[[1,1,1]]);
dim 0:[ 1 .. 3 ]
dim 1:[ [ 1, -1, 0 ], [ 0, 1, -1 ], [ -1, 0, 1 ] ]
dim 2:[ [ 1, 1, 1 ] ]
Basis of 3-cells: [ ]
HomologyGroups(c2);
[ [ 0 ], [ ], [ ] ]
# That of a contracible space
SingularFaces(c2);
[ [ 1, -1, 0 ], [ 0, 1, -1 ], [ -1, 0, 1 ] ]
# the three edges form a border of the disk.
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