[GAP Forum] Chain Complexes: The 2-Disk

[Marc Bogaerts]

# 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.